Related papers: Functional Equations and Fourier Analysis
In this paper, an open problem in the multidimensional complex analysis is pesented that arises in the investigation of the regularity properties of Fourier integral operators and in the regularity theory for hyperbolic partial differential…
A numerical approach to solve the perturbed Lambert's problem is presented. The proposed technique uses the Theory of Functional Connections, which allows the derivation of a constrained functional that analytically satisfies the boundary…
A connection between the Galois-theoretic approach to semi-abelian homology and the homological closure operators is established. In particular, a generalised Hopf formula for homology is obtained, allowing the choice of a new kind of…
The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of…
It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…
These notes briefly discuss Fourier transforms of finite measures and extensions of Fourier integrals to points in complex domains.
Estimating the coefficient functionals on various classes of holomorphic functions traditionally forms an important field of geometric complex analysis and its mathematical and physical applications. These coefficients reflect fundamental…
In this paper, under the Riemann hypothesis, we study the Fourier analysis about the functions $\tilde{\Delta}(x)$ and $N(T)$ .
We show how several important classical problems, with positive definite potential energy, can be solved by starting from the factorization of the total mechanical energy using complex numbers. In particular, we derive in a new way exact…
Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show…
Given a semigroup $S$ equipped with an involutive automorphism $\sigma$, we determine the complex-valued solutions $f,g,h$ of the functional equation \begin{equation*}f(x\sigma(y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\,\,x,y\in S,\end{equation*} in…
Given a semigroup $S$ generated by its squares equipped with an involutive automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of…
In this article, we will showcase some analytical concepts that can be used to tackle Functional Equations (FE) in the positive real numbers domain. Such concepts and related techniques have occasionally appeared in recent High School Math…
Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the…
The problem of three particles interacting through harmonic forces is discussed within the Newtonian formalism. By means of a didactic approach, an exact analytical solution is found, and ways to extend it to the N-body case are pointed…
We present a general approach to establish algebraic functional equations for big Galois representations over multiple $\mathbb{Z}_p$-extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a…
Fourier Transforms is a first in a series of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such…
Using a recently developed approach for solving the three dimensional Dirac equation with spherical symmetry, we obtain simple representations for the Green's function of the Dirac-Oscillator and Dirac-Coulomb problems. This is accomplished…
We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…