Series Representation of Power Function

This paper presents the way to make expansion for the next form function: to the numerical series. The most widely used methods to solve this problem are Newtons Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are inverse operators). The paper provides the other kind of solution, except above described theorems.

Rankin-Selberg convolutions for GL(n)×GL(n)and GL(n)×GL(n-1)for principal series representations

Let k be a local field.Let I_(v) and I_(v′)be smooth principal series representations of GLn(k)and GL_(n-1)(k),respectively.The Rankin-Selberg integrals yield a continuous bilinear map I_(v)×I_(v′)→C with a certain invariance property.We study integrals over a certain open orbit that also yield a continuous bilinear map I_(v)×I_(v′)→C with the same invariance property and show that these integrals equal the Rankin-Selberg integrals up to an explicit constant.Similar results are also obtained for Rankin-Selberg integrals for GLn(k)×GLn(k).

SERIES REPRESENTATION OF DAUBECHIES' WAVELETS

This paper gives a kind of series represeotation of the scaling functions φNand the associated wavelets . constructed by Daubechies. Based on Poission sununation formula, the functions gh. φN(x+N-1), φN (x+N),'''' φN (x+2N-2)(Ox 1) are linearly represented by φN(x), φN(x + 1),''', φN(x + 2N - 2) and some polynomials of order less than N, and φ0(x):= (φN (x), φN (x + 1),''', φN (x + N -2))t is translated into a solution of a nonhomogeneous vectorvalued functional equationwhere A0, A1 are (N - 1) x (N - 1)-dimensional matrices, the components of P0(x), P1 (x) are polynomials of order less than N. By iteration, .φ0(x) is eventualy represented as an (N - 1)-dimensional vector series with vector norm where and

Series Representation of Jointly S˛S Distribution via Symmetric Covariations

We introduce the notion of symmetric covariation,which is a new measure of dependence between two components of a symmetricα-stable random vector,where the stability parameterαmeasures the heavy-tailedness of its distribution.Unlike covariation that exists only whenα∈(1,2],symmetric covariation is well defined for allα∈(0,2].We show that symmetric covariation can be defined using the proposed generalized fractional derivative,which has broader usages than those involved in this work.Several properties of symmetric covariation have been derived.These are either similar to or more general than those of the covariance functions in the Gaussian case.The main contribution of this framework is the representation of the characteristic function of bivariate symmetricα-stable distribution via convergent series based on a sequence of symmetric covariations.This series representation extends the one of bivariate Gaussian.

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.