相关论文: Discrete Fourier analysis with lattices on planar …
A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains.…
A novel basis of discrete analytic polynomials on a rhombic lattice is introduced and the associated convolution product is studied. A class of discrete analytic functions that are rational with respect to this product is also described.
A discrete Fourier analysis on the fundamental domain $\Omega_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $\Omega_2$ is the regular hexagon and $\Omega_3$ is the rhombic dodecahedron, and analogous results on…
We study the Fourier transforms of indicator functions of some special high-dimensional finite type domains and obtain estimates of the associated lattice point discrepancy.
Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in \cite{LSX}. The main results consist of a new derivation of the Gaussian type cubature for the…
It is well-known that the functions $f \in L^1(\mathbb{R}^d)$ whose translates along a lattice $\Lambda$ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a…
This is a survey about tiling by translation only and related questions and methods, especially those that have to do with Fourier Analysis.
We derive results about geometric means of the Fourier modulus of filters and functions related to refinable distributions with arbitrary dilations and translations. Then we develop multi-scale constructions for dilations by…
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $\Lambda \subset \mathbb{R}$, if we have $\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the main results, and prove…
We prove several fundamental results about divisorial integral domains in the setup of multiplicative lattices.
We investigate properties of tempered distributions with discrete or countable supports such that their Fourier transforms are distributions with discrete or countable supports as well. We find sufficient conditions for support of the…
Using supercharacter theory, we identify the matrices that are diagonalized by the discrete cosine and discrete sine transforms, respectively. Our method affords a combinatorial interpretation for the matrix entries.
These notes briefly discuss Fourier transforms of finite measures and extensions of Fourier integrals to points in complex domains.
Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in…
In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in $\R^n$, that generalizes the $n$-th fold DFT of the integer lattice $\Z^n$ to arbitrary lattices. This definition is not applicable for…
A correspondence between arbitrary Fourier series and certain analytic functions on the unit disk of the complex plane is established. The expression of the Fourier coefficients is derived from the structure of complex analysis. The…
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its…
Four families of special functions, depending on n variables, are studied. We call them symmetric and antisymmetric multivariate sine and cosine functions. They are given as determinants or antideterminants of matrices, whose matrix…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…