Author: V. Totik 1
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Пусть
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$K_n (f;x) = \mathop \Sigma \limits_{k = 0}^\infty (\int\limits_{{k \mathord{\left/ {\vphantom {k n}} \right. \kern-\nulldelimiterspace} n}}^{{{(k + 1)} \mathord{\left/ {\vphantom {{(k + 1)} n}} \right. \kern-\nulldelimiterspace} n}} {f(u)du} )e^{ - nx} \frac{{(nx)^k }}{{k!}}(x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0,f \in L^p (0;\infty ))$$ \end{document}
— модификация по Кант оровичу известных оп ератов Саса—Миракяна. Доказ ано, что дляfLP(0; ∞), 1<р<∞ и 0<α<1 условия
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {K_n (f;x) - f(x)} \right\|_{L^p (0;\infty )} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } Kn^{ - \alpha } (n = 1,2,...)$$ \end{document}
эквивалентны, где
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {\Delta _{h\sqrt x }^* (f;x)} \right\|_{L^p (h^2 ;\infty )} + h^\alpha \left\| {\Delta _h^1 (f;x)} \right\|_{L^p (0;\infty )} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } Kh^{2\alpha } (h\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0)$$ \end{document}
и
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\Delta _h^1 (f;x) = f(x + h) - f(x)\Delta _h^* (f;x) = f(x + h) - 2f(x) + f(x - h).$$ \end{document}
При α=1 условие (1) эквива лентно тому, чтоf имее т абсолютно непрерывн ую производную такую, что функцияxf″ (x) принадлежит классуLp(0; ∞).

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Analysis Mathematica
Language English
Size B5
Year of
Foundation
1975
Volumes
per Year
1
Issues
per Year
4
Founder Akadémiai Kiadó
Founder's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0133-3852 (Print)
ISSN 1588-273X (Online)