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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 the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials as mentioned in this paper.
Abstract: In the early 19 century, J. Fourier was an impassioned advocate of the use of such sums, of course writing sines and cosines rather than complex exponentials. Euler, the Bernouillis, and others had used such sums in similar fashions and for similar ends, but Fourier made a claim extravagant for the time, namely that all functions could be expressed in such terms. Unfortunately, in those days there was no clear idea of what a function was, no vocabulary to specificy classes of functions, and no specification of what it would mean to represent a function by such a series. In hindsight, probably issues of pointwise and L convergence, unspecified to some degree, were confused with each other.
92 citations
92 citations
TL;DR: In this paper, an upper bound for the convexity bound of L 2 L-functions with respect to the conductor of the imaginary quadratic field K = Q(√−D) was provided.
Abstract: We continue our study of GL2 L–functions with the aim of providing upper bounds for their order of magnitude. As is familiar it suffices to provide such bounds on the critical line and, both for the sake of applications and for the ideas involved, we are most interested in breaking the convexity bound and this with respect to the conductor. In this paper we are interested primarily in L–functions attached to characters of the class group of the imaginary quadratic field K = Q(√−D). We are motivated by our paper [DFI4]. That work was not included in the current series because the class group L–functions are treated there directly. They may however be viewed as L–functions associated to cusp forms of weight 1, level D and character (the nebentypus) χD(n) = (−D n )
92 citations
TL;DR: In this paper, an analytical solution for the cylindrical bending vibrations of linear piezoelectric laminated plates is obtained by extending the Stroh formalism to the generalized plane strain vibrations of piezolectric materials.
92 citations
TL;DR: In this paper, a fluctuating-surface current formulation of radiative heat transfer between arbitrary shape is proposed, which exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering.
Abstract: We describe a fluctuating-surface current formulation of radiative heat transfer between bodies of arbitrary shape that exploits efficient and sophisticated techniques from the surface-integral-equation formulation of classical electromagnetic scattering. Unlike previous approaches to nonequilibrium fluctuations that involve scattering matrices---relating ``incoming'' and ``outgoing'' waves from each body---our approach is formulated in terms of ``unknown'' surface currents, laying at the surfaces of the bodies, that need not satisfy any wave equation. We show that our formulation can be applied as a spectral method to obtain fast-converging semianalytical formulas in high-symmetry geometries using specialized spectral bases that conform to the surfaces of the bodies (e.g., Fourier series for planar bodies or spherical harmonics for spherical bodies), and can also be employed as a numerical method by exploiting the generality of surface meshes/grids to obtain results in more complicated geometries (e.g., interleaved bodies as well as bodies with sharp corners). In particular, our formalism allows direct application of the boundary-element method, a robust and powerful numerical implementation of the surface-integral formulation of classical electromagnetism, which we use to obtain results in new geometries, such as the heat transfer between finite slabs, cylinders, and cones.
92 citations