Combinatorial orthogonal expansions

Combinatorial Orthogonal Expansions

so that amn is non-negative. In this paper we give in Theorem 1 and Theorem 2 explicit formulas for anm as a polynomial in the α ′ js, β ′ js and the γ ′ js, which give these theorems. The idea is to represent amn as a generating function of paths, whose weights are products of differences. Monotonicity hypotheses on the coefficients force the weights to be individually positive, these are the ...

Quasi-orthogonal expansions for functions in BMO

For {φ_n(x)}, x ε [0,1] an orthonormalsystem of uniformly bounded functions, ||φ_n||_{∞}≤ M

On expansions in orthogonal polynomials

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; final...

Combinatorial Species and Cluster Expansions

This paper will survey recent progress on clarifying the connection between enumerative combinatorics and cluster expansions. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. Cluster expansions, on the other hand, are supposed to give convergent expressions for measures on infinite dimensional spaces, such as those that occ...

Summability of orthogonal expansions of several variables