LEADER 01997nas 2200565-a 450 001 9910229179603321 005 20240116213021.0 035 $a(CKB)958480254665 035 $a(CONSER)---86645870- 035 $a(EXLCZ)99958480254665 100 $a19851205b19851993 --- a 101 0 $aeng 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 00$aSoftware law journal 210 $aManhattan Beach, Calif. $cCenter for Computer/Law$d©1985- 215 $a1 online resource 311 08$aPrint version: Software law journal. 0886-3628 (DLC) 86645870 (OCoLC)12873561 531 $aJOHN MARSHALL JOURNAL OF COMPUTER AND INFORMATION LAW 531 $aSOFTW LAW J 531 0 $aSoftw. law j. 606 $aComputer software industry$xLaw and legislation$zUnited States$vPeriodicals 606 $aComputers$xLaw and legislation$zUnited States$vPeriodicals 606 $aComputer software industry$xLaw and legislation$2fast$3(OCoLC)fst00872619 606 $aComputers$xLaw and legislation$2fast$3(OCoLC)fst00872821 606 $aDatenverarbeitung$2gnd 606 $aRecht$2gnd 606 $aZeitschrift$2gnd 606 $aInformaticarecht$2gtt 607 $aUnited States$2fast$1https://id.oclc.org/worldcat/entity/E39PBJtxgQXMWqmjMjjwXRHgrq 607 $aUSA$2swd 608 $aPeriodicals.$2fast 608 $aPeriodicals.$2lcgft 615 0$aComputer software industry$xLaw and legislation 615 0$aComputers$xLaw and legislation 615 7$aComputer software industry$xLaw and legislation. 615 7$aComputers$xLaw and legislation. 615 7$aDatenverarbeitung. 615 7$aRecht. 615 7$aZeitschrift. 615 17$aInformaticarecht. 676 $a343.73/07800164 676 $a347.3037800164 712 02$aCenter for Computer/Law. 906 $aJOURNAL 912 $a9910229179603321 920 $aexl_impl conversion 996 $aSoftware law journal$91932309 997 $aUNINA LEADER 08296nam 22006615 450 001 9910632485203321 005 20251113211124.0 010 $a3-031-14459-7 024 7 $a10.1007/978-3-031-14459-2 035 $a(MiAaPQ)EBC7144540 035 $a(Au-PeEL)EBL7144540 035 $a(CKB)25456661900041 035 $a(PPN)266355684 035 $a(OCoLC)1351749286 035 $a(DE-He213)978-3-031-14459-2 035 $a(EXLCZ)9925456661900041 100 $a20221122d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMartingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series /$fby Lars-Erik Persson, George Tephnadze, Ferenc Weisz 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2022. 215 $a1 online resource (633 pages) 225 1 $aMathematics and Statistics Series 311 08$aPrint version: Persson, Lars-Erik Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series Cham : Springer International Publishing AG,c2022 9783031144585 320 $aIncludes bibliographical references and index. 327 $aIntro -- Preface -- How to Read the Book? -- Acknowledgements -- Contents -- 1 Partial Sums of Vilenkin-Fourier Series in Lebesgue Spaces -- 1.1 Introduction -- 1.2 Vilenkin Groups and Functions -- 1.3 The Representation of the Vilenkin Groups on the Interval [0,1) -- 1.4 Convex Functions and Classical Inequalities -- 1.5 Lebesgue Spaces -- 1.6 Dirichlet Kernels -- 1.7 Lebesgue Constants -- 1.8 Vilenkin-Fourier Coefficients -- 1.9 Partial Sums -- 1.10 Final Comments and Open Questions -- 2 Martingales and Almost Everywhere Convergence of Partial Sums of Vilenkin-Fourier Series -- 2.1 Introduction -- 2.2 Conditional Expectation Operators -- 2.3 Martingales and Maximal Functions -- 2.4 Calderon-Zygmund Decomposition -- 2.5 Almost Everywhere Convergence of Vilenkin-Fourier Series -- 2.6 Almost Everywhere Divergence of Vilenkin-Fourier Series -- 2.7 Final Comments and Open Questions -- 3 Vilenkin-Fejér Means and an Approximate Identity in Lebesgue Spaces -- 3.1 Introduction -- 3.2 Vilenkin-Fejér Kernels -- 3.3 Approximation of Vilenkin-Fejér Means -- 3.4 Almost Everywhere Convergence of Vilenkin- Fejér Means -- 3.5 Approximate Identity -- 3.6 Final Comments and Open Questions -- 4 Nörlund and T Means of Vilenkin-Fourier Series in Lebesgue Spaces -- 4.1 Introduction -- 4.2 Well-Known and New Examples of Nörlund and TMeans -- 4.3 Regularity of Nörlund and T Means -- 4.4 Kernels of Nörlund Means -- 4.5 Kernels of T Means -- 4.6 Norm Convergence of Nörlund and T Means in Lebesgue Spaces -- 4.7 Almost Everywhere Convergence of Nörlund and T Means -- 4.8 Convergence of Nörlund and T Means in Vilenkin-Lebesgue Points -- 4.9 Riesz and Nörlund Logarithmic Kernels and Means -- 4.10 Final Comments and Open Questions -- 5 Theory of Martingale Hardy Spaces -- 5.1 Introduction -- 5.2 Martingale Hardy Spaces and Modulus of Continuity. 327 $a5.3 Atomic Decomposition of the Martingale Hardy Spaces Hp -- 5.4 Interpolation Between Hardy Spaces Hp -- 5.5 Bounded Operators on Hp Spaces -- 5.6 Examples of p-Atoms and Hp Martingales -- 5.7 Final Comments and Open Questions -- 6 Vilenkin-Fourier Coefficients and Partial Sums in Martingale Hardy Spaces -- 6.1 Introduction -- 6.2 Estimations of Vilenkin-Fourier Coefficients in Hp Spaces -- 6.3 Hardy and Paley Type Inequalities in Hp Spaces -- 6.4 Maximal Operators of Partial Sums on Hp Spaces -- 6.5 Convergence of Partial Sums in Hp Spaces -- 6.6 Convergence of Subsequences of Partial Sums in Hp Spaces -- 6.7 Strong Convergence of Partial Sums in Hp Spaces -- 6.8 Final Comments and Open Questions -- 7 Vilenkin-Fejér Means in Martingale Hardy Spaces -- 7.1 Introduction -- 7.2 Maximal Operator of Vilenkin-Fejér Means on Hp Spaces -- 7.3 Convergence of Vilenkin-Fejér Means in Hp Spaces -- 7.4 Convergence of Subsequences of Vilenkin-Fejér Means in Hp Spaces -- 7.5 Strong Convergence of Vilenkin-Fejér Means in Hp Spaces -- 7.6 Final Comments and Open Questions -- 8 Nörlund and T Means of Vilenkin-Fourier Series in Martingale Hardy Spaces -- 8.1 Introduction -- 8.2 Maximal Operators of Nörlund Means on Hp Spaces -- 8.3 Maximal Operators of T Means on Hp Spaces -- 8.4 Strong Convergence of Nörlund Means in Hp Spaces -- 8.5 Strong Convergence of T Means in Hp Spaces -- 8.6 Maximal Operators of Riesz and Nörlund Logarithmic Means on Hp Spaces -- 8.7 Strong Convergence of Riesz and Nörlund Logarithmic Means in Hp Spaces -- 8.8 Final Comments and Open Questions -- 9 Convergence of Vilenkin-Fourier Series in Variable Martingale Hardy Spaces -- 9.1 Introduction -- 9.2 Variable Lebesgue Spaces -- 9.3 Doob's Inequality in Variable Lebesgue Spaces -- 9.4 The Maximal Operator Us -- 9.5 The Maximal Operator V?,s -- 9.6 Variable Martingale Hardy Spaces. 327 $a9.7 Atomic Decomposition of Variable Hardy Spaces -- 9.8 Martingale Inequalities in Variable Spaces -- 9.9 Partial Sums of Vilenkin-Fourier Series in Variable Lebesgue Spaces -- 9.10 The Maximal Fejér Operator on Hp(·) -- 9.11 Final Comments and Open Questions -- 10 Appendix: Dyadic Group and Walsh and Kaczmarz Systems -- 10.1 Introduction -- 10.2 Walsh Group and Walsh and Kaczmarz Systems -- 10.3 Estimates of the Walsh-Fejér Kernels -- 10.4 Walsh-Fejér Means in Hp -- 10.5 Modulus of Continuity in Hp and Walsh-Fejér Means -- 10.6 Riesz and Nörlund Logarithmic Means in Hp -- 10.7 Maximal Operators of Kaczmarz-Fejér Means on Hp -- 10.8 Modulus of Continuity in Hp and Kaczmarz-Fejér Means -- 10.9 Final Comments and Open Questions -- References -- Notations -- Index. 330 $aThis book discusses, develops and applies the theory of Vilenkin-Fourier series connected to modern harmonic analysis. The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, code theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin-Fourier series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. The first part of the book can be used as an introduction to the subject, and the following chapters summarize the most recent research in this fascinating area and can be read independently. Each chapter concludes with historical remarks and open questions. The book will appeal to researchers working in Fourier and more broad harmonic analysis and will inspire them for their own and their students' research. Moreover, researchers in applied fields will appreciate it as a sourcebook far beyond the traditional mathematical domains. 410 0$aMathematics and Statistics Series 606 $aSequences (Mathematics) 606 $aFourier analysis 606 $aHarmonic analysis 606 $aSequences, Series, Summability 606 $aFourier Analysis 606 $aAbstract Harmonic Analysis 615 0$aSequences (Mathematics) 615 0$aFourier analysis. 615 0$aHarmonic analysis. 615 14$aSequences, Series, Summability. 615 24$aFourier Analysis. 615 24$aAbstract Harmonic Analysis. 676 $a515.2433 676 $a515.2433 700 $aPersson$b Lars-Erik$f1949-$01347641 702 $aTephnadze$b George 702 $aWeisz$b Ferenc$f1964- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910632485203321 996 $aMartingale Hardy spaces and summability of one-dimensional Vilenkin-Fourier series$93084129 997 $aUNINA