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
$$[tex]\Delta _h^kf(x) = \sum\limits_{v = 0}^k {{{( - 1)}^{k - v}}} \left( \begin{array}{l} k \\ 0 \\ \end{array} \right)f(x + vh)[/tex]$$


Fourier Series Fourier Coefficient Lipschitz Condition Trigonometric Polynomial Absolute Convergence 
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Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • Gen-Ichirô Sunouchi
    • 1
  1. 1.Math. InstituteTôhoku UniversitySendaiJapan

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