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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
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.)