Об одном асимптотическом свойстве ядер Дирихле

In this article, the properties of Fourier Series by discussing around the basic properties of integrable functions and kernel are discussed. With the discussion over the function f around discontinuities, it is found that f should at least be Reimann integrable functions to make Fourier series imitate the function successfully. More, about the filters, it is clear that the sharp filters will converge to f under a certain condition, with the analyzation over Dirichlet Kernel, and how the convergent becomes successful with the properties over good kernel.

Proceedings of the London Mathematical SocietyVolume s2-35, Issue 1 p. 445-487 Papers On the Theory of Trigonometric Series (IV) S. Verblunsky, S. VerblunskySearch for more papers by this author S. Verblunsky, S. VerblunskySearch for more papers by this author First published: 1933 https://doi.org/10.1112/plms/s2-35.1.445AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Volumes2-35, Issue11933Pages 445-487 RelatedInformation

The note A constant in the theory of trigonometric series in the October 1964 issue of Mathematics of Computation provided us with a test for our recently constructed algorithms for the computation of roots of functions, and for numerical quadrature in the presence of singularities. The latter algorithm, utilizing the Gaussian 8-point quadrature formula applied to sub-intervals of variable length, involves a sufficiently small number of ordinates that computational labor and round-off error do not become problems. Use of these algorithms indicated the value ao = .3084438, for the root of the equation fJot2 u cos u du = 0, differing from the reported value, .30483, in the third place. To check this result, we made the transformation u = X4 to weaken the character of the singularity at the origin, and obtained the following table by conventional numerical quadrature, confirming our result:

Measure and Probability

CHAPTER I - BASIC CONCEPTS AND THEOREMS IN THE THEORY OF TRIGONOMETRIC SERIES

Orthogonal time frequency space (OTFS) modulation is gaining recognition for its potential to facilitate integrated sensing and communication (ISAC) within future mobile networks. However, computing the sensing channel matrix in orthogonal time frequency space (OTFS), a crucial step for accurate target parameter estimation, presents significant challenges due to its high dimensionality. Therefore, this study introduces an innovative method to reduce such computational complexity by combining two ingredients. First, through algebraic operations, we decompose the sensing channel matrix into four lower-dimensional matrices whose elements can be associated with a Dirichlet kernel. Second, we formulate an analytical criterion, independent of system parameters, that leverages the properties of the Dirichlet kernel and identifies the most informative elements of these matrices that deserve computation. To demonstrate the effectiveness of our approach, we assess the computational complexity of this distilled channel matrix in terms of the number of elementary operations required. Numerical results indicate that our technique markedly decreases receiver complexity by up to three orders of magnitude without compromising sensing performance.

CONTENTS Introduction Chapter I. Convergence of rectangular partial sums §1. Results on the almost everywhere convergence of Fourier series of integrable functions §2. The almost everywhere convergence of Fourier series of functions from §3. A brief survey of new results on the convergence of rectangular partial sums in various metrics Chapter II. New results on rectangular means of multiple Fourier series Chapter III. Dirichlet kernels §1. Lebesgue constants and norms of some trigonometric polynomials §2. Asymptotic behaviour of Dirichlet kernels Chapter IV. Uniqueness problems for multiple trigonometric series (rectangular partial sums) Chapter V. Problems of localization for rectangular partial sums Chapter VI. Special classes of multiple trigonometric series §1. Series with monotone coefficients §2. Multiple lacunary series §3. Fourier series of piecewise monotone functions of several variables Chapter VII. A definition of convergence of multiple series Chapter VIII. Conjugate series and conjugate functions of several variables §1. Conjugate series and conjugate functions in , §2. The conjugation operator in the space §3. Multidimensional analogues of Plessner's theorem Chapter IX. Representation of functions by trigonometric series and “correction” of functions Chapter X. Fourier coefficients §1. Cantor-Lebesgue and Denjoy-Luzin theorems §2. New results on the absolute convergence of Fourier series §3. Transformations of Fourier coefficients and other results References

Research in the theory of trigonometric series has been carried out for over two centuries. The results obtained have greatly influenced various fields of mathematics, mechanics, and physics. Nowadays

We study sampling series and their relationship to frequency band limited functions. Motivated by the theories of multiresolution analyses and subdivision, particular attention is paid to sampling series whose kernels are refinable. After developing the basic properties of the sampling kernels under study, we consider three families of such kernels: (1) damped cardinal sines, (2) the fundamental functions for cardinal spline interpolation, and (3) a family of compactly supported kernels defined in terms of their masks. The limiting kernel of each family is the cardinal sine. In the cases (1) and (2) we present results concerning the limiting behavior of the corresponding sampling series when the data samples {c n} have polynomial growth as n →±∞.

On the Theory of Trigonometric Series (III)

A method of designing robust sampled regulators for open-loop stable, linear, time-invariant, multivariable plants is given. The design procedure requires only the knowledge of the open-loop step response of the system and is independent of its structure. It allows the imposition of constraints on the maximum excursion of the inputs to the plant. The stability of the closed-loop system follows from its passivity and this, in turn, is obtained from a classical theorem of W. H. Young in the theory of trigonometric series. As a result it turns out that the stability of the algorithm is independent of the sampling rate. Moreover, the method of proving stability enables one to estimate the size of plant perturbations and/or measurement errors that can be tolerated before the onset of instability. Finally, a partial answer, leading to a Lyapunov-type equation, is also given to the question of the optimal choice of control parameters.

On a constant in the theory of trigonometric series

Topological results on sequences $\{n_k x\}^\infty_{k=1}$ and their applications in the theory of trigonometric series

The first edition of this book was written almost twenty-five years ago. Since then the theory of trigonometric series has undergone considerable change. It has always been one of the central parts of Analysis, but now we see its notions and methods appearing, in abstract form, in distant fields like the theory of groups, algebra, theory of numbers. These abstract extensions are, however, not considered here and the subject of the second edition of this book is, as before, the classical theory of Fourier series, which may be described as the meeting ground of the Real and Complex Variables.

A Constant in the Theory of Trigonometric Series