We extendF. Holland's definition of the space of resonant classes of functions, on the real line, to the spaceR(Φpq) (1≦p, q≦∞)of resonant classes of measures, on locally compact abelian groups. We characterize this space in terms of transformable measures and establish a realatlonship betweenR(Φpq)and the set of positive definite functions for amalgam spaces. As a consequence we answer the conjecture posed by L. Argabright and J. Gil de Lamadrid in their work on Fourier analysis of unbounded measures.
1.Introduction. Let ℛndenote the set of alln×nmatrices with real elements, and letdenote the subset of ℛnconsisting of all real,n×n, symmetric positive-definite matrices. We shall use the notationto denote that minor of the matrixA= (aij) ∈ ℛnwhich is the determinant of the matrixTheSchur Product(Schur (14)) of two matricesA, B∈ ℛnis denned bywhereA= (aij),B= (bij),C= (cij) andLet ϕ be the mapping of ℛninto the real line defined byfor allA∈ ℛn, where, as in the sequel,.
In this paper we give the solution of Bessis–Moussa–Villani (BMV) conjecture for the generalized Gaussian random variables [Formula: see text] where f is in the real Hilbert space [Formula: see text]. The main examples of generalized Gaussian random variables are q-Gaussian random variables, (-1 ≤ q ≤ 1), related to q-CCR relation and other commutation relations. We will prove that BMV conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function [Formula: see text] is positive-definite function on the real line. The case q = 0, i.e. when G(f) are the free Gaussian (Wigner) random variables and the operators A and B are free with respect to the vacuum trace was proved by Fannes and Petz.23
The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.
AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.
Resonance classes of measures
