Best uniform polynomial error and Chebyshev interpolation operators

Let $C[-1,1]$ be the real continuous functions on $[-1,1]$, with $\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{x\in [-1,1]}\mathrm{\mid }f(x)\mathrm{\mid }$. Let ${\mathcal{P}}_n$ be the restrictions to $[-1,1]$ of real polynomials of degree at most $n$, including the zero polynomial. Interval notation is that of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length. Continuity and polynomial degree have the conventions of Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point and Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials.

For each $n\in \mathbb{N}$ (with naturals in real expressions interpreted by their canonical real images), the first-kind Chebyshev root angles and nodes are ${\theta }_j^{R,n}=\frac{(2j+1)\pi }{2n+2},x_j^{R,n}=\cos {\theta }_j^{R,n},0\le j\le n\mathrm{.}$ They are the roots of $T_{n+1}$ by $T_n(\cos \theta )=\cos (n\theta )$ and $U_n(\cos \theta )\sin \theta =\sin ((n+1)\theta )$ for every $n\in \mathbb{N}$. For $n\ge 1$, the Chebyshev–Lobatto angles and nodes are ${\theta }_j^{L,n}=\frac{j\pi }{n},x_j^{L,n}=\cos {\theta }_j^{L,n},0\le j\le n\mathrm{.}$ They include both endpoints and are the extrema of $T_n$ by the same identity. At $n=0$, define the Lobatto node to be $x_0^{L,0}=0$. The cosine is strictly decreasing on $[0,\pi ]$, so each listed family has distinct nodes; the root family at $n=0$ also consists of $0$.

For either family $\sigma \in \{R,L\}$, write its nodes as $x_0,\dots ,x_n$ and define ${\mathrm{\ell }}_{j,n}^{\sigma }(x)={\prod }_{\begin{array}{c}0\le k\le n\\ k\mathrm{\ne }j\end{array}}\frac{x-x_k}{x_j-x_k},I_n^{\sigma }f={\sum }_{j=0}^{n}f(x_j){\mathrm{\ell }}_{j,n}^{\sigma }\mathrm{.}$ These are the cardinal formula and interpolation sum of Cardinal-polynomial interpolation specialized to this node family. The displayed cosine arrays decrease, whereas that supplier orders nodes increasingly: use $z_i=x_{n-i}$ and data $y_i=f(z_i)$. Its cardinal function with index $n-j$ is exactly the displayed ${\mathrm{\ell }}_{j,n}^{\sigma }$, by reindexing the finite product, and its interpolation sum is $I_n^{\sigma }$. The empty product at $n=0$ is $1$, so the same identification holds for the single node. Distinctness makes every denominator nonzero; Cardinal interpolation functions are polynomials of the stated degrees supplies the degree and cardinal values. The finite sum is linear in the data and maps $C[-1,1]$ into ${\mathcal{P}}_n$. Its operator norm is ${\mathrm{\Lambda }}_n^{\sigma }:=\sup \{\mathrm{\parallel }{I}_n^{\sigma }g{\mathrm{\parallel }}_{\mathrm{\infty }}:g\in C[-1,1],\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\}\mathrm{.}$ If ${\mathrm{\ell }}_{j,n}^{\sigma }(x)={\sum }_{k=0}^n{a}_{jk}{x}^k$, then $\mathrm{\mid }{\mathrm{\ell }}_{j,n}^{\sigma }(x)\mathrm{\mid }\le {\sum }_{k=0}^n\mathrm{\mid }{a}_{jk}\mathrm{\mid }$ for $\mathrm{\mid }x\mathrm{\mid }\le 1$; hence $\mathrm{\parallel }{I}_n^{\sigma }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\sum }_{j,k}\mathrm{\mid }{a}_{jk}\mathrm{\mid }<\mathrm{\infty }$ whenever $\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$. Define the best degree-$n$ uniform approximation error by $E_n(f):={\inf }_{p\in {\mathcal{P}}_n}\mathrm{\parallel }f-p{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$

These quantities are real and well-defined. Polynomial functions are continuous, and sums and scalar multiples of continuous functions remain continuous 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. The interval $[-1,1]\subset \mathbb{R}$ is nonempty, closed, and bounded, so For a nonempty subset of ${\mathbb{R}}^n$ with $n\ge 1$, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent implies that for every such continuous function $h$ there are finite minimum and maximum values $m_h,M_h$. Since $m_h\le h(x)\le M_h$, one has $\mathrm{\mid }h(x)\mathrm{\mid }\le \max (\mathrm{\mid }{m}_h\mathrm{\mid },\mathrm{\mid }{M}_h\mathrm{\mid })$, so each supremum norm above is finite. The set defining ${\mathrm{\Lambda }}_n^{\sigma }$ contains the zero function and is bounded above by the displayed finite coefficient sum. The set defining $E_n(f)$ contains the zero polynomial and is bounded below by zero. Completeness of $\mathbb{R}$ (Complete ordered field (least-upper-bound property)) therefore gives the stated finite supremum and infimum. Also, $n+1$ has positive real image by The canonical natural $\iota (n)=n\cdot 1_F$ of a field and Canonical naturals are positive and strictly increasing, and $n$ has positive real image when $n\ge 1$; thus the root-angle denominator and the Lobatto denominator in its stated range are nonzero.

The displayed root-family nodes are all the real roots of $T_{n+1}$. The degree statement Degrees and leading coefficients of the Chebyshev polynomials gives $\mathrm{deg}{T}_{n+1}=n+1$, and A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots excludes any additional real root beyond the $n+1$ distinct nodes above.