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Summary We establish an improvement of a recent theorem of S. M. Mazhar which is a generalization of our result and studies the embedding relation between the class W r H S ω, including only odd functions and a set of functions defined via the strong means of Fourier series of odd continuous functions (see the precise definitions below).

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ПустьΦN-функция Юнг а со свойствами
\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} $$\Phi (x)x^{ - 1} \downarrow 0, \exists \alpha > 1 \Phi (x)x^{ - \alpha } \uparrow (x \downarrow 0),$$ \end{document}
илиΦ(х)=х, {λk} — положи тельная, неубывающая последовательность 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} $$S_\Phi \{ \lambda \} = \left\{ {f:\left\| {\sum\limits_{k = 0}^\infty \Phi (\lambda _k |f - s_k |)} \right\|_\infty< \infty } \right\}.$$ \end{document}
В работе найдены необ ходимые и достаточны е условия для вложений
\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} $$S_\Phi \{ \lambda \} \subset W^r F(r \geqq 0),$$ \end{document}
, гдеF=C, L , Lip α (0<α≦1). С этой то чки зрения рассматриваются и др угие классы (например, ).
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A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973. A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979. In this work we give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman-Kac formula as well.

Our aim is to show that from an elementary discrete approach, by taking simple limits, one may get the continuous versions. We think that such an approach can be advantageous for both research and applications. Moreover, it is hoped that this approach has pedagogical merits as well: gives insight and seems suitable for teaching students whose mathematical background may not contain e.g. measure theory or stochastic analysis.

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Пустьϕ — возрастающа я непрерывная фцнкци я на [0,π],ϕ(0)=0 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} $$\mathop \smallint \limits_0^h \frac{{\varphi \left( t \right)}}{t}dt = O\left( {\varphi \left( h \right)} \right){\text{ }}\left( {h \to 0} \right).$$ \end{document}
Положим
\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} $$\psi \left( h \right) = h\mathop \smallint \limits_h^\pi \frac{{\varphi \left( t \right)}}{{t^2 }}dt \left( {h \in (0, \pi ]} \right).$$ \end{document}
Доказывается следую щая теорема.Пусть f∈ С[−π, π], ω(f, δ)=О(ϕ(δ))) 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} $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\varphi \left( {\left| h \right|} \right)}}\left| {f\left( {x + h} \right) - f\left( x \right)} \right| = 0$$ \end{document}
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Л. Лейндлер поставил з адачу о том, следует ли при 0<р<1 из 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} $$\mathop {\max }\limits_x \sum\limits_{k = 0}^\infty {\left| {S_k (x) - f(x)} \right|^p< \infty }$$ \end{document}
принадлежность функ цииf классу Lip 1 (здесьS k(x) — сумма Фурье порядкаk функц ииf). В работе дан положите льный ответ на этот во прос. Рассматриваются так же различные обобщен ия этой задачи.
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