Approximations and expansions Rates of approximation: direct and inverse theorems Theorem

Favard's theorem on trigonometric approximation

Builds on Favard constant Trigonometric polynomial Periodic function Error of best approximation

Statement

Let $r \ge 1$ and $n \ge 1$ be integers, and let $K_r$ be the Favard constant. Let $f \colon \R \to \R$ be $2\pi$ -periodic, with $f^{(r-1)}$ absolutely continuous and $\abs{f^{(r)}} \le 1$ almost everywhere. Then there is a real trigonometric polynomial $T$ of degree at most $n - 1$ such that $\max_{x \in \R} \abs{f(x) - T(x)} \le \frac{K_r}{n^r}.$ The constant $K_r$ is sharp. For each $n$ , some $f$ of this class has distance exactly $K_r / n^r$ from the set of all such $T$ .

Remarks

The extremal function is the $r$ -fold periodic integral of the square wave $\sgn \sin(nx)$ , normalised to have mean $0$ . Its error of best approximation equals $K_r / n^r$ , which explains the alternating series in the definition of $K_r$ : they come from the Fourier series of the square wave. The upper bound comes from explicit linear approximation methods, so it is attained by a linear operator.

Example: $r = 1$ , $n = 1$ . A periodic function with $\abs{f'} \le 1$ has $\max f - \min f \le \pi$ , so the best constant has error at most $\pi/2 = K_1$ . The triangle wave equal to $\abs{x}$ on $[-\pi, \pi]$ reaches this bound.

Going deeper

For a student

Jackson's inequality gives the rate $n^{-r}$ with some constant; Favard's theorem gives the exact worst-case constant. It is a model result for the theory of widths and optimal recovery, where one asks for the smallest error over a whole class. The same constants appear in sharp inequalities between a function and its derivatives, such as the Kolmogorov inequalities.

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Error of best approximation concept
Favard constant concept
Periodic function concept
Trigonometric polynomial concept
Favard's theorem on trigonometric approximation statement

Tag	`JYX6N8A2P6`
URI	`m2://JYX6N8A2P6`
Kind	statement, theorem
Writing	W0 stub
MSC	41A44, 42A10, 41A17
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About	Favard constant
Uses	Favard constant, Trigonometric polynomial, Periodic function, Error of best approximation