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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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TL;DR: In this article, it has been shown that a Fourier-Stieltjes transform taking only the values zero and one must either be identically zero or identically one.
Abstract: Introduction. There have been several papers written on the subject of determining all homomorphisms, and more particularly, isomorphisms of group algebras of locally compact abelian groups. Except in certain special cases there does not seem to be much known. One case which has been treated is the case of two groups G and H, where H has a connected dual group L . In this case it has been shown that the only homomorphisms of L1(G) into L1(H) are essentially those induced by homomorphisms of G into H [1]. This result was proved in the case where H is the real line and by means of the structure theory of locally compact abelian groups, extended to the more general case. The crucial point in the proof seems to be the obvious fact that a Fourier-Stieltjes transform taking only the values zero and one must either be identically zero or identically one. Equivalently we may say that the only idempotent measures on H are the zero measure, and Haar measure of the identity subgroup. Another case, that in which H is the circle group, has been solved in [5], [6]. In this case too, the complete analysis of -idempotent measures on the circle achieved in [4], was very heavily used. The author in a previous paper [2], has determined all the idempotent measures on locally compact groups. Thus, it seems reasonable that one should now be able to completely solve the homomorphism problem. In this paper, we shall do exactly that. A very simple, but hitherto unnoticed, relationship between the homomorphism problem and idempotent measures is established in the case of compact G and H. Then a passage to the Bohr compactifications of the groups in question yields the general result. There are certain technical complications which appear, some of which are standard, such as convolutions with approximate identities, which we hope will not confuse the reader. It is perhaps unnecessary to add that at all times the reader should bear in mind the concrete examples of Fourier series and Fourier integrals to better understand what is happening. In the case of Fourier series, our problem is precisely one of determining which mappings of Fourier coefficients in m-variables into coefficients in n-variables, send Fourier coefficients into Fourier coefficients. More specifically, let 7r be a

63 citations

Journal ArticleDOI
TL;DR: In this article, a new analytical method is developed to analyse the response of laminated composite plates subjected to static and dynamic loading, and the modal forms are presented in terms of double Fourier series.

63 citations

Journal ArticleDOI
TL;DR: In this article, the amplitude for the elastic scattering of two spinless particles of equal mass is expanded in terms of eigenfunctions which form a complete set for a certain class of functions of the Mandelstam variables and which display the threshold behavior of the partial-wave amplitudes.
Abstract: The amplitude for the elastic scattering of two spinless particles of equal mass \textonehalf{} is expanded in terms of eigenfunctions which form a complete set for a certain class of functions of the Mandelstam variables $s,t,u(s+t+u=1)$ and which display the threshold behavior of the partial-wave amplitudes. The eigenfunctions are generated by a partial differential operator which commutes with the total angular momentum in any of the three channels and which is invariant under $s$, $t$, $u$ permutations. An infinite number of finite-dimensional crossing relations for the partial-wave amplitudes which are necessary and sufficient for the crossing symmetry of the total amplitude are derived, as well as an explicit form for the corresponding crossing matrices. It is shown that the Fourier coefficients of the expansion satisfy a Froissart-Gribov integral representation whose kernel is determined by the imaginary parts of the partial-wave amplitudes.

63 citations

Journal ArticleDOI
TL;DR: In this article, a new method is introduced for formulating the scattering problem in which the scattered fields and the interior fields in the case of a dielectric scatterer are represented in an expansion in terms of free space modal wave functions in cylindrical coordinates, the coefficients of which are the unknowns.
Abstract: A new method is introduced for formulating the scattering problem in which the scattered fields (and the interior fields in the case of a dielectric scatterer) are represented in an expansion in terms of free-space modal wave functions in cylindrical coordinates, the coefficients of which are the unknowns. The boundary conditions are satisfied using either an analytic continuation procedure, in which the far-field pattern (in Fourier series form) is continued into the near field and the boundary conditions are applied at the surface of the scatterer; or the completeness of the modal wave functions, to approximately represent the fields in the interior and exterior regions of the scatterer directly. The methods were applied to the scattering of two-dimensional cylindrical scatterers of arbitrary cross section and only the TM polarization of the excitation is considered. The solution for the coefficients of the modal wave functions are obtained by inversion of a matrix which depends only on the shape and material of the scatterer. The methods are illustrated using perfectly conducting square and elliptic cylinders and elliptic dielectric cylinders. A solution to the problem of multiple scattering by two conducting scatterers is also obtained using only the matrices characterizing each of the single scatterers. As an example, the method is illustrated by application to a two-body configuration.

63 citations

Journal ArticleDOI
TL;DR: In this paper, the free vibration analysis of functionally graded open shells including cylindrical, conical and spherical ones with arbitrary subtended angle and general boundary conditions is presented, where the modified Fourier series is expressed in the form of the linear superposition of a double cosine series and auxiliary functions which are introduced to ensure and accelerate the convergence of the series representations.

63 citations


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