Относительные средние поперечники пространств Соболевав L ²(ℝ^d)

Analysis Mathematica

Volume/Issue:: Volume 34: Issue 1
DOI:: https://doi.org/10.1007/s10476-008-0107-8

Authors: Yongping Liu and Weiwei Xiao

Abstract

In this paper, in order to consider the problems of relative width on ℝ^d, we proposed definitions of relative average width which combine the ideas of the relative width and the average width. We established the smallest number M which make the following equality

\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} $$\overline K _\sigma (U(W_2^\alpha ),M(W_2^\alpha ),L_2 ({\mathbb{R}}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 ({\mathbb{R}}^d ))$$ \end{document}

hold, where U(W ₂ ^α) is the Riesz potential or Bessel potential of the unit ball in L ²(ℝ^k) and the notations

\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} $$\overline K _\sigma$$ \end{document}

(·, ·,L ₂(ℝ^d)) and

\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} $$\overline d _\sigma$$ \end{document}

(·, L ₂(ℝ^d)) denote respectively the relative average width in the sense of Kolmogorov and the average width in the sense of Kolmogorov in their given order. In 2001, Subbotin and Telyakovskii got similar results on the relative width of Kolmogorov type. We also proved that

\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} $$\overline K _\sigma (U(W_2^\alpha ) \cap B(L_2 (\mathbb{R}^d )),U(W_2^\beta ) \cap B(L_2 (\mathbb{R}^d ))L_2 (\mathbb{R}^d )) = \overline d _\sigma (U(W_2^\alpha ),L_2 (\mathbb{R}^d )),$$ \end{document}

where 0 × β × α.

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Относиелые поперечники классов в L ₂(T), определённых линейным дифференциальным оператором в L _q(T)

Analysis Mathematica

Volume/Issue:: Volume 37: Issue 1
DOI:: https://doi.org/10.1007/s10476-011-0104-1

Author: Weiwei Xiao

Abstract

In this paper, the smallest number M which makes the equality

\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} $$K_n (W_2^{L_r } (T),MW_2^{L_r } (T),L_2 (T)) = d_n (W_2^{L_r } (T),L_2 (T))$$ \end{document}

valid, is established and the asymptotic order of

\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} $$K_n (W_2^{L_r } (T),W_2^{L_r } (T),L_q (T)),1 \leqslant q \leqslant \infty$$ \end{document}

, is obtained, where

\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} $$W_2^{L_r }$$ \end{document}

(T) is a periodic smooth function class which is determined by a linear differential operator, K _n(·, ·, ·) and d _n(·, ·) are the relative width and the width in the sense of Kolmogorov, respectively.

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Certain problems of the approximation of functions in two variables by Fourier-Hermite sums in the space L ₂(Γ², e ^-x ^² ^-y ^²)

Analysis Mathematica

Volume/Issue:: Volume 32: Issue 3
DOI:: https://doi.org/10.1007/s10476-006-0009-6

Authors: V. A. Abilov and M. V. Abilov

Summary

We give an exact estimate of the deviation of the "triangular" partial sums of the double Fourier-Hermite series of functions of the class L ^r ₂(D) in the space L ₂(Γ, e ^-x ^-y ).

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Approximation der Ableitungen von L ₂-Funktionalen durch singuläre integrale

Acta Mathematica Hungarica

Volume/Issue:: Volume 71: Issue 1-2
DOI:: https://doi.org/10.1007/bf00052191

Author: H. Haf

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Приближение сопряже нных функций суммами Фурье в L _2π ^p

Analysis Mathematica

Volume/Issue:: Volume 11: Issue 1
DOI:: https://doi.org/10.1007/bf02633188

Author: S. Baiborodov

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A functional CLT for the L ² modulus of continuity of local time

Periodica Mathematica Hungarica

Volume/Issue:: Volume 61: Issue 1-2
DOI:: https://doi.org/10.1007/s10998-010-3231-8

Author: Jay Rosen

Abstract

We show that as processes in (c, d, t) ∈ C(R ² × R ₊ ¹)

\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} $$\frac{{\int_c^d {(L_t^{x + h} - L_t^x )^2 dx - 4h} \int_c^d {L_t^x dx} }} {{h^{3/2} }}\mathop \Rightarrow \limits^\mathcal{L} \left( {\frac{{64}} {3}} \right)^{1/2} \int_c^d {L_t^x d\eta (x)}$$ \end{document}

as h → 0 for Brownian local time L _t ^x. Here η(x) is an independent two-sided Brownian motion.

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Bundle Convergence of Cesàro Means of Orthogonal Sequences in Noncommutative L ₂-Spaces

Periodica Mathematica Hungarica

Volume/Issue:: Volume 38: Issue 1-2
DOI:: https://doi.org/10.1023/a:1004799115653

Author: Ferenc Móricz

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L ₂(Rⁿ)-экстремальные за дачи на основе непольной и нформации

Analysis Mathematica

Volume/Issue:: Volume 21: Issue 2
DOI:: https://doi.org/10.1007/bf01904272

Author: Liu Yongping

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On the best approximation of periodic functions by trigonometric polynomials and the exact values of widths of function classes in L ₂

Analysis Mathematica

Volume/Issue:: Volume 38: Issue 2
DOI:: https://doi.org/10.1007/s10476-012-0205-5

Authors: М. Шлбоэов and С. Влклрчук

Реэюме

Получены точные неравенства типа Джексона-Стечкина для ос-редненных с весом модулей непрерывности m-го (m ∈ ℕ) порядка. Для классов функций, определенных при помоши мажорант и укаэанных осредненных величин, вычислены точные эначения раэличных n-поперечников при выполнении определенных ограничений на мажоранты.

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The unit balls of $ℒ (^{n} l_{\infty}^{m})$ and $ℒ_{s} (^{n} l_{\infty}^{m})$

Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 57: Issue 3
DOI:: https://doi.org/10.1556/012.2020.57.3.1470

Author: Sung Guen Kim

. Math. (Basel) , 76 ( 2001 ), 73 – 80 . [2] Cavalcante , W. and Pellegrino , D. , Geometry of the closed unit ball of the space of bilinear forms on l ∞ 2 , arXiv : 1603.01535v2 . [3] Choi , Y. S. , Ki , H. and Kim , S. G. , Extreme

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Относительные средние поперечники пространств Соболевав L ²(ℝ^d)

Abstract

Относиелые поперечники классов в L ₂(T), определённых линейным дифференциальным оператором в L _q(T)

Abstract

Certain problems of the approximation of functions in two variables by Fourier-Hermite sums in the space L ₂(Γ², e ^-x ^² ^-y ^²)

Summary

Approximation der Ableitungen von L ₂-Funktionalen durch singuläre integrale

Приближение сопряже нных функций суммами Фурье в L _2π ^p

A functional CLT for the L ² modulus of continuity of local time

Abstract

Bundle Convergence of Cesàro Means of Orthogonal Sequences in Noncommutative L ₂-Spaces

L ₂(Rⁿ)-экстремальные за дачи на основе непольной и нформации

On the best approximation of periodic functions by trigonometric polynomials and the exact values of widths of function classes in L ₂

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The unit balls of $ℒ (^{n} l_{\infty}^{m})$ and $ℒ_{s} (^{n} l_{\infty}^{m})$

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