# Derivatives of a Trigonometric Polynomial of Best Approximation

• Gen-Ichirô Sunouchi
Part of the ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique book series (ISNM, volume 10)

## Abstract

Throughout the paper, we suppose that the function f(x) belongs to the class L p (1 ≦ p < ∞) or C (f(0) = f (2π)) over (0, 2π) and is extended periodically with period 2π. For the sake of simplicity of notation, we may write L instead of C. Let T n (f) be a trigonometric polynomial of best approximation of order n for f(x) with respect to the corresponding norm, that is
$${\left\| {f - {T_n}(f)} \right\|_p} = E_n^{(p)}(f)$$
where E n (p) (f) is the best approximation of f(x) of order n. Moreover for the function which belongs to the class L p (1 ≦ p < ∞), by the integral modulus of smoothness in L p of order k ≧ 1 we mean the expression
$$\omega _n^{(p)}(f;t) = \mathop {\sup }\limits_{0 < \left| h \right| \leqq t} {\left[ {\int\limits_0^{2\pi } {{{\left| {\Delta _h^kf(x)} \right|}^p}dx} } \right]^{1/p}}$$
, where
$$$$\Delta _h^kf(x) = \sum\limits_{v = 0}^k {{{( - 1)}^{k - v}}} \left( \begin{array}{l} k \\ 0 \\ \end{array} \right)f(x + vh)$$$$
.

## Keywords

Fourier Series Fourier Coefficient Lipschitz Condition Trigonometric Polynomial Absolute Convergence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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