Convergence and summability of Fourier transforms and Hardy spaces
著者
書誌事項
Convergence and summability of Fourier transforms and Hardy spaces
(Applied and numerical harmonic analysis / series editor, John J. Benedetto)
Birkhäuser , Springer, c2017
大学図書館所蔵 全5件
  青森
  岩手
  宮城
  秋田
  山形
  福島
  茨城
  栃木
  群馬
  埼玉
  千葉
  東京
  神奈川
  新潟
  富山
  石川
  福井
  山梨
  長野
  岐阜
  静岡
  愛知
  三重
  滋賀
  京都
  大阪
  兵庫
  奈良
  和歌山
  鳥取
  島根
  岡山
  広島
  山口
  徳島
  香川
  愛媛
  高知
  福岡
  佐賀
  長崎
  熊本
  大分
  宮崎
  鹿児島
  沖縄
  韓国
  中国
  タイ
  イギリス
  ドイツ
  スイス
  フランス
  ベルギー
  オランダ
  スウェーデン
  ノルウェー
  アメリカ
注記
Includes bibliographical references (p. 413-427) and index
内容説明・目次
内容説明
This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such as the Fejer, Riesz, Weierstrass, Abel, Picard, Bessel and Rogosinski summations. Following on the classic books by Bary (1964) and Zygmund (1968), this is the first book that considers strong summability introduced by current methodology. A further unique aspect is that the Lebesgue points are also studied in the theory of multi-dimensional summability. In addition to classical results, results from the past 20-30 years - normally only found in scattered research papers - are also gathered and discussed, offering readers a convenient "one-stop" source to support their work. As such, the book will be useful for researchers, graduate and postgraduate students alike.
目次
List of Figures.- Preface.- I One-dimensional Hardy spaces and Fourier transforms.- 1 One-dimensional Hardy spaces.- 1.1 The Lp spaces.- 1.2 Hardy-Littlewood maximal function.- 1.3 Schwartz functions.- 1.4 Tempered distributions and Hardy spaces.- 1.5 Inequalities with respect to Hardy spaces.- 1.6 Atomic decomposition.- 1.7 Interpolation between Hardy spaces.- 1.8 Bounded operators on Hardy spaces.- 2 One-dimensional Fourier transforms.- 2.1 Fourier transforms.- 2.2 Tempered distributions.- 2.3 Partial sums of Fourier series.- 2.4 Convergence of the inverse Fourier transform.- 2.5 Summability of one-dimensional Fourier transforms.- 2.6 Norm convergence of the summability means.- 2.7 Almost everywhere convergence of the summability means.- 2.8 Boundedness of the maximal operator.- 2.9 Convergence at Lebesgue points.- 2.10 Strong summability.- 2.11 Some summability methods.- II Multi-dimensional Hardy spaces and Fourier transforms.- 3 Multi-dimensional Hardy spaces.- 3.1 Multi-dimensional maximal functions.- 3.1.1 Hardy-Littlewood maximal functions.- 3.1.2 Strong maximal functions.- 3.2 Multi-dimensional tempered distributions and Hardy spaces.- 3.3 Inequalities with respect to multi-dimensional Hardy spaces.- 3.4 Atomic decompositions.- 3.4.1 Atomic decomposition of H2p (Rd).- 3.4.2 Atomic decomposition of Hp(Rd).- 3.5 Interpolation between multi-dimensional Hardy spaces.- 3.5.1 Interpolation between the H2p (Rd) spaces.- 3.5.2 Interpolation between the Hp(Rd) spaces.- 3.6 Bounded operators on multi-dimensional Hardy spaces.- 3.6.1 Bounded operators on H2p (Rd).- 3.6.2 Bounded operators on Hp(Rd).- 4 Multi-dimensional Fourier transforms.- 4.1 Fourier transforms.- 4.2 Multi-dimensional partial sums.- 4.3 Convergence of the inverse Fourier transform.- 4.4 Multi-dimensional Dirichlet kernels.- 4.4.1 Triangular Dirichlet kernels.- 4.4.2 Circular Dirichlet kernels.- 5 `q-summability of multi-dimensional Fourier transforms.- 5.1 The `-summability means.- 5.2 Norm convergence of the `q-summability means.- 5.2.1 Proof ofTheorem 5.2.1 for q = 1 and q = 1.- 5.2.1.1 Proof for q = 1 in the two-dimensional case.- 5.2.1.2 Proof for q = 1 in higher dimensions (d 3).- 5.2.1.3 Proof for q = 1 in the two-dimensional case.- 5.2.1.4 Proof for q = 1 in higher dimensions (d 3).- 5.2.2 Some summability methods.- 5.2.3 Further results for the Bochner-Riesz means.- 5.3 Almost everywhere convergence of the `q-summability means.- 5.3.1 Proof of Theorem 5.3.2.- 5.3.1.1 Proof for q = 1 in the two-dimensional case.- 5.3.1.2 Proof for q = 1 in higher dimensions (d 3).- 5.3.1.3 Proof for q = 1 in the two-dimensional case.- 5.3.1.4 Proof for q = 1 in higher dimensions (d 3).- 5.3.2 Proof of Theorem 5.3.3.- 5.3.3 Some summability methods.- 5.3.4 Further results for the Bochner-Riesz means.- 5.4 Convergence at Lebesgue points.- 5.4.1 Circular summability (q = 2).- 5.4.2 Cubic and triangular summability (q = 1 and q = 1).- 5.4.2.1 Proof of the results for q = 1 and d = 2.- 5.4.2.2 Proof of the results for q = 1 and d = 2.- 5.4.2.3 Proof of the results for q = 1 and d 3.- 5.4.2.4 Proof of the results for q = 1 and d 3.- 5.5 Proofs of the one-dimensional strong summability results.- 6 Rectangular summability of multi-dimensional Fourier transforms.- 6.1 Norm convergence of rectangular summability means.- 6.2 Almost everywhere restricted summability.- 6.3 Restricted convergence at Lebesgue points.- 6.4 Almost everywhere unrestricted summability.- 6.5 Unrestricted convergence at Lebesgue points.- Bibliography.- Index.- Notations.
「Nielsen BookData」 より