A refinable function φ(x) : Rn → R or, more generally, a refinable function vector 8(x) = [φ1(x), . . . , φr (x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φj (x − α) : α ∈ Zn, 1 ≤ j ≤ r} form an orthogonal set of functions in L2(Rn)...

Szeg˝ o's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [−1, 1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials Λ, and leads to a new orthogonality structure in the module Λ × Λ. This structure can be interpreted in terms of a 2 × 2 matrix measure on [−1, 1], a...

We characterize the orthogonal frames and orthogonal multiwavelet frames in L2 R with matrix dilations of the form Df x √ |detA|f Ax , where A is an arbitrary expanding d × d matrix with integer coefficients. Firstly, through two arbitrarily multiwavelet frames, we give a simple construction of a pair of orthogonal multiwavelet frames. Then, by using the unitary extension principle, we present ...

This paper is an attempt is made to obtain Generating functions of modified Bessel Function. We can find number of generating functions for various special function and orthogonal polynomials by the application of group-theoretic method introduced by Louis Weisner.The process may also lead to some new generating functions for corresponding special functions. Bessel Function and orthogonal polyn...

Fourier orthogonal series with respect to the weight function (1 − |x|2)μ−1/2 on the unit ball in Rd are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourie...

For the unitary ensembles of N × N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For example the n-point correlation function and the spacing probabilities have nice representations in terms of this kernel. For the orthogonal and symplectic ensembles...

An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graph Kmn,mn (for any values ofm,n) and all graphs in the collection are isomorphic to certa...

Given the n x p orthogonal matrix A and the convex function f : R"-~ R, we find two orthogonal matrices P and Q such that f is almost constant on the convex hull of ± the columns of P, f is sufficiently nonconstant on the column space of Q, and the column spaces of P and Q provide an orthogonal direct sum decomposi t ion of the column space of A. This provides a numerical ly stable algorithm fo...