Related papers: $\mathbf L^1$ completeness for Fourier series
In this paper we show an alternative way of defining Fourier Series and Transform by using the concept of convolution with exponential signals. This approach has the advantage of simplifying proofs of transforms properties and, in our view,…
A completeness theorem is proved involving a system of integro-differential equations with some $\lambda$-depending boundary conditions. Also some sufficient conditions for the root functions to form a Riesz basis are established.
We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,\pi\times\pi')$ over any Galois extension with $\pi$ and $\pi'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at…
The conditions for convergence of square and rectangular Fejer means of functions on the infinite dimensional torus were obtained, also a generalization of the results for the case of abstract measure spaces was formulated.
This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach extends to,…
Determination of linear combination of exponential functions with unknown rate constants from its sampled values is a problem of considerable interest. Here we present a constructive and explicit solution to this problem. Moments of such…
We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. We prove the Frobenius Theorem with sharp regularity estimate when the subbundle is log-Lipschitz: if $\mathcal V$ is a log-Lipschitz…
We present a short proof of the Fabry quotient theorem, which states that for a complex power series with unit radius of convergence, if the quotient of its consecutive coefficients tends to $ s $, then the point $ z=s $ is a singular point…
In this paper, we investigate the convergence properties of Fourier partial sums associated with general orthonormal systems, focusing on functions that belong to specific differentiable function classes. While classical Fourier analysis…
We establish a Liouville type theorem for some conformally invariant fully nonlinear equations
The aim of this paper is to establish various factorization results and then to derive estimates for linear functionals through the use of a generalized Taylor theorem. Additionally, several error bounds are established including…
We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by $\sinh(\pi x)$ or $\cosh(\pi x)$. In many cases, the resulting Fourier transform remains within the same class of functions.…
Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.
Fourier series with power series coefficients for the normal and distance to a point from an ellipse are derived. These expressions are the first of their kind and opens up a range of analysis and computational possibilities.
We prove non-trivial upper and lower bounds for the "Spectrum of Singularities" of Fourier Series with polynomial frequencies. The Spectrum of Singularities of a function f gives the Hausdorff dimension of the set of points with a given…
A function $f:\mathbb{Z}_n \to \mathbb{C}$ can be represented as a linear combination $f(x)=\sum_{\alpha \in \mathbb{Z}_n}\widehat{f}(\alpha) \chi_{\alpha,n}(x)$ where $\widehat{f}$ is the (discrete) Fourier transform of $f$. Clearly, the…
Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible…
We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated…
We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing…
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.