Convolutions power of a characteristic function

International Journal of Mathematics and Mathematical Sciences, 1993

In this paper we establish some new approach to constructing convolution for general Mellin type transforms. This method is based on the theory of double Hellin-Barnes integrals. Some properties of convolutions and several examples are given.

2001

The object of the present paper is to obtain several interest- ing results involving coecient estimates for analytic normalized functions belonging to certain classes defined in terms of the convolution with the extremal function for the class of starlike functions of order , 0 < 1.

Transactions of the American Mathematical Society, 1959

The purpose of this paper is to prove several results in approximations through complex convolution polynomials with Jackson-type rate or with best approximation rate, having the quality of preservation of some properties in geometric function theory, like the preservation of: coefficients' bounds, positive real part, bounded turn, close-to-convexity, starlikeness, convexity, spirallikeness, α-convexity. Also, some sufficient conditions for starlikeness and univalence of analytic functions are preserved.

2000

Abstract. The distribution v\ of the random series Y,±A" is the infinite convolution product of|(6-A"+ 6\ n). These measures have been studied since the 1930s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which A€(i, 1) is v\ absolutely continuous?

Journal of Mathematical Analysis and Applications, 2005

The non-commutative convolution f * g of two distributions f and g in D is defined to be the limit of the sequence {(f τ n ) * g}, provided the limit exists, where {τ n } is a certain sequence of functions in D converging to 1. It is proved that |x| λ * sgn x|x| µ = 2 sin(λπ/2) cos(µπ/2) sin[(λ + µ)π/2] B(λ + 1, µ + 1) sgn x|x| λ+µ+1 , for −1 < λ + µ < 0 and λ, µ = −1, −2, . . . , where B denotes the Beta function.

Stability in Probability, 2010

The paper is, for the most part, devoted to a survey of the analytical properties of generalized convolution algebras and their realizations. This issue appears to be the state of the art until now because intensive research on the generalized convolution and the related models still persists.

Journal of Number Theory, 2013

We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive combinatorics.

Canadian Mathematical Bulletin, 1991

On the set F of complex-valued arithmetic functions we construct an infinite family of convolutions, that is, binary operations ψ of the form so that (F, +, ψ) is a commutative ring, for which the unity is unbounded. Here + denotes pointwise addition.

Bernoulli, 2006

A distribution F on (−∞, ∞) is said to belong to the class S(γ) for some γ ≥ 0 if lim x→∞ F (x − u) /F (x) = e γu holds for all u and lim x→∞ F * 2 (x)/F (x) = 2m F exists and is finite. Let X and Y be two independent random variables, where X has a distribution in the class S(γ) and Y is nonnegative with an endpointŷ = sup {y : Pr(Y ≤ y) < 1} ∈ (0, ∞). We prove that the product XY has a distribution in the class S(γ/ŷ). We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.