Related papers: Fourier series and approximation on hexagonal and …
In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their H\"older…
Fractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration and complex numbers. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has…
We propose a numerical method for the solution of electromagnetic problems on axisymmetric domains, based on a combination of a spectral Fourier approximation in the azimuthal direction with an IsoGeometric Analysis (IGA) approach in the…
We make the case for neural network objects and extend an already existing neural network calculus explained in detail in Chapter 2 on \cite{bigbook}. Our aim will be to show that, yes, indeed, it makes sense to talk about neural network…
Frames and Bessel sequences in Fr\'echet spaces and their duals are defined and studied. Their relation with Schauder frames and representing systems is analyzed. The abstract results presented here, when applied to concrete spaces of…
We have discovered three non-power infinite series representations for Bessel functions of the first kind of integer orders and real arguments. These series contain only elementary functions and are remarkably simple. Each series was…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…
The main objective of this article is a constructive generalization of the holomorphic power and Laurent series expansions in C to dimension 3 using the framework of hypercomplex function theory. For this reason, deals the first part of…
We consider row sequences of simultaneous rational approximations constructed in terms of Fourier expansions and prove a Montessus de Ballore type theorem.
The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete…
Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly…
In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as…
This paper introduces a weighted generalized inverse framework for Fourier extensions, designed to suppress spurious oscillations in the extended region while maintaining high approximation accuracy on the original interval. By formulating…
In this paper we study linear and non-linear Fourier-Pad\'e approximation for Angelesco systems of functions. This construction is similar to that of Hermite-Pad\'e approximation. Instead of considering power series expansions of the…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by…
The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…
Linear combinations of translations of a single Gaussian, e^{-x^2}, are shown to be dense in L^2(R). Two algorithms for determining the coefficients for the approximations are given, using orthogonal Hermite functions and least squares.…
In view of recent developments of the study of reproducing kernel Hilbert spaces, in particular with the context the Hardy spaces on tubes, aspects of rational approximation for functions of finite energy in several complex and several real…