John N. McDonald

Reproducing kernel structure and sampling on time-warped spaces with application to warped wavelets

Time-warped signal spaces have received attention in the literature. Among the topics of particular interest are sampling of time-warped signals and signal analysis using warped analysis functions, including wavelets. This correspondence introduces a reproducing kernel (RK) structure for time-warped signal spaces that unifies multiple perspectives on sampling in such spaces.

A sequence of extremal problems for trigonometric polynomials

The sequence of extremal problems In = sup{(2π)−1 ∝02π¦p(θ)¦2 dθ¦pϵ Pn}, where Pn denotes the set of nonnegative trigonometric polynomials of degree ⩽n having constant term 1, is studied. It is shown that (n + 1) C1 ⩽ In < 1 + (n + 1) C1, where C1 = 0.686981293….

On Some Zero Configurations Associated with the van der Waerden Conjecture

Abstract Let A be a permanent minimizing doubly stochastic matrix. This paper discusses the maximum number of zeros which can occur in any row or column of A. The results are applied to reaffirming the van der Waerden conjecture in the cases n⩽4.

Polynomial extreme elements of the closed unit ball of H ∞(B)

Properties of homogeneous polynomials which are extreme points of the closed unit ball of the space H ∞(B) of bounded analytic functions on the open unit ball in C n are studied. A complete characterization is given in the case of degree 2 and applied to the study of holomorphic functions on B having positive real part. A sufficient condition for extremality is also given.

A windowing condition for characterization of finite signals from spectral phase or magnitude

Reconstruction of a signal from its spectral phase or magnitude is in general an ill-posed problem. Various conditions restricting the class of signals under consideration have been shown to be sufficient to regularize the problem so that a unique or essentially unique signal corresponds to any given spectral magnitude or spectral phase function. This paper shows that a finite discrete-time signal is characterized by its spectral magnitude (or phase) and the spectral magnitude (or phase) of an ancillary signal obtained by windowing the original signal.

A sampling approach to region-selective image compression

This paper proposes a region-selective image compression method that is motivated by the nonuniform sampling structure of the human eye. The approach is based on the nonuniform sampling theorem of Clark et al. (1985) for time-warped bandlimited functions. The theoretical underpinnings of the method are explained and examples are presented to illustrate its implementation and performance.

Holomorphic mappings into convex subsets of Cn

Holomorphic mappings f V→W are studied, where: V is a complete circular domain ⊆Cn W⊆Cn is convex and openf(0) = 0 and f′(0) = I. A classical result due to Caratheodory, namely , is used to obtain estimates on the terms of the homogeneous expansion of 〈f(Z),u〉. These estimates are used to give a new proof that W must contain a ball of radius 1/2 about 0 when V is the unit ball of Cn . Extremal cases of the estimates are studied. In particular, new information is obtained about the case where the 1/2 is the radius of the largest ball around 0 contained in W.

Extreme operators on H

Let A 1 and A 2 be sup-norm algebras, each containing the constant functions. Let P ( A 1 , A 2 ) denote the set of bounded linear operators from A 1 to A 2 which carry 1 into 1 and have norm 1. Several authors have considered the problem of describing the extreme points of P ( A 1 , A 2 ). In the case where A 1 is the algebra of continuous complex functions on some compact Hausdorff space, and A 2 is the algebra of complex scalars, Arens and Kelley proved that the extreme operators in P ( A 1 , A 2 ) are exactly the multiplicative ones (see [1]). It was shown by Phelps in [6] that if A 1 is self-adjoint, then every extreme point of P ( A 1 , A 1 ) is multiplicative. In [4], Lindenstrauss, Phelps, and Ryff exhibited non-multiplicative extreme points of P ( A , A ) and P ( H ∞ , H ∞ ), where A and H ∞ are, respectively, the disk algebra, and the algebra of bounded analytic functions on the open unit disk D . The extreme multiplicative operators in P ( A , A ) were described in [6]. Rochberg proved in [8] that, if T is a member of P ( A , A ) which carries the identity on D into an extreme point of the unit ball of A , then T is multiplicative and is an extreme point of P ( A , A ). Rochbergs paper [9] is a study of certain extremal subsets of P ( A , A ), namely, those of the form K ( F , G ) = { T ∊ P ( A , A ): TF = G }, where F and G are inner functions in A . We proved in [5] that, if F is non-constant, then K ( F , G ) contains an extreme point of P ( A , A ).