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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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60 citations
TL;DR: In this article, the authors consider the homogeneous one-dimensional wave equation on [0,π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0π] of Lebesgue measure Lπ.
Abstract: In this paper, we consider the homogeneous one-dimensional wave equation on [0,π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0,π]. Let L∈(0,1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0,π] of Lebesgue measure Lπ. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L=1/2. When L≠1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following Hebrard and Henrot (Syst. Control Lett., 48:199–209, 2003; SIAM J. Control Optim., 44:349–366, 2005), we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.
60 citations
TL;DR: Here it is shown that for both operations this naive upper bound may be reduced to O(q 4 log q), where the implied constant is universal, depending only on the complexity of the “classical” fast Fourier transform.
Abstract: We study the complexity of performing Fourier analysis for the group SL2(F q ), where F q is the finite field of q elements. Direct computation of a complete set of Fourier transforms for a complex-valued function f onSL2(F q ) requires q 6 operations. A similar bound holds for performing Fourier inversion. Here we show that for both operations this naive upper bound may be reduced to O(q 4 log q), where the implied constant is universal, depending only on the complexity of the “classical” fast Fourier transform. The techniques we use depend strongly on explicit construct io ns of matrix representationsof the group. Additionally, the ability to construct the matrix representations permits certain numerical experiments. By quite general methods, recent work of others has shown that certain families of Cayley graphs on SL2 (F q ) are expanders. However, little is known about their exact spectra. Computation of the relevant Four ier transform permits extensive numerical investigations of the spectra of these...
60 citations
01 Jan 2006
TL;DR: In this paper, the properties of Fourier integral operators on the torus were studied by using the global representations by Fourier series instead of local representations, which can be applied in studying hyperbolic partial differential equations.
Abstract: Basic properties of Fourier integral operators on the torus \( \mathbb{T}^n = (\mathbb{R}/2\pi \mathbb{Z})^n \) are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations.
60 citations