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Lebesgue constants and approximation of functions by Fourier and Fejér sums. (English. Russian original) Zbl 0587.42008
Sov. Math., Dokl. 30, 182-185 (1984); translation from Dokl. Akad. Nauk SSSR 277, 1033-1035 (1984).
Given a bounded region D in $${\mathbb{R}}^ m$$, the partial sum $$S_ R(f,D)$$ of the Fourier series $$\sum_{\nu \in {\mathbb{Z}}^ m}c_{\nu}e^{i\nu x}$$ for a function $$f\in L(T^ m)$$ is defined to be the trigonometric polynomial $$S_ R(f,x,D)=\sum_{\nu \in RD\cap {\mathbb{Z}}^ m}c_{\nu}e^{i\nu x}$$ where $$RD=\{x\in {\mathbb{R}}^ m:x/R\in D\}$$ as $$R>0$$ and $$0D=\{0\}$$. Then the map $$f\mapsto S_ R(f,D)$$ gives an operator from $$L^ p$$ to $$L^ p$$ (1$$\leq p\leq \infty$$ and $$L^{\infty}$$ means C). The norm of this operator is called the Lebesgue constant and denoted by $${\mathcal L}_ R(p,m,D)$$ which is an important quantity in the theory of Fourier series.
In this note the author states the results giving the upper estimate of $${\mathcal L}_ R(p,m,\Pi)$$ with $$\Pi$$ any closed polyhedron and the lower estimate of $${\mathcal L}_ R(p,m,D)$$ with D any convex body. When D is a convex centrally symmetric body, the de la Vallée Poussin sum corresponding with $$S_ R(f,D)$$ is defined in a similar way and denoted by $$\sigma_{R,s}(f,D)$$. For a function $$\epsilon =\epsilon_ r$$ such that $$\epsilon_ r\downarrow 0$$ (r$$\uparrow \infty)$$ the class $$X^ m_{\epsilon}$$ is defined by $$X^ m_{\epsilon}=\{f\in X^ m:E_ r(f)x^ m\leq \epsilon_ r$$, $$r>0\}$$ where $$X^ m$$ denotes $$L^ p$$ $$(1\leq p<\infty)$$ or C and $$E_ r(f)_{X^ m}$$ denotes the best rth- order approximation of f in the metric of $$X^ m$$. The author gives some results related to the interesting problem (posed by S. B. Stechkin) of estimating the quantity $$\sup \{\| f-\sigma_{R,s}(f,D)\|_{X^ m}:f\in X^ m_{\epsilon}\}$$. For $$1<p<\infty$$, a one-dimensional $$L^ p$$ result for conjugate approximation is given.
Reviewer: K.Wang
##### MSC:
 42B05 Fourier series and coefficients in several variables 42A10 Trigonometric approximation 46B20 Geometry and structure of normed linear spaces 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
Lebesgue constant; convex centrally symmetric body