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Fourier Analysis and Approximation of Functions

In Fourier Analysis and Approximation of Functions basics of classical Fourier Analysis are given as well as those of approximation by polynomials, splines and entire functions of exponential type. In Chapter 1 which has an introductory nature, theorems on convergence, in that or another sense, of integral operators are given. In Chapter 2 basic properties of simple and multiple Fourier series are discussed, while in Chapter 3 those of Fourier integrals are studied. The first three chapters as well as partially Chapter 4 and classical Wiener, Bochner, Bernstein, Khintchin, and Beurling theorems in Chapter 6 might be interesting and available to all familiar with fundamentals of integration theory and elements of Complex Analysis and Operator Theory. Applied mathematicians interested in harmonic analysis and/or numerical methods based on ideas of Approximation Theory are among them. In Chapters 6-11 very recent results are sometimes given in certain directions. Many of these results have never appeared as a book or certain consistent part of a book and can be found only in periodics; looking for them in numerous journals might be quite onerous, thus this book may work as a reference source. The methods used in the book are those of classical analysis, Fourier Analysis in finite-dimensional Euclidean space Diophantine Analysis, and random choice.

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EAN (ISBN-13): 9789048166411
ISBN (ISBN-10): 9048166411
Gebonden uitgave
pocket book
Verschijningsjaar: 2010
Uitgever: Springer
600 Bladzijden
Gewicht: 0,894 kg
Taal: eng/Englisch

Boek bevindt zich in het datenbestand sinds 2012-12-18T07:37:50+01:00 (Amsterdam)
Detailpagina laatst gewijzigd op 2022-01-04T14:27:09+01:00 (Amsterdam)
ISBN/EAN: 9789048166411

ISBN - alternatieve schrijfwijzen:
90-481-6641-1, 978-90-481-6641-1


Gegevens van de uitgever

Auteur: Roald M. Trigub; Eduard S. Belinsky
Titel: Fourier Analysis and Approximation of Functions
Uitgeverij: Springer; Springer Netherland
586 Bladzijden
Verschijningsjaar: 2010-12-06
Dordrecht; NL
Gedrukt / Gemaakt in
Gewicht: 0,908 kg
Taal: Engels
123,04 € (DE)
126,49 € (AT)
136,00 CHF (CH)
POD
XIV, 586 p.

BC; Previously published in hardcover; Hardcover, Softcover / Mathematik/Analysis; Differentialrechnung und -gleichungen; Verstehen; Fourier transform; Hilbert space; Operator theory; bigfin; calculus; harmonic analysis; B; Approximations and Expansions; Fourier Analysis; Sequences, Series, Summability; Measure and Integration; Functional Analysis; Approximations and Expansions; Fourier Analysis; Sequences, Series, Summability; Measure and Integration; Functional Analysis; Mathematics and Statistics; Funktionalanalysis und Abwandlungen; Mathematische Analysis, allgemein; Integralrechnung und -gleichungen; Funktionalanalysis und Abwandlungen; BB

1. Representation Theorems.- 1.1 Theorems on representation at a point.- 1.2 Integral operators. Convergence in Lp-norm and almost everywhere.- 1.3 Multidimensional case.- 1.4 Further problems and theorems.- 1.5 Comments to Chapter 1.- 2. Fourier Series.- 2.1 Convergence and divergence.- 2.2 Two classical summability methods.- 2.3 Harmonic functions and functions analytic in the disk.- 2.4 Multidimensional case.- 2.5 Further problems and theorems.- 2.6 Comments to Chapter 2.- 3. Fourier Integral.- 3.1 L-Theory.- 3.2 L2-Theory.- 3.3 Multidimensional case.- 3.4 Entire functions of exponential type. The Paley-Wiener theorem.- 3.5 Further problems and theorems.- 3.6 Comments to Chapter 3.- 4. Discretization. Direct and Inverse Theorems.- 4.1 Summation formulas of Poisson and Euler-Maclaurin.- 4.2 Entire functions of exponential type and polynomials.- 4.3 Network norms. Inequalities of different metrics.- 4.4 Direct theorems of Approximation Theory.- 4.5 Inverse theorems. Constructive characteristics. Embedding theorems.- 4.6 Moduli of smoothness.- 4.7 Approximation on an interval.- 4.8 Further problems and theorems.- 4.9 Comments to Chapter 4.- 5. Extremal Problems of Approximation Theory.- 5.1 Best approximation.- 5.2 The space Lp. Best approximation.- 5.3 Space C. The Chebyshev alternation.- 5.4 Extremal properties for algebraic polynomials and splines.- 5.5 Best approximation of a set by another set.- 5.6 Further problems and theorems.- 5.7 Comments to Chapter 5.- 6. A Function as the Fourier Transform of A Measure.- 6.1 Algebras A and B. The Wiener Tauberian theorem.- 6.2 Positive definite and completely monotone functions.- 6.3 Positive definite functions depending only on a norm.- 6.4 Sufficient conditions for belonging to Ap and A*.- 6.5 Further problems and theorems.- 6.6 Comments to Chapter 6.- 7. Fourier Multipliers.- 7.1 General properties.- 7.2 Sufficient conditions.- 7.3 Multipliers of power series in the Hardy spaces.- 7.4 Multipliers and comparison of summability methods of orthogonal series.- 7.5 Further problems and theorems.- 7.6 Comments to Chapter 7.- 8. Summability Methods. Moduli of Smoothness.- 8.1 Regularity.- 8.2 Applications of comparison. Two-sided estimates.- 8.3 Moduli of smoothness and K-functionals.- 8.4 Moduli of smoothness and strong summability in Hp(D), 0erences.- Author Index.- Topic Index.

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9781402028762 Fourier Analysis and Approximation of Functions (Roald M. Trigub#Eduard S. Belinsky)


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