Related papers: A Remark on Fourier Transform
Mathematical diffraction theory is concerned with the analysis of the diffraction measure of a translation bounded complex measure $\omega$. It emerges as the Fourier transform of the autocorrelation measure of $\omega$. The mathematically…
An overview of some basic notions is given, especially with an eye towards somewhat "fractal" examples, such as infinite products of cyclic groups, p-adic numbers, and solenoids.
A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton-Wigner representation we discuss the spectral decomposition…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
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 give the converse to Dirichlet's theorem on primes in arithmetic progressions by generalizing an old result of Guinand.
Analogue of Springer's formula for the Poincar\'e series of the algebra invariants of ternary form is found.
In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights…
In this short note we show the equivalence of Fourier expansion and Poisson summation approaches for the series approximation of the exponential function $\exp ({-{t^2}/4})$. The application of the Poisson summation formula is shown to…
The proof of the theorem concerning to the inverse cyclotomic Discrete Fourier Transform algorithm over finite field is provided.
In this note we propose a generalization of the Laplace and Fourier transforms which we call symmetric Laplace transform. It combines both the advantages of the Fourier and Laplace transforms. We give the definition of this generalization,…
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,…
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical…
In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between…
An explicit formula is derived for the Fourier transform of a Gaussian measure on the Heisenberg group at the Schrodinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two…
New sufficient conditions for representation of a function via the absolutely convergent Fourier integral are obtained in the paper. In the main result, Theorem 1.1, this is controlled by the behavior near infinity of both the function and…
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…
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…
We investigate several infinite product of cosines and find the closed form using the Fourier transform. The answers provide limiting distributions for some elementary probability experiments.
We discuss the notion of the orbifold transform, and illustrate it on simple examples. The basic properties of the transform are presented, including transitivity and the exponential formula for symmetric products. The connection with the…