| Título : |
Convergence and Summability of Fourier Transforms and Hardy Spaces |
| Tipo de documento: |
documento electrónico |
| Autores: |
Weisz, Ferenc, Autor |
| Mención de edición: |
1 ed. |
| Editorial: |
[s.l.] : Springer |
| Fecha de publicación: |
2017 |
| Número de páginas: |
XXII, 435 p. 34 ilustraciones |
| ISBN/ISSN/DL: |
978-3-319-56814-0 |
| Nota general: |
Libro disponible en la plataforma SpringerLink. Descarga y lectura en formatos PDF, HTML y ePub. Descarga completa o por capítulos. |
| Palabras clave: |
Secuencias (Matemáticas) análisis de Fourier Análisis armónico Secuencias Series Sumabilidad Análisis armónico abstracto |
| Índice Dewey: |
515.24 |
| Resumen: |
Este libro investiga la convergencia y sumabilidad de las transformadas de Fourier unidimensionales y multidimensionales, así como la teoría de los espacios de Hardy. Para ello, se estudia un método de sumabilidad general conocido como theta-suma, que engloba todos los métodos de sumabilidad conocidos, como las sumaciones de Fejér, Riesz, Weierstrass, Abel, Picard, Bessel y Rogosinski. Siguiendo los libros clásicos de Bary (1964) y Zygmund (1968), este es el primer libro que considera la fuerte sumabilidad introducida por la metodología actual. Otro aspecto singular es que los puntos de Lebesgue también se estudian en la teoría de la sumabilidad multidimensional. Además de los resultados clásicos, también se recopilan y analizan resultados de los últimos 20 o 30 años (normalmente sólo se encuentran en artículos de investigación dispersos), ofreciendo a los lectores una fuente conveniente y única para respaldar su trabajo. Como tal, el libro será útil tanto para investigadores como para estudiantes de grado y posgrado. |
| Nota de contenido: |
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. |
| En línea: |
https://link-springer-com.biblioproxy.umanizales.edu.co/referencework/10.1007/97 [...] |
| Link: |
https://biblioteca.umanizales.edu.co/ils/opac_css/index.php?lvl=notice_display&i |
Convergence and Summability of Fourier Transforms and Hardy Spaces [documento electrónico] / Weisz, Ferenc, Autor . - 1 ed. . - [s.l.] : Springer, 2017 . - XXII, 435 p. 34 ilustraciones. ISBN : 978-3-319-56814-0 Libro disponible en la plataforma SpringerLink. Descarga y lectura en formatos PDF, HTML y ePub. Descarga completa o por capítulos.
| Palabras clave: |
Secuencias (Matemáticas) análisis de Fourier Análisis armónico Secuencias Series Sumabilidad Análisis armónico abstracto |
| Índice Dewey: |
515.24 |
| Resumen: |
Este libro investiga la convergencia y sumabilidad de las transformadas de Fourier unidimensionales y multidimensionales, así como la teoría de los espacios de Hardy. Para ello, se estudia un método de sumabilidad general conocido como theta-suma, que engloba todos los métodos de sumabilidad conocidos, como las sumaciones de Fejér, Riesz, Weierstrass, Abel, Picard, Bessel y Rogosinski. Siguiendo los libros clásicos de Bary (1964) y Zygmund (1968), este es el primer libro que considera la fuerte sumabilidad introducida por la metodología actual. Otro aspecto singular es que los puntos de Lebesgue también se estudian en la teoría de la sumabilidad multidimensional. Además de los resultados clásicos, también se recopilan y analizan resultados de los últimos 20 o 30 años (normalmente sólo se encuentran en artículos de investigación dispersos), ofreciendo a los lectores una fuente conveniente y única para respaldar su trabajo. Como tal, el libro será útil tanto para investigadores como para estudiantes de grado y posgrado. |
| Nota de contenido: |
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. |
| En línea: |
https://link-springer-com.biblioproxy.umanizales.edu.co/referencework/10.1007/97 [...] |
| Link: |
https://biblioteca.umanizales.edu.co/ils/opac_css/index.php?lvl=notice_display&i |
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