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Chebyshev-root and Lobatto interpolation norms have logarithmic upper bounds

Statement

For $n \geq 1$ , let $Λ_{n}^{R}$ be the operator norm of interpolation at the $n + 1$ roots of $T_{n + 1}$ , and let $Λ_{n}^{L}$ be the operator norm at the $n + 1$ Chebyshev–Lobatto nodes. With the conventions in Best uniform polynomial error and Chebyshev interpolation operators, $Λ_{n}^{R} \leq C \log (n + 1), Λ_{n}^{L} \leq C \log (n + 1), C : = 4 + \frac{π + 4}{\log 2} .$ Here $\log$ is the natural logarithm of The natural logarithm as the inverse of the exponential function. For $n = 0$ , both one-node operators have norm $1$ ; the logarithmic estimate is stated for $n \geq 1$ .

Facts & Assumptions

Given: An integer $n \geq 1$ , $θ \in [0, π]$ , and either angular node family.

[F1]

Best uniform polynomial error and Chebyshev interpolation operators defines $I_{n}^{σ}$ , $Λ_{n}^{σ}$ , and the equally spaced root and Lobatto angles.

[F2]

Chebyshev cardinal interpolation reproduces polynomials and its norm is the cardinal sum: $Λ_{n}^{σ} = \sup_{x \in [- 1, 1]} \sum_{j} ∣ ℓ_{j, n}^{σ} (x) ∣$ .

[F3]

Trigonometric formulas for Chebyshev cardinal polynomials: for roots, $∣ ℓ_{j} (\cos θ) ∣ = ∣ \cos ((n + 1) θ) ∣ \sin θ_{j} / ((n + 1) ∣ \cos θ - \cos θ_{j} ∣)$ ; for Lobatto nodes, $∣ ℓ_{j} (\cos θ) ∣ = c_{j} ∣ \sin θ \sin (n θ) ∣ / (n ∣ \cos θ - \cos θ_{j} ∣)$ , with $c_{j} \leq 1$ .

[F4]

Sine chord and Lipschitz bounds: $\sin u \geq 2 u / π$ for $0 \leq u \leq π / 2$ , and $∣ \sin u - \sin v ∣ \leq ∣ u - v ∣$ for all real $u, v$ .

[F5]

The addition formulas for sine and cosine, The subtraction formulas for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, and Quarter-turn values and shifts by pi/2 and pi: the addition/subtraction formulas give $\cos θ - \cos ϕ = - 2 \sin ((θ + ϕ) / 2) \sin ((θ - ϕ) / 2)$ and $\sin (s \pm t) = \sin s \cos t \pm \cos s \sin t$ ; on $[0, π]$ , sine is nonnegative, increasing up to $π / 2$ , and satisfies $\sin (π - s) = \sin s$ , while $∣ \cos u ∣ \leq 1$ for every real $u$ .

[F6]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: for $x > 0$ , $\log x = \int_{1}^{x} d t / t$ .

[F7]

The natural logarithm as the inverse of the exponential function: $\log$ is the natural logarithm; The canonical natural $ι (n) = n \cdot 1_{F}$ of a field and Canonical naturals are positive and strictly increasing: natural indices in real expressions use their canonical images, and positive natural indices have positive real image.

[F8]

Cosine is continuous on $[0, π]$ by The derivatives of sine and cosine are cosine and minus sine and A function differentiable at $c$ is continuous at $c$ . Its endpoint values are $\cos 0 = 1$ and $\cos π = - 1$ by Quarter-turn values and shifts by pi/2 and pi, and $π > 0$ by Pi as twice the smallest positive zero of cosine. The intermediate value theorem Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ therefore shows that every value in $[- 1, 1]$ is $\cos θ$ for some $θ \in [0, π]$ . Conversely, cosine has range $[- 1, 1]$ by Signs, monotonicity intervals, and ranges of sine and cosine, so $\cos ([0, π]) = [- 1, 1]$ .

Proof

1.1givenF1choosealgebra

In either angular grid the spacing is $h = π / (n + 1)$ for roots and $h = π / n$ for Lobatto nodes. Choose a nearest grid angle $θ_{k}$ and put $d_{j} = ∣ θ - θ_{j} ∣$ . Then $d_{k} \leq h / 2$ ; if $∣ j - k ∣ = r \geq 1$ , the triangle inequality gives $d_{j} \geq r h - d_{k} \geq (r - \frac{1}{2}) h$ . At most two indices occur for each $r$ .

2.1givenF4F5step 1.1algebra

For $θ \neq θ_{j}$ , put $s = (θ + θ_{j}) / 2$ and $d = θ - θ_{j}$ . Since the two angles lie in $[0, π]$ , $∣ d ∣ / 2 \leq \min (s, π - s)$ . The addition/subtraction formulas in [F5] give $\sin θ_{j}, \sin θ \leq \sin s + ∣ \sin (d / 2) ∣$ ; the monotonicity and supplementary identity in [F5] then give $∣ \sin (d / 2) ∣ \leq \sin s$ , so each numerator is at most $2 \sin s$ . Those same formulas give $∣ \cos θ - \cos θ_{j} ∣ = 2 \sin s \sin (∣ d ∣ / 2)$ , and [F4] gives $\sin (∣ d ∣ / 2) \geq ∣ d ∣ / π$ . Thus both ratios $\sin θ_{j} / ∣ \cos θ - \cos θ_{j} ∣$ and $\sin θ / ∣ \cos θ - \cos θ_{j} ∣$ are at most $π / d_{j}$ .

3.1givenF3F4F5step 1.1step 2.1algebra

For roots, [F3] and step 2.1 give $∣ ℓ_{j, n}^{R} (\cos θ) ∣ \leq π ∣ \cos ((n + 1) θ) ∣ / ((n + 1) d_{j})$ . At the nearest node $k$ , $\cos ((n + 1) θ_{k}) = 0$ , and the cosine addition formula plus [F4] gives $∣ \cos ((n + 1) θ) ∣ \leq (n + 1) d_{k}$ , so its term is at most $π$ . For other nodes use $∣ \cos ((n + 1) θ) ∣ \leq 1$ and step 1.1; summing over at most two nodes per distance rank yields $\sum_{j} ∣ ℓ_{j, n}^{R} (\cos θ) ∣ \leq π + 2 \sum_{r = 1}^{n} 1 / (r - \frac{1}{2})$ . At a node the cardinal sum is $1$ .

3.2givenF3F4F5step 1.1step 2.1algebra

For Lobatto nodes, [F3] and step 2.1 give $∣ ℓ_{j, n}^{L} (\cos θ) ∣ \leq π c_{j} ∣ \sin (n θ) ∣ / (n d_{j})$ . At the nearest node $k$ , $\sin (n θ_{k}) = 0$ , so [F4] gives $∣ \sin (n θ) ∣ \leq n d_{k}$ and its term is at most $π$ because $c_{k} \leq 1$ . For other nodes use $∣ \sin (n θ) ∣ \leq 1$ and step 1.1; summing over at most two nodes per distance rank yields $\sum_{j} ∣ ℓ_{j, n}^{L} (\cos θ) ∣ \leq π + 2 \sum_{r = 1}^{n} 1 / (r - \frac{1}{2})$ . At a node the cardinal sum is $1$ .

4.1givenF2F6F7F8step 3.1step 3.2algebra∎

Since $1 / (r - \frac{1}{2}) \leq 2 / r$ , the shifted sum is at most $2 H_{n}$ , where $H_{n} = \sum_{r = 1}^{n} 1 / r$ . For $r \geq 2$ , $1 / r \leq \int_{r - 1}^{r} d t / t$ , so [F6] gives $H_{n} \leq 1 + \log n$ . Positivity of the real image of $n$ is [F7], so these logarithms are defined. The integral formula also gives $0 = \log 1 \leq \log n \leq \log (n + 1)$ and $\log 2 \leq \log (n + 1)$ by integrating $1 / t > 0$ over $[1, n]$ , $[n, n + 1]$ , and $[2, n + 1]$ . Since $\log 2 = \int_{1}^{2} d t / t > 0$ and $π > 0$ , we have $C > 0$ and $1 < C \log 2$ . Thus both sums are at most $π + 4 + 4 \log n \leq (4 + (π + 4) / \log 2) \log (n + 1)$ . By [F8], the angular supremum is over exactly $[- 1, 1]$ ; taking it and using [F2] proves both bounds. At $n = 0$ , each one-node operator has norm $1$ directly.

Depends on

Used by

An exponential best-error rate transferred to Chebyshev interpolation Example

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Sources

Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chs. 2–6 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Polynomial interpolation and conditioning (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Problems and conditioning (standard reference, not scraped)