Nonsaturated Estimates of the Kotelnikov Formula Error

摘要

The error of approximation by the Kotelnikov sumsUTfx=∑j∈?fjTsincTx?j.T>0,sincz=sinπzπzdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {U}_Tf(x)=sum_{jin mathbb{Z}}fleft(frac{j}{T}right)operatorname{sinc}left( Tx-jright).T>0,operatorname{sinc} z=frac{sin pi z}{pi z} $$end{document}is estimated. Let f ∈ A, i.e.,fx=∫?gyeixydy,g∈L1?,documentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ f(x)=underset{mathbb{R}}{int }g(y){e}^{ixy} dy,gin {L}_1left(mathbb{R}right), $$end{document} and let fA=∫?gdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {leftVert frightVert}_{textbf{A}}=underset{mathbb{R}}{int}leftgright $$end{document} be the Wiener norm of f. Then the sharp inequalityf?UTfA≤2ATπfAdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {leftVert f-{U}_TfrightVert}_{textbf{A}}le 2{A}_{Tpi}{(f)}_{textbf{A}} $$end{document}holds, where Aσ(f)A is the best approximation of f in the Wiener norm by entire functions of exponential type not exceeding . Several non-saturated uniform estimates are also established.