Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

The Segal Algebra {\bf S}_0({\Bbb R}^d) and Norm Summability of Fourier Series and Fourier Transforms

Abstract: A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L1-norm convergence of the θ-means σnθf to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case θ is an element of Feichtinger’s Segal algebra \({\bf S}_0({\Bbb R}^d)\), then these convergence results hold. Some new sufficient conditions are given for θ to be in \({\bf S}_0({\Bbb R}^d)\). A long list of concrete special cases of the θ-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case.

Fourier smoother and additive models

Abstract: Suppose the observations (ti,yi), i = 1,… n, follow the model where gj are unknown functions. The estimation of the additive components can be done by approximating gj, with a function made up of the sum of a linear fit and a truncated Fourier series of cosines and minimizing a penalized least-squares loss function over the coefficients. This finite-dimensional basis approximation, when fitting an additive model with r predictors, has the advantage of reducing the computations drastically, since it does not require the use of the backfitting algorithm. The cross-validation (CV) [or generalized cross-validation (GCV)] for the additive fit is calculated in a further 0(n) operations. A search path in the r-dimensional space of degrees of freedom is proposed along which the CV (GCV) continuously decreases. The path ends when an increase in the degrees of freedom of any of the predictors yields an increase in CV (GCV). This procedure is illustrated on a meteorological data set.

Laplacians on fractals with spectral gaps have nicer Fourier series

Abstract: On the Sierpinski gasket and related fractals, partial sums of Fourier series (spectral expansions of the Laplacian) converge along certain special subsequences. This is related to the existence of gaps in the spectrum. Laplacians on fractals have been studied intensively using both probabilistic and analytic tools, as a “rough” counterpart to Laplacians on smooth Riemannian manifolds ([B], [Ki], [S1]). This research has succeeded in establishing many “expected” analogs of results from the smooth theory, but has also turned up some startling differences. For example: there exist localized eigenfunctions [FS]; the square of a nonconstant function in the domain of the Laplacian is never in the domain of the Laplacian [BST]; the energy measure is singular [Ku]; the wave equation has infinite propagation speed [DSV]; the Weyl ratio does not have a limit [FS], [KL]; the Laplacian does not behave like a second order operator [S2]; to mention just a few. One might be tempted to say that the fractal world resembles the smooth world to some degree, but everything is worse. On the other hand, recent numerical experiments hint that when it comes to convergence of Fourier series, things might be better on fractals. To be specific, consider the standard Laplacican ∆ on the Sierpinski gasket SG. With either Dirichlet or Neumann boundary conditions, there is a complete orthonormal basis of eigenfunctions, say −∆uj = λjuj , j = 1, 2, 3, . . . , and every L function f has a Fourier series

Nonequispaced Hyperbolic Cross Fast Fourier Transform

Abstract: A straightforward discretization of problems in $d$ spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives. We propose a nonequispaced hyperbolic cross FFT based on one hyperbolic cross FFT and a dedicated interpolation by splines on sparse grids. Analogously to the nonequispaced FFT for trigonometric polynomials with Fourier coefficients supported on the full grid, this allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes.

Basic analog of Fourier series on a $q$-quadratic grid

Abstract: We prove orthogonality relations for some analogs of trigonometric functions on a g-quadratic grid and introduce the corresponding g-Fourier series. We also discuss several other properties of this basic trigonometric system and the g-Fourier series.