Discrete Harmonic Analysis Associated with Ultraspherical Expansions

Calderón-zygmund Operators Associated to Ultraspherical Expansions

On Rapid Computation of Expansions in Ultraspherical Polynomials

We present an O(N log2 N) algorithm for the computation of the first N coefficients in the expansion of an analytic function in ultraspherical polynomials. We first represent expansion coefficients as an infinite linear combination of derivatives and then as an integral transform with a hypergeometric kernel along the boundary of a Bernstein ellipse. Following a transformation of the kernel, we...

Berry-ess Een-type Inequalities for Ultraspherical Expansions

This paper contains several variants of Berry-Ess een-type inequalities for ultraspher-ical expansions of probability measures on 0; ]. Similar to the classical results on IR , proofs will be based in some cases on ultraspherical analogues of Fej er-kernels. The inequalities in this paper in particular lead to relations between the spherical-cap-distance of probability measures on unit spheres ...

Discrete Analogues in Harmonic Analysis

These notes were taken in the second week of the 2011 Study Analysis and Geometry summer program at Princeton.

Mod–discrete Expansions

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the n’th random variable Xn is by a particular member Rn of a given family of distributions, whose variance increases with n. The basic assumption is that the ratio ...