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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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130 citations
TL;DR: In this paper, a class of nonanalytic automorphic functions which were first mentioned to A. Selberg by C. L. Siegel are considered, and they have Fourier coefficients which are closely connected with the Fourier coefficient of analytic automomorphic forms and they are also eigenfunctions of the Laplace operator derived from the hyperbolic metric.
Abstract: In this paper we consider a class of nonanalytic automorphic functions which were first mentioned to A. Selberg by C. L. Siegel. These functions have Fourier coefficients which are closely connected with the Fourier coefficients of analytic automorphic forms, and they are also eigenfunctions of the Laplace operator derived from the hyperbolic metric. We shall show how this latter property gives new results in the classical theory of automorphic forms.
129 citations
TL;DR: In this article, a method for modal analysis of non-linear and non-conservative mechanical systems is proposed, in particular, dry-friction nonlinearities are considered although the method is not restricted to these.
128 citations
TL;DR: In this paper, the orthogonal nature of the Fourier transform (FT) is maintained by using the trapezoidal rule for the mechanical quadrature of the FT of one, two, and three dimensions.
128 citations
TL;DR: In this paper, a truncated Fourier series expansion for a 2π-periodic function of finite regularity is used to accurately reconstruct the corresponding function, and an algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question.
Abstract: Kowledge of a truncated Fourier series expansion for a 2π-periodic function of finite regularity, which is assumed to be piecewise smooth in each period, is used to accurately reconstruct the corresponding function. An algebraic equation of degree M is constructed for the M singularity locations in each period for the function in question. The M coefficients in this algebraic equation are obtained by solving an algebraic system of M equations determined by the coefficients in the known truncated expansion. If discontinuities in the derivatives of the function are considered, in addition to discontinuities in the function itself, that algebraic system will be nonlinear with respect to the M unknown coefficients. The degree of the algebraic system will depend on the desired order of accuracy for the reconstruction, i.e., a higher degree will normally lead to a more accurate determination of the singularity locations. By solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations, we are able to reconstruct the 2π-periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order
128 citations