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Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.


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Journal Article
TL;DR: In this paper, the basic eigenfunctions of Dk and the spectral decomposition of f>fc for the trivial group, used in the construction of an automorphic ''prime-form'' and automomorphic functions with prescribed eigen functions are introduced and used to give summation methods for the classical kernel functions on a compact Riemann surface.
Abstract: *~T + ~T~2~)~2ifc.y-r— acting on a Hubert space §k of automorphic forms dx dy) dx of weight k e IR. In this paper, we present the basic eigenfunction expansions of Gs k(z, z') and discuss applications to conditionally convergent Poincare series and series of Dirichlet type for Fuchsian groups of the first kind, and to the spectral decomposition of §* for groups of the second kind. The outline of the paper is äs follows: in § l we set up the basic eigenfunctions of Dk and the spectral decomposition of f>fc for the trivial group, used in the construction of an automorphic \"prime-form\" and automorphic functions with prescribed automorphic eigenfunctions are introduced in § 2 and are used to give summation methods for the classical kernel functions on a compact Riemann surface, äs well äs for the construction of an automorphic \"prime-form\" and automorphic function with prescribed singularities. The Fourier coefficients of the resolvent at a parabolic cusp are worked out in § 3 and include many special cases of historical interest; the use of the resolvent here explicates certain multiplicative relations of Hecke type and \"expansions of zero\" associated to analytic forms of positive dimension for the modular group. Finally, in § 4 we consider Fourier developments at the hyperbolic fixed-points, including a summation method for the period matrix of a compact Riemann surface; for a Fuchsian group of the second kind with a single free side, the continuous spectral measure for §k is given in terms of a Poisson kernel and Fourier coefficients analogous to the Eisenstein series for horocyclic groups.

232 citations

Book
11 Mar 2013
TL;DR: In this article, a two-volume text in harmonic analysis introduces a wealth of analytical results and techniques, including Fourier series, harmonic functions, Hilbert transform, and Weyl calculus.
Abstract: This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderon–Zygmund and Littlewood–Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman–Meyer theory; Carleson's resolution of the Lusin conjecture; Calderon's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.

231 citations

Journal ArticleDOI
TL;DR: A general formula for computing changes in the signal-to-noise ratio of a spectrum resulting from the Fourier self-deconvolution procedure is derived and the rate of decrease in the SNR as a function of K for eight different smoothing (apodization) functions is studied.
Abstract: A general formula for computing changes in the signal-to-noise ratio of a spectrum resulting from the Fourier self-deconvolution procedure is derived. Self-deconvolution reduces the intrinsic halfwidths of lines by a factor K, which is in practice limited by the noise in the spectrum. With the help of the derived formula, the rate of decrease in the SNR as a function of K for eight different smoothing (apodization) functions is studied. With high K values there are significant differences in the SNR as a result of the use of different smoothing functions. With K = 4 difference of more than 1 order of magnitude between two extreme cases is demonstrated, and with K = 5 a difference of almost 2 orders of magnitude in the SNR is predicted.

230 citations

Book
30 Jun 2003
TL;DR: In this paper, the authors present a general theory of weakly almost periodic functions and apply it to almost periodic types of functions, such as vector-valued almost-periodic functions and weakly-almost-constant functions.
Abstract: 1 Almost periodic type functions.- 1.1.- 1.1.1 Numerical almost periodic functions.- 1.1.2 Uniform almost periodic functions.- 1.1.3 Vector-valued almost periodic functions.- 1.2 Asymptotically almost periodic functions.- 1.3 Weakly almost periodic functions.- 1.3.1 Vector-valued weakly almost periodic functions.- 1.3.2 Ergodic theorem.- 1.3.3 Invariant mean and mean convolution.- 1.3.4 Fourier series of WAV(?, H).- 1.3.5 Uniformly weakly almost periodic functions.- 1.4 Approximate theorem and applications.- 1.4.1 Numerical approximate theorem.- 1.4.2 Vector-valued approximate theorem.- 1.4.3 Unique decomposition theorem.- 1.5 Pseudo almost periodic functions.- 1.5.1 Pseudo almost periodic functions.- 1.5.2 Generalized pseudo almost periodic functions.- 1.6 Converse problems of Fourier expansions.- 1.7 Almost periodic type sequences.- 1.7.1 Almost periodic sequences.- 1.7.2 Other almost periodic type sequences.- 2 Almost periodic type differential equations.- 2.1 Linear differential equations.- 2.1.1 Ordinary differential equations.- 2.1.2 Abstract differential equations.- 2.1.3 Integration of almost periodic type functions.- 2.2 Partial differential equations.- 2.2.1 Dirichlet Problems.- 2.2.2 Parabolic equations.- 2.2.3 Second-order equations with gradient operators.- 2.3 Means, introversion and nonlinear equations.- 2.3.1 General theory of means and introversions.- 2.3.2 Applications to (weakly) almost periodic functions.- 2.3.3 Nonlinear differential equations.- 2.3.4 Implications of almost periodic type solutions.- 2.4 Regularity and exponential dichotomy.- 2.4.1 General theory of regularity.- 2.4.2 Stability of regularity.- 2.4.3 Almost periodic type solutions.- 2.5 Equations with piecewise constant argument.- 2.5.1 Exponential dichotomy for difference equations.- 2.5.2 Equations with piecewise constant argument.- 2.5.3 Almost periodic difference equations.- 2.6 Equations with unbounded forcing term.- 2.7 Almost periodic structural stability.- 2.7.1 Topological equivalence and structural stability.- 2.7.2 Exponential dichotomy and structural stability.- 3 Ergodicity and abstract differential equations.- 3.1 Ergodicity and regularity.- 3.1.1 Ergodicity and regularity.- 3.1.2 Solutions of almost periodic type equations.- 3.2 Ergodicity and nonlinear equations.- 3.3 Semigroup of operators and applications.- 3.3.1 Semigroup of operators.- 3.3.2 Almost periodic type solutions.- 3.4 Delay differential equations.- 3.4.1 Introduction of delay differential equations.- 3.4.2 Linear autonomous equations.- 3.4.3 Linear nonautonomous equations.- 3.5 Spectrum of functions.- 3.6 Abstract Cauchy Problems.- 3.6.1 Harmonic analysis of solutions.- 3.6.2 Asymptotic behavior of solutions.- 3.6.3 Mild solutions.- 3.6.4 Weakly almost periodic solutions.- 4 Ergodicity and averaging methods.- 4.1 Ergodicity and its properties.- 4.2 Quantitative theory.- 4.2.1 Introduction.- 4.2.2 Quantitative theory of averaging methods.- 4.2.3 Example and comments.- 4.3 Perturbations of noncritical linear systems.- 4.4 Qualitative theory of averaging methods.- 4.4.1 Almost periodic type solutions of nonlinear equations.- 4.4.2 Some examples.- 4.5 Averaging methods for functional equations.- 4.5.1 Averaging for functional differential equations.- 4.5.2 Averaging for delay difference equations.- Notations.

230 citations

Book
07 Sep 2004
TL;DR: In this article, the authors present a discussion on representation at a point, including convergence and divergence, convergence in Lp-norm and almost everywhere, and convergence in the space C. The Paley-Wiener theorem, the Chebyshev alternation, and the Wiener Tauberian theorem.
Abstract: 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.

229 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023270
2022702
2021511
2020510
2019589
2018580