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Spectral collocation and Galerkin methods

1 · Prerequisites

2 · Summary

This page defines periodic Fourier and Chebyshev–Gauss–Lobatto collocation, including the even-grid Nyquist convention, exact differentiation of the chosen interpolants, and one finite two-point Dirichlet collocation solve with explicit singular-matrix status. The BVP item identifies the resulting algebraic system; it does not assert unconditional nonsingularity, stability, or PDE convergence.

The weighted Galerkin items give finite orthogonal-polynomial spaces and coefficient-defined spectral scales. Chebyshev and Legendre projections have algebraic coefficient-tail rates; Chebyshev–Lobatto interpolation has an algebraic rate under the stated weighted spectral regularity and an exponential rate for data holomorphic near a Bernstein ellipse. A separate pointwise remainder estimate uses the Lagrange remainder theorem. These coefficient-defined scales make no derivative-Sobolev equivalence claim.

The quadrature items compare exact Gaussian mass assembly with composite-Simpson assembly and its fourth-derivative entrywise bound. Fourier and Lobatto aliasing are explicit sample identities; they do not establish nonlinear stability or a de-aliasing guarantee. The source-specific $O (N^{4})$ second-derivative spectral-radius observation is recorded only as conditioning context for its cited Dirichlet setup.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Periodic Fourier collocation grid and even-grid Nyquist convention

Definition

Let $N \geq 2$ be an integer, let $x_{j} = 2 π j / N$ for $j = 0, \dots, N - 1$ , and identify $x_{N}$ with $x_{0}$ periodically. The real trigonometric interpolation space is $T_{N} = {\begin{cases} {span}_{R} {1, \cos (k x), \sin (k x) : 1 \leq k \leq (N - 1) / 2}, & N odd, \\ {span}_{R} {1, \cos (k x), \sin (k x) : 1 \leq k < N / 2} \oplus {span}_{R} {\cos (N x / 2)}, & N even . \end{cases}$ For real nodal data $U = (U_{0}, \dots, U_{N - 1})$ , define its periodic Fourier interpolant by

I_{N} U (x) = \sum_{j = 0}^{N - 1} U_{j} L_{j} (x),

where, for $N = 2 K + 1$ ,

L_{j} (x) = \frac{1}{N} (1 + 2 \sum_{k = 1}^{K} \cos (k (x - x_{j}))),

and, for $N = 2 K$ ,

L_{j} (x) = \frac{1}{N} (1 + 2 \sum_{k = 1}^{K - 1} \cos (k (x - x_{j})) + \cos (K (x - x_{j}))) .

Thus the even-grid Nyquist component is the real cosine $\cos (N x / 2)$ only. Its sampled derivative is defined by differentiating this interpolant; the resulting Nyquist derivative values are zero at every grid point.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Periodic Fourier modes alias modulo the grid size

Statement

Let $N \geq 2$ and $x_{j} = 2 π j / N$ . If integers $k, r$ satisfy $k \equiv r (m o d N)$ , then

e^{i k x_{j}} = e^{i r x_{j}} (0 \leq j < N) .

Consequently, sampling a product of Fourier modes with frequencies $k_{1}, \dots, k_{s}$ gives the same nodal data as the single mode whose frequency is the sum $\sum_{ℓ = 1}^{s} k_{ℓ}$ reduced modulo $N$ .

The frequency space in Periodic Fourier collocation grid and even-grid Nyquist convention is real-valued. For complex data, let $T_{N}^{C}$ be the space obtained by allowing complex coefficients in that same real basis, and extend its cardinal formula by

I_{N}^{C} V (x) : = \sum_{j = 0}^{N - 1} V_{j} L_{j} (x), V \in C^{N} .

Then $I_{N}^{C} V$ is the unique member of $T_{N}^{C}$ with nodal data $V$ . On real data this formula agrees with $I_{N}$ ; in the even case the complex Nyquist samples are interpolated by the real basis function $\cos (N x / 2)$ .

Facts & Assumptions

Given: An integer $N \geq 2$ , its periodic grid, and integers $k, r$ with $k - r = q N$ for some integer $q$ .

[F1]

Periodic Fourier collocation grid and even-grid Nyquist convention: the grid nodes are $x_{j} = 2 π j / N$ , the cardinal functions are $L_{j}$ , and on the grid the complexified basis modes give one complete residue system modulo $N$ (the even-grid Nyquist cosine has the samples of the residue $N / 2$ ); the complexification and its cardinal formula are specified in the Statement.

[F2]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential: $e^{z + w} = e^{z} e^{w}$ ; $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ : $e^{w} = 1$ when $w \in 2 π i Z$ .

[F3]

Orthogonality of the characters $x \mapsto e^{2 π i k x / N}$ on $Z / N Z$ : the sum over a complete residue system of the characters at $i - j$ is $N$ when $i = j$ and $0$ otherwise.

[F4]

Finite Fourier inversion for the unitary transform on $Z / N Z$ : complex nodal data on $Z / N Z$ have a unique finite Fourier representation.

[F5]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : $e^{i t} = \cos t + i \sin t$ for real $t$ .

[F6]

The derivatives of sine and cosine are cosine and minus sine and Quarter-turn values and shifts by pi/2 and pi: $\sin 0 = 0$ and $\sin (x + π) = - \sin x$ , hence $\sin (m π) = 0$ for every integer $m$ .

Proof

1.1givenF1F2algebra

Since $k - r = q N$ , for every grid index $j$ we have $e^{i (k - r) x_{j}} = e^{2 π i q j} = 1$ by [F2]. Multiplying by $e^{i r x_{j}}$ gives $e^{i k x_{j}} = e^{i r x_{j}}$ .

2.1givenF2step 1.1algebra

For frequencies $k_{1}, \dots, k_{s}$ , multiplication of the sampled modes gives $\prod_{ℓ = 1}^{s} e^{i k_{ℓ} x_{j}} = e^{i (\sum_{ℓ} k_{ℓ}) x_{j}}$ by the exponential addition law in [F2]. Applying step 1.1 to this sum and any congruent representative proves the product aliasing identity.

3.1F1F2F3F4F5F6step 2.1algebra∎

For $Δ = x_{i} - x_{j}$ , Euler's identity [F5] writes the odd-grid kernel as $L_{j} (x_{i}) = N^{- 1} \sum_{k = - K}^{K} e^{i k Δ}$ when $N = 2 K + 1$ . For an even grid $N = 2 K$ , it writes the kernel as $L_{j} (x_{i}) = N^{- 1} \sum_{k = - K + 1}^{K} e^{i k Δ}$ , because $K Δ = π (i - j)$ and [F6] makes the sine term at this Nyquist frequency vanish. In either branch $e^{i N Δ} = 1$ by [F2], so the sum over these complete residue systems equals the sum over $k = 0, \dots, N - 1$ . Character orthogonality [F3] therefore gives $L_{j} (x_{i}) = 1$ for $i = j$ and $0$ otherwise, so the displayed complex-linear cardinal formula interpolates every $V \in C^{N}$ . For uniqueness inside $T_{N}^{C}$ , expand each low-frequency cosine/sine pair by [F5] into its two characters. On an odd grid these are the residues $- K, \dots, K$ ; on an even grid they are $- K + 1, \dots, K - 1$ together with the Nyquist sample $\cos (K x_{j}) = e^{i K x_{j}}$ . The real-basis coefficients map invertibly to these character coefficients (the low modes use the two Euler coefficients and the Nyquist coefficient is unchanged). If a member of $T_{N}^{C}$ has zero samples, inversion [F4] makes every character coefficient zero, hence the member is zero. The product samples from step 2.1 therefore have the unique cardinal interpolant. In the even Nyquist case the interpolant is $\cos (N x / 2)$ , not the complex exponential as a function away from the grid.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Periodic Fourier collocation differentiation matrix

Definition

Input an integer $N \geq 2$ and real data $U_{0}, \dots, U_{N - 1}$ on the grid of Periodic Fourier collocation grid and even-grid Nyquist convention. Set $K = (N - 1) / 2$ when $N$ is odd and $K = N / 2$ when $N$ is even. Form the $(N \times N)$ matrix $(D_{N})_{i j} = {\begin{cases} - \frac{2}{N} \sum_{k = 1}^{K} k \sin (k (x_{i} - x_{j})), & N odd, \\ - \frac{2}{N} \sum_{k = 1}^{K - 1} k \sin (k (x_{i} - x_{j})), & N even, \end{cases} 0 \leq i, j < N .$ The algorithm returns $D_{N}$ and the derivative vector $D_{N} U$ . In the even case it omits the derivative of the Nyquist cosine because $\sin ((N / 2) (x_{i} - x_{j})) = 0$ for grid indices $i, j$ . All operations are in exact real arithmetic; no floating-point stability or error guarantee is asserted.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Periodic Fourier collocation differentiates its interpolant exactly

Statement

Let $N \geq 2$ , let $U = (U_{j})_{j = 0}^{N - 1} \in R^{N}$ , and let $p = I_{N} U \in T_{N}$ be the interpolant defined in Periodic Fourier collocation grid and even-grid Nyquist convention. The matrix Periodic Fourier collocation differentiation matrix satisfies

(D_{N} U)_{i} = p^{'} (x_{i}), 0 \leq i < N .

For even $N$ , this statement includes the Nyquist cosine mode, whose derivative vanishes at the grid nodes.

Facts & Assumptions

Given: An integer $N \geq 2$ , the periodic grid and interpolant, and real nodal data $U$ .

[F1]

Periodic Fourier collocation grid and even-grid Nyquist convention: $I_{N} U = \sum_{j} U_{j} L_{j}$ on the $N$ equally spaced nodes; its frequency set is a complete set of residues modulo $N$ , with only the cosine Nyquist mode when $N$ is even.

[F2]

Periodic Fourier collocation differentiation matrix: $(D_{N})_{i j}$ is the termwise derivative of $L_{j}$ evaluated at $x_{i}$ ; in the even case the Nyquist sine term is zero at these nodes.

[F3]

Orthogonality of the characters $x \mapsto e^{2 π i k x / N}$ on $Z / N Z$ : summing all $N$ characters over $Z / N Z$ gives $N$ when the two indices agree modulo $N$ and $0$ otherwise.

[F4]

Finite Fourier inversion for the unitary transform on $Z / N Z$ : the finite Fourier transform on $Z / N Z$ has an inverse, so the frequency coefficients representing nodal data are unique.

[F5]

The derivatives of sine and cosine are cosine and minus sine: $(\cos k x)^{'} = - k \sin (k x)$ and $(\sin k x)^{'} = k \cos (k x)$ .

[F6]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ : $e^{i t} = \cos t + i \sin t$ for real $t$ .

[F7]

Quarter-turn values and shifts by pi/2 and pi: $\sin (k π) = 0$ for every integer $k$ .

[F8]

The subtraction formulas for sine and cosine: $\cos (u - v) = \cos u \cos v + \sin u \sin v$ , so each $\cos (k (x - x_{j}))$ expands into cosine and sine modes with frequency $k$ .

[F9]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : differentiating a trigonometric function composed with $x \mapsto k (x - x_{j})$ multiplies its derivative by $k$ .

[F10]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : differentiating a finite sum of differentiable functions gives the sum of their derivatives.

Proof

1.1F1F3F6F7F8algebra

By [F8], each low-frequency cosine difference in $L_{j}$ expands into the corresponding cosine/sine modes in $T_{N}$ . For even $N = 2 K$ , its Nyquist term expands into $\cos (K x) \cos (K x_{j}) + \sin (K x) \sin (K x_{j})$ ; [F7] makes the second coefficient zero because $K x_{j} = π j$ . At a grid difference $x_{i} - x_{j} = 2 π (i - j) / N$ , Euler's identity [F6] rewrites the cosine sum defining $L_{j} (x_{i})$ as the normalized sum of all characters in a complete residue system modulo $N$ : for odd $N = 2 K + 1$ the residues are $- K, \dots, K$ , and for even $N = 2 K$ they are $- K + 1, \dots, K$ . Character orthogonality [F3] gives $L_{j} (x_{i}) = 1$ when $i = j$ and $0$ otherwise. Thus $I_{N} U$ takes the prescribed nodal values and lies in $T_{N}$ .

1.2F1F4F6algebra

Let $q \in T_{N}$ . Euler's formula expands each low-frequency cosine/sine pair into the characters $e^{i k x_{j}}$ and $e^{- i k x_{j}}$ . If $N = 2 K + 1$ , the resulting residues are $- K, \dots, K$ ; if $N = 2 K$ , they are $- K + 1, \dots, K - 1$ together with the Nyquist residue $K$ , because $\cos (K x_{j}) = e^{i K x_{j}}$ . In either case this is one coefficient for each residue modulo $N$ , and the change from the real basis coefficients to these character coefficients is invertible (for each low mode it is the usual two-by-two Euler change, and the Nyquist coefficient is unchanged). Thus if $q$ vanishes at the nodes, its finite Fourier transform is zero and inversion [F4] makes every character coefficient, hence every real basis coefficient, zero. This proves uniqueness.

2.1F2F5F7F9F10step 1.1algebra

Differentiate the finite sum $p (x) = \sum_{j} U_{j} L_{j} (x)$ term by term using [F10]. By [F5] and [F9], $p^{'} (x_{i}) = \sum_{j} U_{j} L_{j}^{'} (x_{i})$ . When $N$ is odd this is exactly the sine sum defining $D_{N} U$ in [F2]. When $N = 2 K$ the derivative of the remaining Nyquist term is proportional to $\sin (K (x_{i} - x_{j})) = \sin (π (i - j)) = 0$ by [F7], so the same equality holds with the Nyquist term omitted.

3.1step 1.1step 1.2step 2.1∎

Steps 1.1–2.1 establish the nodal interpolation and derivative identity for every data vector, including both parity branches of $N$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Chebyshev–Gauss–Lobatto collocation grid

Definition

Let $N \geq 1$ be an integer and put $x_{j} = \cos (j π / N)$ for $j = 0, \dots, N$ . The ordering is decreasing: $x_{0} = 1$ and $x_{N} = - 1$ . These are the Chebyshev–Gauss–Lobatto nodes; $T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ gives $T_{N} (x_{j}) = (- 1)^{j}$ . They are distinct because cosine is strictly decreasing on $[0, π]$ by Signs, monotonicity intervals, and ranges of sine and cosine.

For interpolation using Cardinal-polynomial interpolation, use the increasing reindexing $z_{k} = x_{N - k}$ , $k = 0, \dots, N$ . For descending-grid real data $U_{j}$ , the corresponding increasing data are $Y_{k} = U_{N - k}$ .

For real nodal data $U_{0}, \dots, U_{N} \in R$ , define $I_{N}^{L} U$ to be the cardinal interpolant of $(Y_{k})_{k = 0}^{N}$ at $(z_{k})_{k = 0}^{N}$ . For a function $f : [- 1, 1] \to R$ , write $I_{N}^{L} f$ for the interpolant of its nodal values. The increasing-node interpolation theorem Existence and uniqueness of the cardinal-polynomial interpolant makes this polynomial unique.

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Chebyshev–Lobatto interpolation folds polynomial modes

Statement

Let $N \geq 1$ , let $x_{j} = \cos (j π / N)$ , and let $I_{N}^{L}$ denote degree-at-most- $N$ interpolation on these nodes. For every integer $n \geq 0$ , let $ρ_{N} (n)$ be the unique integer $r \in {0, \dots, N}$ congruent to either $n$ or $- n$ modulo $2 N$ . Then

I_{N}^{L} T_{n} = T_{ρ_{N} (n)} .

In particular, sampling a Chebyshev series on the Lobatto grid can fold every degree $n > N$ to a degree at most $N$ .

Facts & Assumptions

Given: An integer $N \geq 1$ , its Chebyshev–Gauss–Lobatto grid, and an integer $n \geq 0$ .

[F1]

Chebyshev–Gauss–Lobatto collocation grid: $x_{j} = \cos (j π / N)$ for $0 \leq j \leq N$ , with distinct nodes.

[F2]

Chebyshev polynomials of the first and second kinds by their three-term recurrences defines the polynomials $T_{n}$ ; $T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ gives $T_{n} (\cos θ) = \cos (n θ)$ .

[F3]

Existence and uniqueness of the cardinal-polynomial interpolant: a polynomial of degree at most $N$ that agrees with the data at the $N + 1$ distinct increasing nodes is their unique interpolant; reversing node order and data order preserves the same interpolation polynomial.

[F4]

Degrees and leading coefficients of the Chebyshev polynomials: $T_{r}$ has degree $r$ for every $r \geq 1$ , and $T_{0} = 1$ .

[F5]

Applying the cosine shift by $π$ in Quarter-turn values and shifts by pi/2 and pi twice gives $\cos (α + 2 q π) = \cos α$ for every integer $q$ ; Parity and the Pythagorean identity for sine and cosine gives $\cos (- α) = \cos α$ .

Proof

1.1givenF1choosealgebra

Reduce $n$ modulo $2 N$ to a residue $s \in {0, \dots, 2 N - 1}$ and put $r = \min (s, 2 N - s)$ . Then $0 \leq r \leq N$ , $r \equiv n$ or $- n (m o d 2 N)$ , and this $r$ is unique: the classes ${\pm r}$ are distinct for $0 < r < N$ , with singleton classes at $r = 0, N$ .

2.1F2F5step 1.1algebra

For every node $x_{j}$ , [F2] gives $T_{n} (x_{j}) = \cos (n j π / N)$ . Since $n \equiv \pm r (m o d 2 N)$ , [F5] gives $\cos (n j π / N) = \cos (r j π / N) = T_{r} (x_{j})$ . Thus $T_{n}$ and $T_{r}$ have the same nodal values.

3.1F1F3F4step 2.1∎

By [F4], $T_{r}$ has degree $r \leq N$ . Interpolation uniqueness [F3] therefore gives $I_{N}^{L} T_{n} = T_{r}$ .

AlgorithmAlgorithm: Literature-sourcedProof: Not applicableOpen item page →

Chebyshev–Gauss–Lobatto polynomial differentiation matrix

Definition

Input $N \geq 1$ and real nodal values $U_{0}, \dots, U_{N}$ at the descending grid $x_{j} = \cos (j π / N)$ from Chebyshev–Gauss–Lobatto collocation grid. Set $z_{k} = x_{N - k}$ and $Y_{k} = U_{N - k}$ , so $z_{0} < \dots < z_{N}$ . Form the cardinal polynomials $ℓ_{k}$ for the increasing nodes $z_{0}, \dots, z_{N}$ by Cardinal-polynomial interpolation. Define

(D_{N})_{i j} = ℓ_{N - j}^{'} (x_{i}), 0 \leq i, j \leq N,

and return the derivative vector $D_{N} U$ . Equivalently, form $p (x) = \sum_{k = 0}^{N} Y_{k} ℓ_{k} (x)$ and return $(p^{'} (x_{i}))_{i = 0}^{N}$ . The procedure is exact-real arithmetic and makes no stability claim for a floating-point implementation.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Chebyshev–Gauss–Lobatto differentiation is exact on the interpolant

Statement

Let $N \geq 1$ , let $U_{j} \in R$ be data on the nodes $x_{j} = \cos (j π / N)$ , and let $p$ be the unique polynomial of degree at most $N$ with $p (x_{j}) = U_{j}$ . The matrix Chebyshev–Gauss–Lobatto polynomial differentiation matrix satisfies

(D_{N} U)_{i} = p^{'} (x_{i}), 0 \leq i \leq N .

In particular, applying $D_{N}$ twice gives $p^{''} (x_{i})$ at each node.

Facts & Assumptions

Given: $N \geq 1$ , distinct Chebyshev–Gauss–Lobatto nodes, and real nodal data.

[F1]

Chebyshev–Gauss–Lobatto collocation grid: the nodes are distinct; reversing their order gives increasing nodes $z_{k} = x_{N - k}$ .

[F2]

Chebyshev–Gauss–Lobatto polynomial differentiation matrix: $D_{i j} = ℓ_{N - j}^{'} (x_{i})$ for the cardinal polynomials of the increasing reindexing.

[F3]

Existence and uniqueness of the cardinal-polynomial interpolant: at $N + 1$ distinct ordered nodes, the cardinal sum is the unique polynomial of degree at most $N$ with the prescribed values.

[F4]

Proof

1.1givenF1F3construct

Reindex the data by $Y_{k} = U_{N - k}$ . By [F1], the nodes $z_{k} = x_{N - k}$ are strictly increasing, so [F3] gives $p (x) = \sum_{k = 0}^{N} Y_{k} ℓ_{k} (x)$ as the unique degree-at-most- $N$ interpolant. Reversing the finite sum back to the descending index gives $p (x) = \sum_{j = 0}^{N} U_{j} ℓ_{N - j} (x)$ .

2.1F2F4step 1.1algebra

Differentiate the finite polynomial sum in step 1.1 term by term using [F4]. At $x_{i}$ this gives $p^{'} (x_{i}) = \sum_{j = 0}^{N} U_{j} ℓ_{N - j}^{'} (x_{i}) = \sum_{j = 0}^{N} (D_{N})_{i j} U_{j} = (D_{N} U)_{i}$ by [F2]. This proves the first identity for each node.

3.1F1F3step 2.1algebra∎

The polynomial $p^{'}$ has degree at most $N - 1 \leq N$ and its nodal values are $D_{N} U$ by step 2.1. Uniqueness [F3] therefore makes $p^{'}$ the interpolant of the vector $D_{N} U$ . Applying the already proved first-derivative identity of step 2.1 to that vector gives $D_{N} (D_{N} U) = (p^{''} (x_{i}))_{i = 0}^{N}$ .

AlgorithmAlgorithm: Literature-sourcedProof: Not applicableOpen item page →

Boundary-eliminated Chebyshev collocation for a two-point Dirichlet BVP

Definition

Input an integer $N \geq 2$ , real functions $a, b, f$ given at the Chebyshev–Gauss–Lobatto nodes $x_{j} = \cos (j π / N)$ , and endpoint data $g_{-}, g_{+} \in R$ . The continuous BVP being discretized is

- u^{''} (x) + a (x) u^{'} (x) + b (x) u (x) = f (x), - 1 < x < 1, u (- 1) = g_{-}, u (1) = g_{+} .

The finite algorithm uses only the values at the nodes and does not assume a continuous BVP solvability theorem. Construct the matrix used internally by Chebyshev–Gauss–Lobatto polynomial differentiation matrix explicitly: form $L_{j} (x) = \prod_{0 \leq k \leq N, k \neq j} (x - x_{k}) / (x_{j} - x_{k})$ and set $(D_{N})_{i j} = L_{j}^{'} (x_{i})$ for $0 \leq i, j \leq N$ . Then set $D_{N}^{(2)} = D_{N} D_{N}$ . For each interior row $i = 1, \dots, N - 1$ and each column $j = 0, \dots, N$ , put

C_{i j} = - (D_{N}^{(2)})_{i j} + a (x_{i}) (D_{N})_{i j} + b (x_{i}) δ_{i j} .

Let $A = (C_{i j})_{1 \leq i, j \leq N - 1}$ and $r_{i} = f (x_{i}) - C_{i 0} g_{+} - C_{i N} g_{-}$ . If $A$ is singular, return status singular-interior-matrix and the matrix $A$ . If $A$ is invertible, solve $A U_{I} = r$ , set $U_{0} = g_{+}$ and $U_{N} = g_{-}$ , and return success with $U = (U_{0}, U_{1}, \dots, U_{N})$ . This exact-arithmetic finite solve asserts no unconditional nonsingularity, stability, convergence, or PDE error bound.

TheoremStatement: Literature-sourcedProof: AI-adaptedOpen item page →

Boundary elimination returns exactly the finite Chebyshev collocation system

Statement

For the algorithm Boundary-eliminated Chebyshev collocation for a two-point Dirichlet BVP, a success output is the unique nodal vector with $U_{0} = g_{+}$ , $U_{N} = g_{-}$ satisfying

- (D_{N}^{(2)} U)_{i} + a (x_{i}) (D_{N} U)_{i} + b (x_{i}) U_{i} = f (x_{i}), 1 \leq i < N,

where $D_{N}$ is the exact Chebyshev–Gauss–Lobatto differentiation matrix and $D_{N}^{(2)} = D_{N} D_{N}$ . If the interior matrix is singular, the algorithm returns singular-interior-matrix and makes no claim about existence or uniqueness for the given right-hand side.

Facts & Assumptions

Given: An integer $N \geq 2$ , node values of $a, b, f$ , endpoints $g_{-}, g_{+}$ , the boundary-elimination algorithm, and its interior matrix $A$ .

[F1]

Boundary-eliminated Chebyshev collocation for a two-point Dirichlet BVP: $C_{i j} = - (D_{N}^{2})_{i j} + a (x_{i}) (D_{N})_{i j} + b (x_{i}) δ_{i j}$ , $A = (C_{i j})_{1 \leq i, j < N}$ , and $r_{i} = f (x_{i}) - C_{i 0} g_{+} - C_{i N} g_{-}$ .

[F2]

Chebyshev–Gauss–Lobatto differentiation is exact on the interpolant: for the interpolant $p$ through the full nodal vector $U$ , $D_{N} U = (p^{'} (x_{i}))_{i}$ ; applying $D_{N}$ again gives $(p^{''} (x_{i}))_{i}$ .

[F3]

Invertible matrix theorem: invertibility, full pivot rank, RREF $I$ , trivial nullspace and unique solvability are equivalent: an invertible square matrix has exactly one solution for every right-hand side.

Proof

1.1givenF1

On a success branch the algorithm sets $U_{0} = g_{+}$ and $U_{N} = g_{-}$ and solves $A U_{I} = r$ for the interior vector $U_{I} = (U_{1}, \dots, U_{N - 1})$ .

1.2givenF2

By [F2], $D_{N} U$ and $D_{N}^{(2)} U$ are respectively the first and second derivatives of the degree-at-most- $N$ nodal interpolant evaluated at the nodes.

2.1givenF1step 1.1step 1.2algebra

Expanding the $i$ th row of the full expression using [F1] gives $\sum_{j = 1}^{N - 1} C_{i j} U_{j} + C_{i 0} g_{+} + C_{i N} g_{-} = f (x_{i}),$ which is exactly the reduced equation $A U_{I} = r$ with the definition of $r_{i}$ . Combined with step 1.2, this is the stated collocation equation at every interior node.

2.2givenF1F3step 1.1algebra

Since the success branch has invertible $A$ , [F3] gives exactly one interior vector solving $A U_{I} = r$ . The prescribed endpoint values then give exactly one full nodal vector. The singular branch returns its declared status and asserts nothing about solvability of its right-hand side.

3.1step 1.1step 1.2step 2.1step 2.2∎

Steps 1.1–2.2 prove the success claim and the stated singular-matrix limit.

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Arcsine-weighted inner product on the Chebyshev interval

Definition

For continuous real functions $f, g$ on $[- 1, 1]$ , define

⟨ f, g ⟩_{C} : = \int_{0}^{π} f (\cos θ) g (\cos θ) d θ, ∥ f ∥_{C}^{2} : = ⟨ f, f ⟩_{C} .

This is the arcsine-weighted pairing on the compact Chebyshev interval: in the variable $x = \cos θ$ its weight is $ω_{C} (x) = (1 - x^{2})^{- 1 / 2}$ for $- 1 < x < 1$ . The angular integral is the endpoint convention, so no endpoint value of the singular density is required.

The displayed integral is defined for every such pair: cosine is continuous and maps $[0, π]$ into $[- 1, 1]$ ; composition and multiplication preserve continuity, so the integrand is continuous. Since $0 < π$ , it is Riemann integrable by A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion.

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Chebyshev polynomials are orthogonal for the arcsine weight

Statement

Under the pairing of Arcsine-weighted inner product on the Chebyshev interval, $⟨ T_{m}, T_{n} ⟩_{C} = {\begin{cases} π, & m = n = 0, \\ π / 2, & m = n \geq 1, \\ 0, & m \neq n . \end{cases}$ The polynomials $T_{0}, T_{1}, \dots$ form a basis of the real polynomial space, and the pairing is positive definite on continuous real functions on $[- 1, 1]$ . If a continuous real function $h$ satisfies $⟨ h, T_{n} ⟩_{C} = 0$ for every $n \geq 0$ , then $h = 0$ on $[- 1, 1]$ .

Facts & Assumptions

Given: Natural numbers $m, n$ , the Chebyshev polynomials $T_{k}$ , and the arcsine-weighted pairing.

[F1]

Arcsine-weighted inner product on the Chebyshev interval: $⟨ f, g ⟩_{C} = \int_{0}^{π} f (\cos θ) g (\cos θ) d θ$ .

[F2]

$T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ : $T_{k} (\cos θ) = \cos (k θ)$ for every natural $k$ and real $θ$ .

[F3]

The addition formulas for sine and cosine: $2 \cos u \cos v = \cos (u + v) + \cos (u - v)$ .

[F4]

The derivatives of sine and cosine are cosine and minus sine and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ give $(\sin (k θ) / k)^{'} = \cos (k θ)$ for nonzero integer $k$ . The interval is nondegenerate since $0 < π$ by Pi as twice the smallest positive zero of cosine, and each cosine integrand is continuous, hence integrable by A function differentiable at $c$ is continuous at $c$ and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. The second fundamental theorem The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ therefore gives $\int_{0}^{π} \cos (k θ) d θ = (\sin (k π) - \sin 0) / k$ .

[F10]

Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ : integrals of finite sums and scalar multiples of integrable functions are the corresponding sums and scalar multiples; in particular it gives the integral bilinearity used in steps 1.1, 3.1, and 4.1.

[F5]

Quarter-turn values and shifts by pi/2 and pi: $\sin π = 0$ and $\sin (x + π) = - \sin x$ , so $\sin (k π) = 0$ for every integer $k$ ; Parity and the Pythagorean identity for sine and cosine: cosine is even.

[F6]

Chebyshev polynomials of the first and second kinds by their three-term recurrences: $T_{0} = 1$ , $T_{1} = X$ , and $T_{n + 1} = 2 X T_{n} - T_{n - 1}$ for $n \geq 1$ .

[F7]

Polynomials are uniformly dense in $C ([a, b], R)$ for every closed interval: every continuous real function on $[- 1, 1]$ is a uniform limit of polynomials.

[F8]

Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact and Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value: a continuous real function on $[- 1, 1]$ is bounded.

[F9]

A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs and Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function make the integrands in steps 3.1 and 4.1 continuous; A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion makes them integrable on $[0, π]$ , and If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ passes pointwise lower and upper bounds to their integrals.

[F11]

Cosine is continuous by [F4], with $\cos 0 = 1$ and $\cos π = - 1$ by The derivatives of sine and cosine are cosine and minus sine and [F5]. Thus Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ applied on $[0, π]$ gives, for every $x \in [- 1, 1]$ , a $θ \in [0, π]$ with $\cos θ = x$ .

Proof

1.1F1F2F3F4F5F10algebra

By [F1]–[F2], $⟨ T_{m}, T_{n} ⟩_{C} = \int_{0}^{π} \cos (m θ) \cos (n θ) d θ$ . If $m \neq n$ , [F3], cosine parity [F5], and $\sin (k π) = 0$ from [F5] make the integral of each nonzero-frequency cosine vanish by [F4]. If $m = n \geq 1$ , [F3] gives $\frac{1}{2} (1 + \cos (2 m θ))$ , whose integral is $π / 2$ . If $m = n = 0$ , the integrand is $1$ and its integral is $π$ . Linearity [F10] passes from the product-to-sum identity to these integrals.

2.1F6step 1.1algebrainduction

The recurrence [F6] shows inductively that $T_{n}$ has degree $n$ and nonzero leading coefficient: this holds for $T_{0}, T_{1}$ , and the leading term of $2 X T_{n}$ cannot cancel with the lower-degree $T_{n - 1}$ . Eliminating the leading term of any polynomial successively proves that $T_{0}, T_{1}, \dots$ span all polynomials; their distinct degrees also give linear independence. Step 1.1 gives positive squared norms and pairwise orthogonality.

3.1F1F7F8F9F10F11step 2.1choosealgebra

Suppose continuous $h$ pairs to zero with every $T_{n}$ . By step 2.1 it pairs to zero with every polynomial. Fix $ε > 0$ and choose a polynomial $q$ with $∥ q - h ∥_{\infty} < ε$ by [F7]. The bound $M = ∥ h ∥_{\infty} < \infty$ follows from [F8]. The pointwise bounds $- M ε \leq h (\cos θ) (h - q) (\cos θ) \leq M ε$ and [F9] give $0 \leq ⟨ h, h ⟩_{C} = ⟨ h, h - q ⟩_{C} \leq π M ε$ , where the equality uses bilinearity [F10]. The lower inequality uses the nonnegative integrand $h (\cos θ)^{2}$ and [F9]. Since this holds for every $ε > 0$ , $⟨ h, h ⟩_{C} = 0$ . If $h (x_{0}) \neq 0$ at some $x_{0} \in [- 1, 1]$ , [F11] supplies $θ_{0} \in [0, π]$ with $\cos θ_{0} = x_{0}$ ; continuity makes $h (\cos θ)^{2}$ bounded below by a positive constant on a one-sided neighborhood if $θ_{0}$ is an endpoint, or a two-sided neighborhood otherwise. [F9] then gives a positive integral, contradicting zero norm. Therefore $h = 0$ .

4.1F1F9F10F11givenalgebra∎

For any continuous $h$ , if $⟨ h, h ⟩_{C} = 0$ but $h (\cos θ_{0}) \neq 0$ at some $θ_{0} \in [0, π]$ , continuity gives a one-sided interval at an endpoint, or a two-sided interval in the interior, on which $h (\cos θ)^{2}$ is bounded below by a positive number. [F9] makes its integral positive, a contradiction. Thus $h (\cos θ) = 0$ for every $θ$ , and [F11] gives $h = 0$ on $[- 1, 1]$ . Symmetry and bilinearity follow from the integral definition [F1] and linearity [F10], proving positive definiteness.

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Chebyshev weighted spectral Sobolev scale and projection

Definition

For $s \geq 0$ , define $H_{C}^{s}$ to be the completion of finite real Chebyshev sums $p = \sum_{n = 0}^{M} c_{n} T_{n}$ in the norm

∥ p ∥_{H_{C}^{s}}^{2} = π ∣ c_{0} ∣^{2} + \frac{π}{2} \sum_{n = 1}^{M} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} .

By Chebyshev polynomials are orthogonal for the arcsine weight, the coefficients of a finite sum are unique and this is a norm. At $s = 0$ , this norm on finite sums is the arcsine-weighted norm from Arcsine-weighted inner product on the Chebyshev interval. This is the spectral Sobolev scale for the named weight and domain; it is defined by Chebyshev coefficients, and no equivalence with an unweighted derivative-based Sobolev norm is asserted.

The weights are positive by Real powers for positive bases, with the zero-base positive-exponent convention and The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ . For fixed $s \geq 0$ , $x \mapsto x^{s}$ is nondecreasing on $(0, \infty)$ : the derivative formula Continuity and derivatives of positive-base real powers gives $s x^{s - 1} \geq 0$ , and the mean-value theorem The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ gives monotonicity. Thus $(1 + n^{2})^{s} \geq 1$ . For $s > 0$ , the derivative formula also makes $x \mapsto x^{- s}$ nonincreasing on $(0, \infty)$ .

The completion is isometric to the weighted coefficient sequence space

{(c_{n})_{n \geq 0} : c_{n} \in R for every n, π ∣ c_{0} ∣^{2} + \frac{π}{2} \sum_{n \geq 1} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} < \infty},

as follows. Put $w_{0} = π$ and $w_{n} = \frac{π}{2} (1 + n^{2})^{s}$ for $n \geq 1$ . The map $(c_{n}) \mapsto (\sqrt{w_{n}} c_{n})$ is an isometry onto the real square-summable sequence space; it is onto because every $w_{n} > 0$ . Finite-support sequences are dense by truncation, and The real square-summable sequence space is complete gives completeness. Therefore the displayed weighted sequence space is exactly the completion. The notation $u = \sum_{n \geq 0} c_{n} T_{n}$ denotes its coefficient representation and does not assert pointwise convergence of the series. Its norm is the displayed weighted coefficient sum.

At $s = 0$ , denote this completion by $L_{C}^{2} : = H_{C}^{0}$ and define

∥ u ∥_{C}^{2} : = ∥ u ∥_{H_{C}^{0}}^{2} = π ∣ c_{0} ∣^{2} + \frac{π}{2} \sum_{n \geq 1} ∣ c_{n} ∣^{2} .

By the orthogonality lemma, this is the unique norm extension from finite sums of the arcsine-weighted norm of Arcsine-weighted inner product on the Chebyshev interval. Since $(1 + n^{2})^{s} \geq 1$ for $s \geq 0$ , every $H_{C}^{s}$ embeds continuously in $L_{C}^{2}$ , so $∥ u ∥_{C}$ is defined for every $u \in H_{C}^{s}$ , including when $s = 0$ .

If an element has a continuous representative given by a uniformly convergent Chebyshev series, its arcsine integral norm agrees with this completion norm: for continuous $h$ , the defining pairing gives $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ , so uniform convergence implies convergence in the arcsine norm as well.

For $u = \sum_{n \geq 0} c_{n} T_{n}$ in this coefficient completion, define the degree- $N$ orthogonal projection by

Π_{N} u = \sum_{n = 0}^{N} c_{n} T_{n} .

The projection is well-defined by coefficient truncation and the positive coefficient norm above.

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Weighted Chebyshev projection has its spectral Sobolev rate

Statement

Let $s \geq 0$ , $N \geq 0$ , and $u \in H_{C}^{s}$ as defined in Chebyshev weighted spectral Sobolev scale and projection. Then

∥ u - Π_{N} u ∥_{C} \leq (1 + (N + 1)^{2})^{- s / 2} ∥ u ∥_{H_{C}^{s}} .

Thus the weighted projection error is $O (N^{- s})$ when the data belong to this explicitly defined weighted spectral Sobolev space.

Facts & Assumptions

Given: $s \geq 0$ , an integer $N \geq 0$ , and a Chebyshev coefficient expansion $u = \sum_{n \geq 0} c_{n} T_{n}$ with finite $H_{C}^{s}$ norm.

[F1]

Chebyshev weighted spectral Sobolev scale and projection: $Π_{N} u = \sum_{n = 0}^{N} c_{n} T_{n}$ and the $H_{C}^{s}$ norm weights coefficient $c_{n}$ by $(1 + n^{2})^{s}$ ; these weights are nondecreasing in $n$ for $s \geq 0$ .

[F2]

Chebyshev polynomials are orthogonal for the arcsine weight: the squared $C$ -norm of $T_{0}$ is $π$ and that of each $T_{n}$ , $n \geq 1$ , is $π / 2$ .

Proof

1.1F1F2algebra

By [F1], $u - Π_{N} u = \sum_{n > N} c_{n} T_{n}$ . Orthogonality from [F2] gives $∥ u - Π_{N} u ∥_{C}^{2} = \frac{π}{2} \sum_{n > N} ∣ c_{n} ∣^{2} .$ Since $n > N$ implies $(1 + n^{2})^{s} \geq (1 + (N + 1)^{2})^{s}$ , this is at most $(1 + (N + 1)^{2})^{- s} \frac{π}{2} \sum_{n > N} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} \leq (1 + (N + 1)^{2})^{- s} ∥ u ∥_{H_{C}^{s}}^{2} .$ Taking square roots proves the result for finite sums.

2.1F1step 1.1algebra∎

Finite sums are dense in the completion defining $H_{C}^{s}$ . The truncation $Π_{N}$ is a contraction in the coefficient norm and in the $C$ -norm, so both sides of the finite-sum inequality pass to the limit along any Cauchy sequence of finite sums. This yields the stated estimate for every $u \in H_{C}^{s}$ .

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Chebyshev–Lobatto interpolation inherits a weighted near-best bound

Statement

For $f \in C [- 1, 1]$ and $N \geq 0$ , let $E_{N} (f)$ and $Λ_{N}^{L}$ be the best uniform error and the Lebesgue constant from Best uniform polynomial error and Chebyshev interpolation operators. Then

∥ f - I_{N}^{L} f ∥_{C} \leq \sqrt{π} (1 + Λ_{N}^{L}) E_{N} (f),

where $∥ \cdot ∥_{C}$ is the arcsine-weighted norm of Arcsine-weighted inner product on the Chebyshev interval.

Facts & Assumptions

Given: A continuous real function $f$ on $[- 1, 1]$ and $N \geq 0$ .

[F1]

Chebyshev interpolation error is controlled by best polynomial error: $∥ f - I_{N}^{L} f ∥_{\infty} \leq (1 + Λ_{N}^{L}) E_{N} (f)$ .

[F2]

Arcsine-weighted inner product on the Chebyshev interval: $∥ h ∥_{C}^{2} = \int_{0}^{π} ∣ h (\cos θ) ∣^{2} d θ$ , so $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ .

Proof

1.1givenF2algebra

For every $θ \in [0, π]$ , $∣ h (\cos θ) ∣ \leq ∥ h ∥_{\infty}$ . Integrating its square in [F2] gives $∥ h ∥_{C}^{2} \leq π ∥ h ∥_{\infty}^{2}$ and hence $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ .

2.1givenF1step 1.1algebra∎

Apply step 1.1 to $h = f - I_{N}^{L} f$ , then use the uniform near-best bound [F1]. The result is $∥ f - I_{N}^{L} f ∥_{C} \leq \sqrt{π} (1 + Λ_{N}^{L}) E_{N} (f)$ .

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Chebyshev–Lobatto interpolation has an algebraic weighted spectral-Sobolev rate

Statement

Let $s > 1 / 2$ , $N \geq 1$ , and $u = \sum_{n \geq 0} c_{n} T_{n} \in H_{C}^{s}$ . Then the series converges uniformly, its nodal interpolant $I_{N}^{L} u$ is defined, and $∥ u - I_{N}^{L} u ∥_{C} \leq C_{s} N^{- s} ∥ u ∥_{H_{C}^{s}}, C_{s} = 1 + 2 \sqrt{1 + \frac{1}{2 s - 1}} .$

Facts & Assumptions

Given: A real $s > 1 / 2$ , an integer $N \geq 1$ , and a Chebyshev coefficient sequence in $H_{C}^{s}$ .

[F1]

Chebyshev weighted spectral Sobolev scale and projection: $∥ u ∥_{H_{C}^{s}}^{2} = π ∣ c_{0} ∣^{2} + \frac{π}{2} \sum_{n \geq 1} (1 + n^{2})^{s} ∣ c_{n} ∣^{2}$ .

[F2]

Chebyshev–Lobatto interpolation folds polynomial modes: $I_{N}^{L} T_{n} = T_{ρ_{N} (n)}$ for the unique folded degree $ρ_{N} (n) \in {0, \dots, N}$ .

[F3]

Cardinal-polynomial interpolation: $I_{N}^{L} f$ is the finite Lagrange cardinal sum at the $N + 1$ distinct Lobatto nodes, hence is linear and continuous in the uniform norm.

[F4]

Chebyshev polynomials are orthogonal for the arcsine weight: $∥ T_{0} ∥_{C}^{2} = π$ and $∥ T_{r} ∥_{C}^{2} = π / 2$ for $r \geq 1$ , with distinct modes orthogonal.

[F5]

$T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ and Signs, monotonicity intervals, and ranges of sine and cosine: $T_{n} (\cos θ) = \cos (n θ)$ and $∣ \cos t ∣ \leq 1$ for real $t$ .

[F6]

The p-series for a real exponent p converges exactly when p is greater than one: $\sum_{n \geq 1} n^{- 2 s}$ converges because $2 s > 1$ .

[F7]

The Cauchy-Schwarz inequality for finite sums: for finite real lists, $∣ \sum a_{k} b_{k} ∣^{2} \leq (\sum a_{k}^{2}) (\sum b_{k}^{2})$ .

[F8]

The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents gives $(a b)^{r} = a^{r} b^{r}$ and $(a^{r})^{s} = a^{r s}$ for positive bases; the monotonicity of $x^{s}$ for $s \geq 0$ and of $x^{- s}$ for $s > 0$ is recorded with the coefficient weights in Chebyshev weighted spectral Sobolev scale and projection.

[F9]

For $p = 2 s > 1$ , $x^{- p}$ is continuous and decreasing on $[1, \infty)$ by Continuity and derivatives of positive-base real powers and the scale's real-power monotonicity. Its antiderivative $x^{1 - p} / (1 - p)$ follows from the same power rule, so The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ evaluates its integral on each finite interval. A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum identifies a bounded sequence of its nonnegative partial sums with a finite series sum.

Proof

1.1givenF1F5F6F7algebra

Let $w_{n} = (1 + n^{2})^{s}$ . By [F1] the weighted coefficient sum $\sum_{n \geq 0} w_{n} ∣ c_{n} ∣^{2}$ is finite. For $n \geq 1$ , monotonicity of $x^{- s}$ from [F1] gives $w_{n}^{- 1} \leq n^{- 2 s}$ , so [F6] implies $\sum_{n \geq 0} w_{n}^{- 1} < \infty$ . For every finite $Q$ , apply [F7] to $\sqrt{w_{n}} ∣ c_{n} ∣$ and $w_{n}^{- 1 / 2}$ , $0 \leq n \leq Q$ ; this bounds $\sum_{n = 0}^{Q} ∣ c_{n} ∣$ uniformly in $Q$ . Nonnegative partial sums therefore give $\sum_{n \geq 0} ∣ c_{n} ∣ < \infty$ by A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum. By [F5], $∣ T_{n} (x) ∣ \leq 1$ on $[- 1, 1]$ , so the series converges uniformly and its nodal values are defined.

2.1F2F3step 1.1algebra

Write $u = S_{N} + R_{N}$ , where $S_{N} = \sum_{n = 0}^{N} c_{n} T_{n}$ and $R_{N} = \sum_{n > N} c_{n} T_{n}$ . The Lagrange operator [F3] reproduces $S_{N}$ and is continuous in the uniform norm; therefore $u - I_{N}^{L} u = R_{N} - I_{N}^{L} R_{N}$ . By [F2], uniform convergence and the finite cardinal sum give $I_{N}^{L} R_{N} = \sum_{r = 0}^{N} b_{r} T_{r}$ , where $b_{r} = \sum_{n > N : ρ_{N} (n) = r} c_{n}$ .

3.1givenF2F8F9step 2.1algebra

Fix $r \in {0, \dots, N}$ . Since $n \equiv r$ or $- r (m o d 2 N)$ , every index in its alias class has form $2 q N + r$ or $2 q N - r$ . When $n > N$ the corresponding integer $q$ is positive. For each $q$ there are at most two candidates; the plus candidate is at least $q N$ , and the minus candidate is at least $(2 q - 1) N \geq q N$ . Thus $1 + n^{2} \geq (q N)^{2}$ ; since $x^{- s}$ is nonincreasing for $s > 0$ , and by [F8], $(1 + n^{2})^{- s} \leq (q N)^{- 2 s} = N^{- 2 s} q^{- 2 s}$ . For every integer $Q \geq 2$ , monotonicity gives $q^{- 2 s} \leq \int_{q - 1}^{q} x^{- 2 s} d x$ for $2 \leq q \leq Q$ , so [F9] yields $\sum_{q = 1}^{Q} q^{- 2 s} \leq 1 + \int_{1}^{Q} x^{- 2 s} d x = 1 + \frac{1 - Q^{1 - 2 s}}{2 s - 1} \leq 1 + \frac{1}{2 s - 1}$ . By nonnegativity and [F9], the infinite sum has the same bound. Therefore $\sum_{n > N : ρ_{N} (n) = r} (1 + n^{2})^{- s} \leq 2 (1 + \frac{1}{2 s - 1}) N^{- 2 s}$ .

4.1F1F2F4F7step 1.1step 3.1algebra

Let $A_{s} = 1 + 1 / (2 s - 1)$ . Apply [F7] to finite partial sums over an alias class, use step 3.1, and pass to the limit; the coefficient sums converge absolutely by step 1.1. This gives $∣ b_{r} ∣^{2} \leq 2 A_{s} N^{- 2 s} \sum_{n > N : ρ_{N} (n) = r} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} .$ The alias classes partition $n > N$ . By [F4], $∥ I_{N}^{L} R_{N} ∥_{C}^{2} \leq π \sum_{r = 0}^{N} ∣ b_{r} ∣^{2} \leq 2 π A_{s} N^{- 2 s} \sum_{n > N} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} \leq 4 A_{s} N^{- 2 s} ∥ u ∥_{H_{C}^{s}}^{2},$ where the last inequality uses [F1]. Also [F1] and [F4] give $∥ R_{N} ∥_{C}^{2} = \frac{π}{2} \sum_{n > N} ∣ c_{n} ∣^{2} \leq (N + 1)^{- 2 s} ∥ u ∥_{H_{C}^{s}}^{2} .$ Thus $∥ R_{N} ∥_{C} \leq (N + 1)^{- s} ∥ u ∥_{H_{C}^{s}}$ .

5.1step 2.1step 4.1algebra∎

By step 2.1 and the triangle inequality, $∥ u - I_{N}^{L} u ∥_{C} \leq ∥ R_{N} ∥_{C} + ∥ I_{N}^{L} R_{N} ∥_{C} \leq (1 + 2 \sqrt{A_{s}}) N^{- s} ∥ u ∥_{H_{C}^{s}}$ because $(N + 1)^{- s} \leq N^{- s}$ . This is the claimed estimate.

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Chebyshev–Lobatto interpolation remainder bound

Statement

Let $N \geq 1$ and $f \in C^{N + 1} ([- 1, 1])$ . Let $I_{N}^{L} f$ be interpolation at the nodes $x_{j} = \cos (j π / N)$ . Then $∥ f - I_{N}^{L} f ∥_{C} \leq \sqrt{π} \frac{2^{1 - N}}{(N + 1)!} \max_{x \in [- 1, 1]} ∣ f^{(N + 1)} (x) ∣ .$

Facts & Assumptions

Given: $N \geq 1$ and $f \in C^{N + 1} ([- 1, 1])$ .

[F1]

Chebyshev–Gauss–Lobatto collocation grid: the $N + 1$ distinct nodes are $x_{j} = \cos (j π / N)$ , including both endpoints.

[F2]

Chebyshev polynomials of the first and second kinds by their three-term recurrences and Degrees and leading coefficients of the Chebyshev polynomials: $T_{N}$ has degree $N$ and leading coefficient $2^{N - 1}$ for $N \geq 1$ .

[F3]

$T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ , The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , and The derivatives of sine and cosine are cosine and minus sine: for $x = \cos θ$ , $0 < θ < π$ , $T_{N}^{'} (x) = N \sin (N θ) / \sin θ$ .

[F4]

Quarter-turn values and shifts by pi/2 and pi: $\sin (k π) = 0$ for every integer $k$ , by repeated shifts by $π$ .

[F5]

Parity and the Pythagorean identity for sine and cosine: $∣ \sin t ∣ \leq 1$ for every real $t$ .

[F6]

Cardinal-polynomial interpolation: the node polynomial is $ω_{N} (x) = \prod_{j = 0}^{N} (x - x_{j})$ and $I_{N}^{L} f$ is its degree-at-most- $N$ Lagrange interpolant after reversing the decreasing node order.

[F7]

Pointwise Lagrange interpolation remainder: for $x$ off the nodes, $f (x) - I_{N}^{L} f (x) = f^{(N + 1)} (ξ) ω_{N} (x) / (N + 1)!$ for some $ξ$ in the interval between the extreme nodes and $x$ ; at a node the error is zero.

[F8]

Arcsine-weighted inner product on the Chebyshev interval: $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ because the angular measure has total mass $π$ .

[F9]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ , Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact, and Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value: $f^{(N + 1)}$ is continuous on the compact interval, so the displayed finite maximum exists.

[F10]

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots: a polynomial with more distinct roots than its degree is the zero polynomial.

Proof

1.1F2algebra

By the recurrence in [F2], $T_{N}$ has leading coefficient $2^{N - 1}$ for every $N \geq 1$ . Hence $q_{N} (x) : = \frac{2^{1 - N}}{N} (x^{2} - 1) T_{N}^{'} (x)$ is monic of degree $N + 1$ .

1.2F1F2F3F4F6F10algebra

For $1 \leq j < N$ , put $θ_{j} = j π / N$ . Then [F3] and [F4] give $T_{N}^{'} (x_{j}) = N \sin (j π) / \sin θ_{j} = 0$ . Also $q_{N} (1) = q_{N} (- 1) = 0$ . Thus $q_{N}$ vanishes at all $N + 1$ distinct nodes in [F1]. The node product $ω_{N}$ is also monic of degree $N + 1$ by [F6]; their difference has degree at most $N$ and $N + 1$ distinct roots, so [F10] gives $q_{N} = ω_{N}$ .

2.1F1F3F5step 1.2algebra

For $x = \cos θ$ with $0 < θ < π$ , the derivative formula in [F3] gives $∣ ω_{N} (x) ∣ = 2^{1 - N} ∣ \sin θ \sin (N θ) ∣ \leq 2^{1 - N} .$ The inequality uses [F5]. At the endpoints $ω_{N} = 0$ , so this bound holds on all of $[- 1, 1]$ .

3.1F1F7F9step 2.1algebra

At nodes the interpolation error is zero; off the nodes apply [F7] and step 2.1. Since the maximum in [F9] is finite, this gives $∥ f - I_{N}^{L} f ∥_{\infty} \leq \frac{2^{1 - N}}{(N + 1)!} \max_{[- 1, 1]} ∣ f^{(N + 1)} ∣ .$

4.1F8step 3.1algebra∎

Apply the weighted-norm estimate [F8] to the uniform error in step 3.1. This proves the stated inequality.

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Analytic ellipse data give exponential Chebyshev–Lobatto interpolation error

Statement

Let $ρ > 1$ , and let $E_{ρ}$ be the closed Bernstein ellipse

E_{ρ} = {\frac{ρ e^{i t} + ρ^{- 1} e^{- i t}}{2} : 0 \leq t \leq 2 π}

together with its interior. Suppose $f$ is real-valued on $[- 1, 1]$ , extends holomorphically to an open neighborhood of $E_{ρ}$ , and satisfies $∣ f (z) ∣ \leq M$ on $E_{ρ}$ . Then for every $N \geq 1$ ,

∥ f - I_{N}^{L} f ∥_{C} \leq \frac{4 \sqrt{π} M}{ρ - 1} ρ^{- N} .

Facts & Assumptions

Given: $ρ > 1$ , a function $f$ holomorphic near $E_{ρ}$ with $∣ f ∣ \leq M$ there, and $N \geq 1$ .

[F1]

A continuous function holomorphic away from one point on a star-shaped domain has a primitive and zero closed-contour integrals: a holomorphic function on a star-shaped open set has zero integral around every closed rectifiable contour in that set.

[F2]

$T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ and Quarter-turn values and shifts by pi/2 and pi: $T_{n} (\cos θ) = \cos (n θ)$ and cosine is $2 π$ -periodic on the real line; Parity and the Pythagorean identity for sine and cosine gives its evenness.

[F3]

Chebyshev–Lobatto interpolation folds polynomial modes: $I_{N}^{L} T_{n} = T_{ρ_{N} (n)}$ for the folded degree $0 \leq ρ_{N} (n) \leq N$ .

[F4]

Chebyshev polynomials are orthogonal for the arcsine weight: the Chebyshev modes have squared norms $π$ at degree zero and $π / 2$ at positive degrees, and a continuous function pairing to zero with every mode is zero. The coefficient convention used in this proof is defined explicitly in step 2.1.

[F5]

Arcsine-weighted inner product on the Chebyshev interval: the angular measure has total mass $π$ , so $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ .

[F6]

Signs, monotonicity intervals, and ranges of sine and cosine: $∣ \cos t ∣ \leq 1$ for real $t$ .

[F7]

For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges: for $r \in R$ with $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ .

[F8]

Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential and Complex sine, cosine, hyperbolic sine, and hyperbolic cosine are entire with their standard derivatives: complex cosine is entire and $\cos z = (\exp (i z) + \exp (- i z)) / 2$ . The complex exponential is entire (The complex exponential is entire and its complex derivative is itself); its sums, products, and compositions are complex differentiable by The chain rule for complex derivatives and Linearity, product, reciprocal, and quotient rules for complex derivatives, and complex differentiability implies continuity (Complex differentiability at a point implies continuity there).

[F9]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential and $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ : the complex exponential adds and $\exp (2 π i k) = 1$ for every integer $k$ . $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ gives its Cartesian form and modulus; Parity and the Pythagorean identity for sine and cosine gives the real sine and cosine parity used in the coefficient formula.

[F10]

The natural logarithm as the inverse of the exponential function and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for $L = \log ρ$ with $ρ > 1$ , $L > 0$ and $e^{L} = ρ$ . The real exponential is strictly increasing and $e^{0} = 1$ (The exponential function is strictly increasing, The real exponential function and the number $e$ by a power series).

[F11]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , Integer powers $a^{m}$ , and Laws of integer exponents: $e^{- n L} = (e^{- L})^{n} = (1 / ρ)^{n} = ρ^{- n}$ for each natural $n$ .

Proof

1.1givenF8F9F10algebra

Put $L = \log ρ > 0$ . For $z = θ - i t$ , $0 \leq t \leq L$ , let $r = e^{t}$ , so $1 \leq r \leq ρ$ by [F10]. The definitions in [F8] and Cartesian formula [F9] give $\cos (θ - i t) = A_{r} \cos θ + i B_{r} \sin θ$ , where $A_{r} = (r + r^{- 1}) / 2$ and $B_{r} = (r - r^{- 1}) / 2$ . For $1 \leq r \leq ρ$ , $A_{r} \leq A_{ρ}$ and $B_{r} \leq B_{ρ}$ : the differences are $(ρ - r) (1 - 1 / (ρ r)) / 2$ and $(ρ - r) (1 + 1 / (ρ r)) / 2$ . Therefore $x^{2} / A_{ρ}^{2} + y^{2} / B_{ρ}^{2} \leq \cos^{2} θ + \sin^{2} θ = 1$ for $r > 1$ , so the image lies in the filled ellipse $E_{ρ}$ ; at $r = 1$ it lies on $[- 1, 1] \subset E_{ρ}$ . The closed rectangle $R_{0} = {- π \leq ℜ z \leq π, - L \leq ℑ z \leq 0}$ is compact and lies in the open set $\cos^{- 1} (V)$ , where $V$ is a neighborhood on which $f$ is holomorphic. Hence a slightly larger open rectangle $R$ still lies in $\cos^{- 1} (V)$ ; it is convex and therefore star-shaped. By [F8], $g = f \circ \cos$ is holomorphic on $R$ , and each $H_{n} (z) = g (z) e^{- i n z}$ is holomorphic and continuous there by [F8]. On the lower edge $∣ g (θ - i L) ∣ \leq M$ .

2.1F1F2F8F9F11step 1.1algebra

For each $n \geq 1$ define $c_{n} = \frac{2}{π} \int_{0}^{π} f (\cos θ) \cos (n θ) d θ$ , and define $c_{0} = \frac{1}{π} \int_{0}^{π} f (\cos θ) d θ$ . For real $θ$ , $g$ is even and $f (\cos θ)$ is real; Euler's identity [F9] makes the sine part of $\int_{- π}^{π} g (θ) e^{- i n θ} d θ$ vanish, so this integral is $π c_{n}$ . Also $H_{n} (z) = g (z) e^{- i n z}$ is $2 π$ -periodic: $g$ is periodic by [F8]--[F9], and $e^{- 2 π i n} = 1$ by [F9]. Apply [F1] to the boundary of $R_{0}$ ; the vertical integrals cancel by periodicity, so the top and lower horizontal integrals, both oriented from left to right, are equal. On the lower edge, [F9] and [F11] give $∣ e^{- i n (θ - i L)} ∣ = e^{- n L} = ρ^{- n}$ , hence $∣ c_{n} ∣ \leq 2 M ρ^{- n}$ . Also $∣ c_{0} ∣ \leq M$ by the defining integral and the bound $∣ f ∣ \leq M$ .

3.1F2F4F6step 2.1algebra

The coefficient bounds in step 2.1 make $\sum_{n \geq 0} c_{n} T_{n}$ uniformly convergent, since [F2] and [F6] give $∣ T_{n} (x) ∣ \leq 1$ . Uniform convergence permits termwise integration against each bounded $T_{k}$ ; orthogonality [F4] shows that this series and $f$ have the same coefficients as defined in step 2.1. Their continuous difference therefore pairs to zero with every Chebyshev mode, so [F4] gives $f = \sum_{n \geq 0} c_{n} T_{n}$ on $[- 1, 1]$ .

4.1F2F3F6F7step 3.1algebra

By [F3], every mode $T_{n}$ with $n > N$ interpolates to a Chebyshev mode. Each such mode has sup norm at most one by [F2] and [F6]. Thus the series tail and its interpolant each have sup norm at most $\sum_{n > N} ∣ c_{n} ∣$ , and $∥ f - I_{N}^{L} f ∥_{\infty} \leq 2 \sum_{n > N} ∣ c_{n} ∣ \leq 4 M \sum_{n > N} ρ^{- n} = \frac{4 M}{ρ - 1} ρ^{- N},$ where [F7] evaluates the geometric tail.

5.1F5step 4.1algebra∎

Apply [F5] to the uniform error in step 4.1. This yields $∥ f - I_{N}^{L} f ∥_{C} \leq 4 \sqrt{π} M (ρ - 1)^{- 1} ρ^{- N}$ , as claimed.

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Composite-Simpson assembly of the Chebyshev weighted mass matrix

Definition

Input a polynomial degree $N \geq 0$ and an even integer $M \geq 2$ specifying the number of uniform subintervals of $[0, π]$ . For $0 \leq i, j \leq N$ , form

g_{i j} (θ) = T_{i} (\cos θ) T_{j} (\cos θ)

and return

M_{i j}^{S} : = Q_{S} (g_{i j}; 0, π, M),

where $Q_{S}$ is the composite Simpson procedure from Composite Simpson rule on an even uniform grid. By the Chebyshev multiple-angle identity, $g_{i j} (θ) = \cos (i θ) \cos (j θ)$ , so this is the composite-Simpson approximation of the corresponding arcsine-weighted mass entry. The procedure uses exact real arithmetic and asserts no quadrature or floating-point accuracy beyond a separately proved bound.

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Composite Simpson error bound for Chebyshev weighted mass entries

Statement

Let $N \geq 0$ , let $M \geq 2$ be even, put $h = π / M$ , and let $G_{i j} = ⟨ T_{i}, T_{j} ⟩_{C}$ be the exact weighted mass entry from Arcsine-weighted inner product on the Chebyshev interval. The approximation $M_{i j}^{S}$ from Composite-Simpson assembly of the Chebyshev weighted mass matrix satisfies

∣ G_{i j} - M_{i j}^{S} ∣ \leq \frac{π h^{4}}{180} (i + j)^{4}, 0 \leq i, j \leq N .

Facts & Assumptions

Given: Integers $N \geq 0$ , even $M \geq 2$ , $h = π / M$ , and indices $0 \leq i, j \leq N$ .

[F1]

Arcsine-weighted inner product on the Chebyshev interval: $⟨ T_{i}, T_{j} ⟩_{C} = \int_{0}^{π} T_{i} (\cos θ) T_{j} (\cos θ) d θ$ .

[F2]

Composite-Simpson assembly of the Chebyshev weighted mass matrix: $M_{i j}^{S}$ is the composite Simpson value on $[0, π]$ for $g_{i j} (θ) = T_{i} (\cos θ) T_{j} (\cos θ)$ with step $h$ .

[F3]

Composite Simpson error bound for a four-times continuously differentiable function: if $g \in C^{4} ([0, π])$ , then $∣ \int_{0}^{π} g - Q_{S} (g; 0, π, M) ∣ \leq π h^{4} \max ∣ g^{(4)} ∣ / 180$ .

[F4]

$T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ and The addition formulas for sine and cosine: $T_{k} (\cos θ) = \cos (k θ)$ and $2 \cos u \cos v = \cos (u + v) + \cos (u - v)$ .

[F5]

The derivatives of sine and cosine are cosine and minus sine and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ give the derivatives of each $\cos (k θ)$ ; Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ distributes these rules over the finite sum in step 1.1. The resulting derivatives are finite linear combinations of sine and cosine functions, which are continuous by differentiability and A function differentiable at $c$ is continuous at $c$ ; Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function preserves continuity under those sums. Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ is the $C^{4}$ convention.

Proof

1.1givenF4F5algebra

By [F4], $g_{i j} (θ) = \frac{1}{2} {\cos ((i + j) θ) + \cos ((i - j) θ)} .$ By [F5] this finite trigonometric expression is $C^{4}$ on $[0, π]$ , and its fourth derivative is $g_{i j}^{(4)} (θ) = \frac{1}{2} {(i + j)^{4} \cos ((i + j) θ) + (i - j)^{4} \cos ((i - j) θ)} .$ Therefore $\max_{[0, π]} ∣ g_{i j}^{(4)} ∣ \leq \frac{1}{2} ((i + j)^{4} + ∣ i - j ∣^{4}) \leq (i + j)^{4} .$ This also covers $i = j = 0$ , when the fourth derivative is zero.

2.1F1F2F3step 1.1∎

Apply [F3] to $g_{i j}$ and use [F1]–[F2] to identify the exact integral and the Simpson approximation. The derivative bound in step 1.1 gives the claimed entrywise error.

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Weighted orthogonal-polynomial Galerkin spaces and spectral scale

Definition

Let $μ$ be a finite positive Borel measure on a compact interval $[a, b]$ with infinite relative support. For each integer $N \geq 0$ , the moment data $(a, b, m = N + 1, r = 2 N + 1, μ)$ satisfy the hypotheses of Positive-measure Gaussian quadrature moment data. Let $p_{0}, \dots, p_{N}$ be the monic orthogonal polynomials supplied by Monic orthogonal polynomials exist uniquely through the quadrature order, and write

h_{n} = \int_{[a, b]} p_{n} (x)^{2} d μ (x) > 0.

The finite-dimensional weighted Galerkin space is

V_{N} (μ) = span {p_{0}, \dots, p_{N}} \subset L^{2} (μ) .

Here $L^{2} (μ)$ means measurable functions with finite integral of the squared absolute value, modulo equality $μ$ -almost everywhere.

For a coefficient sequence $(c_{n})_{n \geq 0}$ with $\sum_{n \geq 0} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} h_{n} < \infty$ , define its weighted spectral Sobolev norm by ${∥ \sum_{n \geq 0} c_{n} p_{n} ∥}_{H_{μ}^{s}}^{2} = \sum_{n \geq 0} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} h_{n}, s \geq 0.$ The real powers are those of Real powers for positive bases, with the zero-base positive-exponent convention; since every base $1 + n^{2}$ is positive, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ makes every weight positive. The associated orthogonal Galerkin projection is $Π_{N} (\sum_{n \geq 0} c_{n} p_{n}) = \sum_{n = 0}^{N} c_{n} p_{n}$ . This coefficient-defined scale is specific to the weight $μ$ and the selected orthogonal-polynomial family; no equivalence with a derivative-based Sobolev norm is asserted here.

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The weighted orthogonal-polynomial Galerkin projection is well-defined

Statement

Under the hypotheses of Weighted orthogonal-polynomial Galerkin spaces and spectral scale, for every real measurable $u$ such that $\int ∣ u p_{j} ∣ d μ < \infty$ for $0 \leq j \leq N$ , the formula

Π_{N} u = \sum_{j = 0}^{N} \frac{\int u (x) p_{j} (x) d μ (x)}{h_{j}} p_{j}

is finite and defines the unique element of $V_{N} (μ)$ whose residual is orthogonal to every $p_{i}$ , $0 \leq i \leq N$ .

Facts & Assumptions

Given: A finite positive measure $μ$ with the support condition in Weighted orthogonal-polynomial Galerkin spaces and spectral scale, an integer $N \geq 0$ , and a real measurable $u$ with $\int ∣ u p_{j} ∣ d μ < \infty$ for $0 \leq j \leq N$ .

[F1]

Weighted orthogonal-polynomial Galerkin spaces and spectral scale: the basis polynomials $p_{0}, \dots, p_{N}$ are monic orthogonal polynomials and $h_{j} = \int p_{j}^{2} d μ > 0$ .

[F2]

The polynomial moment pairing is a real inner product: on each finite polynomial space allowed by the moment data, the moment pairing is a positive-definite real inner product.

[F3]

The given integrability hypotheses make each numerator $\int u p_{j} d μ$ a finite real number.

[F4]

The Lebesgue integral is linear on $L^{1} (μ)$ : finite sums and scalar multiples of integrable functions are integrable and may be integrated term by term.

Proof

1.1givenF1F2F3algebra

Each numerator is finite by [F3], and each denominator $h_{j}$ is finite and strictly positive by [F1]–[F2]. Thus every coefficient in the displayed finite sum is a defined real number.

2.1F1F4step 1.1algebra

The integrability hypotheses and [F1] make every term in the finite sum integrable, and [F4] distributes its integral. Orthogonality of the $p_{j}$ then gives, for each $i \leq N$ , $\int (u - Π_{N} u) p_{i} d μ = \int u p_{i} d μ - \sum_{j = 0}^{N} \frac{\int u p_{j} d μ}{h_{j}} \int p_{j} p_{i} d μ = 0,$ because all terms with $j \neq i$ vanish and the $j = i$ term equals $\int u p_{i} d μ$ . Thus $Π_{N} u \in V_{N} (μ)$ has the required residual orthogonality.

3.1F1F2step 2.1algebra∎

If $v \in V_{N} (μ)$ has the same residual property, then $w = v - Π_{N} u$ lies in $V_{N} (μ)$ and is orthogonal to every basis vector, hence to itself. Positive definiteness in [F2] gives $\int w^{2} d μ = 0$ , so $w = 0$ in $L^{2} (μ)$ and $v = Π_{N} u$ . This proves uniqueness.

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Assemble a weighted Galerkin mass matrix from Gaussian nodes

Definition

Input a degree $N \geq 0$ , the monic orthogonal basis $p_{0}, \dots, p_{N}$ and the $(N + 1)$ -node Gaussian rule for the same positive measure $μ$ from The m-node Gaussian quadrature rule for a positive measure. With Gaussian nodes $ξ_{1}, \dots, ξ_{N + 1}$ and weights $w_{1}, \dots, w_{N + 1}$ , output the matrix

M_{i j}^{G} : = \sum_{r = 1}^{N + 1} w_{r} p_{i} (ξ_{r}) p_{j} (ξ_{r}), 0 \leq i, j \leq N .

The inputs are admissible only when the moment data have order at least $2 N + 1$ and the rule nodes, weights, and basis all use the same measure. The algorithm performs the displayed finite exact-arithmetic sums; no floating-point conditioning claim is made.

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Gaussian quadrature exactly assembles the orthogonal Galerkin mass matrix

Statement

Let $N \geq 0$ , take moment data of order $2 N + 1$ for a finite positive measure $μ$ with support containing at least $N + 2$ points, and use its $(N + 1)$ -node Gaussian rule and orthogonal polynomials $p_{0}, \dots, p_{N}$ . The matrix from Assemble a weighted Galerkin mass matrix from Gaussian nodes has entries $M_{i j}^{G} = \int p_{i} (x) p_{j} (x) d μ (x) = {\begin{cases} 0, & i \neq j, \\ h_{i}, & i = j . \end{cases}$

Facts & Assumptions

Given: $N \geq 0$ , moment data of order $2 N + 1$ , its $(N + 1)$ -node Gaussian rule, and the corresponding orthogonal polynomials.

[F1]

Gaussian quadrature is exact through degree 2m−1 and has positive weights: for an $m$ -node rule from moment data of order $2 m - 1$ , every polynomial of degree at most $2 m - 1$ integrates exactly.

[F2]

Positive-measure Gaussian quadrature moment data and Monic orthogonal polynomials exist uniquely through the quadrature order with $m = N + 1$ : $p_{i}$ is the monic degree- $i$ polynomial orthogonal to every lower-degree polynomial for $0 \leq i \leq N$ . Write $h_{i} : = \int p_{i}^{2} d μ$ ; these integrals are finite since $2 i \leq 2 N < 2 N + 1$ , within the supplied moment order. Only the stated $N + 2$ support points are needed, not infinite support.

[F3]

Assemble a weighted Galerkin mass matrix from Gaussian nodes: $M_{i j}^{G}$ is the Gaussian sum of $p_{i} p_{j}$ over the $N + 1$ nodes.

Proof

1.1givenF1F2F3algebra

By [F2], the product $p_{i} p_{j}$ has degree $i + j \leq 2 N < 2 N + 1 = 2 (N + 1) - 1$ . Thus [F1] applies with $m = N + 1$ and gives $M_{i j}^{G} = \sum_{r = 1}^{N + 1} w_{r} p_{i} (ξ_{r}) p_{j} (ξ_{r}) = \int p_{i} (x) p_{j} (x) d μ (x)$ by [F3].

2.1F2step 1.1algebra∎

If $i \neq j$ , the higher-degree polynomial is orthogonal to the lower-degree one by [F2], so the integral in step 1.1 is zero. If $i = j$ , it is $h_{i}$ by [F2]. This is the displayed diagonal matrix identity.

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Legendre polynomials by Rodrigues' formula and their unweighted pairing

Definition

For each integer $n \geq 0$ , define the Legendre polynomial

P_{n} (x) : = \frac{1}{2^{n} n!} \frac{d^{n}}{d x^{n}} (x^{2} - 1)^{n},

with $n!$ as in The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ and repeated derivatives as in Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ . Define the unweighted pairing of real polynomials by

⟨ p, q ⟩_{L} : = \int_{- 1}^{1} p (x) q (x) d x, ∥ p ∥_{L}^{2} : = ⟨ p, p ⟩_{L} .

The integral is the ordinary integral of a continuous function; polynomial products are continuous and hence integrable by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. The normalization and all polynomial identities needed for the orthogonal Galerkin scale are proved in Legendre polynomials have explicit norms and weighted derivative orthogonality.

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Legendre polynomials have explicit norms and weighted derivative orthogonality

Statement

For $m, n \geq 0$ , the polynomials of Legendre polynomials by Rodrigues' formula and their unweighted pairing satisfy $\int_{- 1}^{1} P_{n} (x) P_{m} (x) d x = \frac{2}{2 n + 1} δ_{m n}, \int_{- 1}^{1} (1 - x^{2}) P_{n}^{'} (x) P_{m}^{'} (x) d x = \frac{2 n (n + 1)}{2 n + 1} δ_{m n},$ where $δ_{m n} = 1$ if $m = n$ and $0$ otherwise. Each $P_{n}$ has degree $n$ and nonzero leading coefficient, so $P_{0}, \dots, P_{N}$ form a basis of polynomials of degree at most $N$ .

Facts & Assumptions

Given: Nonnegative integers $m, n$ and the Rodrigues polynomials $P_{k}$ .

[F1]

Legendre polynomials by Rodrigues' formula and their unweighted pairing: $P_{n} = (2^{n} n!)^{- 1} D^{n} (x^{2} - 1)^{n}$ and $⟨ p, q ⟩_{L} = \int_{- 1}^{1} p q$ .

[F2]

If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ : for differentiable factors with integrable derivatives on a closed interval, $\int u v^{'} = [u v] - \int u^{'} v$ .

[F3]

For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term: $D (x^{k}) = k x^{k - 1}$ for positive integer $k$ , and constants differentiate to zero; repeated application gives the derivatives of monomials.

[F4]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : polynomial sums, products, and scalar multiples obey the usual derivative rules.

Proof

1.1F1F2F3F4giveninduction

Write $(x^{2} - 1)^{n} = (x - 1)^{n} (x + 1)^{n}$ . For every $j < n$ , its $j$ th derivative vanishes at both endpoints: by the product rule, every term differentiates each factor at most $j < n$ times and retains a positive power of the factor that vanishes at that endpoint. Thus $n$ integrations by parts in [F2] give, for every polynomial $q$ of degree less than $n$ , $\int_{- 1}^{1} P_{n} q = \frac{(- 1)^{n}}{2^{n} n!} \int_{- 1}^{1} (x^{2} - 1)^{n} q^{(n)} = 0,$ because $q^{(n)} = 0$ by [F3]. Hence $P_{n}$ is orthogonal to every lower-degree polynomial.

2.1F1F3step 1.1algebra

The leading term of $(x^{2} - 1)^{n}$ is $x^{2 n}$ ; differentiating it $n$ times gives $(2 n)! / n!$ times $x^{n}$ by [F3]. All other terms have lower degree after differentiation. Therefore $P_{n}$ has degree $n$ and leading coefficient $κ_{n} = (2 n)! / (2^{n} (n!)^{2}) > 0$ . The distinct degrees imply the family $P_{0}, \dots, P_{N}$ is linearly independent and spans every polynomial of degree at most $N$ , by successive elimination of leading terms.

3.1F1F2F3step 1.1step 2.1algebra

Since $P_{n} - κ_{n} x^{n}$ has degree less than $n$ , step 1.1 gives $\int_{- 1}^{1} P_{n}^{2} = κ_{n} \int_{- 1}^{1} P_{n} x^{n} .$ Apply [F2] $n$ times to the Rodrigues formula against $x^{n}$ . The endpoint terms vanish as in step 1.1 and $(x^{n})^{(n)} = n!$ , so $\int_{- 1}^{1} P_{n} x^{n} = \frac{1}{2^{n}} \int_{- 1}^{1} (1 - x^{2})^{n} d x .$

4.1F2F3F4step 1.1step 2.1step 3.1inductionalgebra

Put $J_{n} = \int_{- 1}^{1} (1 - x^{2})^{n} d x$ . Then $J_{0} = 2$ . For $n \geq 1$ , the endpoint values of $x (1 - x^{2})^{n}$ are zero, so integrating its derivative gives $0 = J_{n} - 2 n (J_{n - 1} - J_{n}), J_{n} = \frac{2 n}{2 n + 1} J_{n - 1} .$ Induction yields $J_{n} = 2^{2 n + 1} (n!)^{2} / (2 n + 1)!$ . Substituting this and $κ_{n}$ from step 2.1 into step 3.1 gives $\int P_{n}^{2} = 2 / (2 n + 1)$ . Together with step 1.1 this proves the first orthogonality formula, including $n = 0$ .

5.1F1F2F3F4step 1.1step 2.1step 4.1algebra

When $n = 0$ , $P_{0} = 1$ and $Q_{0} = - ((1 - x^{2}) P_{0}^{'})^{'} = 0$ , so $Q_{0} = 0 \cdot P_{0}$ . Now take $n \geq 1$ and define $Q_{n} = - ((1 - x^{2}) P_{n}^{'})^{'}$ . Its leading term is $n (n + 1) κ_{n} x^{n}$ , so $Q_{n} - n (n + 1) P_{n}$ has degree less than $n$ . For any polynomial $q$ of degree less than $n$ , two integrations by parts give $\int_{- 1}^{1} Q_{n} q = \int_{- 1}^{1} (1 - x^{2}) P_{n}^{'} q^{'} = - \int_{- 1}^{1} P_{n} ((1 - x^{2}) q^{'})^{'} = 0,$ where both boundary terms vanish because of the factor $1 - x^{2}$ , and the last polynomial paired with $P_{n}$ has degree less than $n$ , so step 1.1 applies. Therefore $Q_{n} - n (n + 1) P_{n}$ is a lower-degree polynomial orthogonal to every lower-degree polynomial. Expand it in the basis from step 2.1; the first formula and the positive norms in step 4.1 force every coefficient to be zero. Thus $Q_{n} = n (n + 1) P_{n}$ .

6.1F1F2step 1.1step 4.1step 5.1algebra∎

One integration by parts and step 5.1 give $\int_{- 1}^{1} (1 - x^{2}) P_{n}^{'} P_{m}^{'} = \int_{- 1}^{1} Q_{n} P_{m} = n (n + 1) \int_{- 1}^{1} P_{n} P_{m} .$ The first formula, already proved in step 4.1 and step 1.1, yields the displayed weighted derivative identity for every $m, n \geq 0$ .

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Legendre weighted spectral-Sobolev scale and projection

Definition

For $s \geq 0$ , define $H_{L}^{s}$ as the completion of finite Legendre sums $p = \sum_{n = 0}^{M} c_{n} P_{n}$ in the coefficient norm

∥ p ∥_{H_{L}^{s}}^{2} = \sum_{n = 0}^{M} (1 + n (n + 1))^{s} \frac{2}{2 n + 1} ∣ c_{n} ∣^{2} .

The coefficients are unique because the polynomials have distinct degrees by Legendre polynomials have explicit norms and weighted derivative orthogonality. At $s = 0$ this norm is exactly the unweighted $L^{2} ([- 1, 1])$ norm on finite sums, by the same lemma; its completion is denoted $L_{L}^{2}$ . Define the degree- $N$ orthogonal projection on finite sums by

Π_{N} u = \sum_{n = 0}^{N} c_{n} P_{n} .

It extends to the coefficient completion by truncation. This explicitly coefficient-defined scale uses the Legendre weight $1$ on $[- 1, 1]$ ; no equivalence with an unweighted derivative-based Sobolev norm is asserted here.

For fixed $s \geq 0$ , $x \mapsto x^{s}$ is nondecreasing on $(0, \infty)$ : its derivative is $s x^{s - 1} \geq 0$ by Continuity and derivatives of positive-base real powers and real powers are positive by Real powers for positive bases, with the zero-base positive-exponent convention and The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , so the mean-value theorem The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ applies. Thus the spectral factors $(1 + n (n + 1))^{s}$ are positive and nondecreasing with $n$ ; the full weights $w_{n}$ below need not be nondecreasing.

For $w_{n} = (1 + n (n + 1))^{s} \frac{2}{2 n + 1} > 0$ , the coefficient map $(c_{n}) \mapsto (\sqrt{w_{n}} c_{n})$ is an isometry onto real $ℓ^{2}$ . Thus the coefficient model of $H_{L}^{s}$ is exactly ${(c_{n})_{n \geq 0} : \sum_{n \geq 0} (1 + n (n + 1))^{s} \frac{2}{2 n + 1} ∣ c_{n} ∣^{2} < \infty} .$ Finite-support sequences are dense by truncation, and completeness of $ℓ^{2}$ from The real square-summable sequence space is complete makes this the completion.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

Legendre weighted projection has its spectral-Sobolev rate

Statement

Let $s \geq 0$ , $N \geq 0$ , and $u = \sum_{n \geq 0} c_{n} P_{n} \in H_{L}^{s}$ from Legendre weighted spectral-Sobolev scale and projection. Then

∥ u - Π_{N} u ∥_{L_{L}^{2}} \leq (1 + (N + 1) (N + 2))^{- s / 2} ∥ u ∥_{H_{L}^{s}} .

Facts & Assumptions

Given: A real $s \geq 0$ , an integer $N \geq 0$ , and a coefficient sequence in $H_{L}^{s}$ .

[F1]

Legendre weighted spectral-Sobolev scale and projection: $Π_{N} u = \sum_{n = 0}^{N} c_{n} P_{n}$ and the $H_{L}^{s}$ coefficient weight is $(1 + n (n + 1))^{s} 2 / (2 n + 1)$ ; its spectral factor is nondecreasing in $n$ for $s \geq 0$ .

[F2]

Legendre polynomials have explicit norms and weighted derivative orthogonality: $\int_{- 1}^{1} P_{n} P_{m} = 2 δ_{m n} / (2 n + 1)$ .

Proof

1.1F1F2algebra

For a finite Legendre sum, [F1] gives $u - Π_{N} u = \sum_{n > N} c_{n} P_{n}$ . Orthogonality [F2] therefore gives $∥ u - Π_{N} u ∥_{L_{L}^{2}}^{2} = \sum_{n > N} \frac{2}{2 n + 1} ∣ c_{n} ∣^{2} .$ For every $n > N$ , $1 + n (n + 1) \geq 1 + (N + 1) (N + 2)$ ; multiplying each term by the inverse weight gives the stated squared bound for finite sums.

2.1F1step 1.1algebra∎

Finite sums are dense in the completion defining $H_{L}^{s}$ . The truncation $Π_{N}$ is contractive in both the coefficient norm and the $s = 0$ norm, so the finite-sum inequality passes to the limit along any Cauchy sequence of finite sums. Taking square roots proves the claimed estimate for every $u \in H_{L}^{s}$ .

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

Chebyshev collocation spectral-radius growth is setup-specific conditioning context

Scope note

For the cited Chebyshev–Gauss–Lobatto second-derivative collocation matrix with the Dirichlet boundary rows treated as in the source, Shen and Tang report spectral-radius growth of order $N^{4}$ . This is a source-specific conditioning observation for that matrix and boundary treatment. It is not a stability estimate for every BVP discretization, a convergence theorem for the algorithm Boundary-eliminated Chebyshev collocation for a two-point Dirichlet BVP, or evidence about any continuum PDE regularity question. This carrier records the source's reported scope and does not claim to prove the spectral-radius asymptotic.

5 · Examples, counterexamples and false statements

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