Topics in Classical Analysis and Applications in Honor of Daniel Waterman

The following sections are included:

• PREFACE

• CONTENTS

The following sections are included:

• REMINISCENCES

• RESEARCH
  • High Indices
  • Reflexivity and Summability
  • Harmonic Analysis
  • Change of Variable
  • Fourier Series and Generalized Variation
  • Representation of Functions, Orthogonal Series
  • Real Analysis
  • Summability
  • Survey Papers

• PUBLICATIONS

• DOCTORAL STUDENTS

We were asked by the organizers of the Waterman conference on which this commemorative volume is based to collect reminiscences for inclusion. As colleagues of Dan's during his thirty years at Syracuse University, we were a reasonable choice, but it was a daunting assignment in view of Dan's many doctoral students (19 at last count), friends (too numerous to count) and colleagues. However, we contacted all those we could locate and received the responses given below. Several more wished to contribute but time constraints prevented their doing so.

A sum of exponentials of the form f(x) = exp(2πiN₁x) + exp(2πiN₂x) + … + exp (2πiN_mx), where the N_k are distinct integers is called an idempotent trigonometric polynomial or, simply, an idempotent. It is known that for every p > 1, and every set S of the torus 𝕋 = ℝ/ℤ with │S│ > 0, there are idempotents concentrated on S in the L^p sense. We sketch how this concentration phenomenon originated as a reformulation of a functional analysis problem, and, in turn, studying concentration led to some interesting questions about L^p norms of Dirichlet kernels associated with multiple trigonometric series. Some counterexamples involving linear operators not of convolution type are given.

Let T be a nonempty subset of ℝ, X a metric space with metric d and X^T the set of all functions mapping T into X. Given ε > 0 and f ∈ X^T, we denote by N(ε, f, T) the least upper bound of those n ∈ ∕, for which there exist numbers s₁, …, s_n, t₁, …, t_n from T such that s₁ < t₁ ≤ s₂ < t₂ ≤ … < s_n < t_n and d(f(s_i), f(t_i)) > ε for all i = 1,…, n (N(ε, f, T) = 0 if there are no such n's). The following pointwise selection principle is proved: If a sequence of functions is such that the closure in X of the sequence is compact for each t ∈ T and lim sup_j→∞N(ε, f_j, T) < ∞ for all ε > 0, then contains a subsequence, converging pointwise on T to a function f ∈ X^T, such that N(ε, f, T) < ∞ for all ε > 0. We establish several variants of this result for functions with values in a metric semigroup and reflexive separable Banach space as well as for the weak pointwise and almost everywhere convergence of extracted subsequences, and comment on the necessity of conditions in the selection principles. We show that many Helly-type pointwise selection principles are consequences of our results, which can be applied to sequences of non-regulated functions, and compare them with recent results by Chistyakov [J. Math. Anal. Appl. 310 (2005) 609–625] and Chistyakov and Maniscalco [J. Math. Anal. Appl. 341 (2008) 613–625].

In this paper we prove new L^p estimates for Gegenbauer polynomials . We let be the measure in (−1, 1) which makes the polynomials orthogonal, and we compare the L^p(dµs) norm of with that of xⁿ. We also prove new L^p(dµs) estimates of the restriction of these polynomials to the intervals [0, z_n] and [z_n, 1] where z_n denotes the largest zero of

In this paper we study convergence results of different types (uniform, L_p, almost everywhere, etc.) for one- and multidimensional trigonometric series. The sufficient conditions for these results to hold are written for the series with general monotone coefficients. The sharpness of these results is examined.

Analogous to the use of sums of squares of certain real-valued special functions to prove the reality of the zeros of the Bessel functions J_α(z) when α ≥ −1, confluent hypergeometric functions ₀F₁(c > z) when c > 0, Laguerre polynomials when α ≥ −2, Jacobi polynomials when α ≥ −1 and β ≥ −1, and some other entire special functions considered in G. Gasper [Using sums of squares to prove that certain entire functions have only real zeros, in Fourier Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojević and C. V. Stanojević, eds. (Marcel Dekker, Inc., 1994) pp. 171–186.], integrals of squares of certain real-valued special functions are used to prove the reality of the zeros of the Pólya 𝚵*(z) function, the K_iz(a) functions when a > 0, and some other entire functions.

We start with a measure space L²[ℝ, dμ] and give a lower bound for the norm of functions in this space whose first N moments form a geometric progression. Several consequences are investigated including a new criterion for the determinacy of the moment problem. The corresponding questions on the unit circle are also investigated. In particular we give a lower bound for the L² norm of interpolatory functions in the disk algebra.

This is partly a survey article. We present a survey of results related with the characterization of scaling functions of multiresolution analyses. Given a linear invertible map A : ℝⁿ → ℝⁿ such that A(ℤⁿ) ⊂ ℤⁿ and all (complex) eigenvalues of A have absolute value greater than one we give a characterization of scaling functions of a frame multiresolution analysis on some closed subspaces of L²(ℝⁿ) denoted by . As a corollary we obtain that if the density number of the set G, |G|_n > 0 at the origin is less than one then no any function g ∈ L¹(ℝⁿ) can be a scaling function of an -FMRA associated with A. Putting together results obtained in [22], [12] one observes that for any measurable A*-invariant set G of positive measure always exists some -MRA associated with A.

We formulate and prove a version of the celebrated Coifman-Rochberg-Weiss commutator theorem for the real method of interpolation

There has recently been much interest in approximation of functions by m-term approximants with regard to a basis (or minimal system). We discuss the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where Λ ⊂ ℤ^d is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients of function f. Note that G_m(f) gives the best m-term approximant in the L₂-norm and, therefore, for each f ∈ L₂, ║f − G_m(f)║₂ → 0 as m → ∞. It is known from previous results that in the case of p ≠ 2 the condition f ∈ L_p does not guarantee the convergence ║ f − G_m(f)║_p → 0 as m → ∞. The main goal of this paper is to complement a survey [11] by a discussion of recent results in the following setting: find an additional (to f ∈ L_p) condition on f to guarantee that ║f − G_m(f)║_p → 0 as m→∞.

This paper surveys the results on general properties of functions of bounded Λ-variation in one variable, including some not yet published. The extensions to several variables and applications to Fourier series are not discussed. The relation between the class ΛBVof functions of bounded Λ-variation and its important subclass ΛBV_c of functions continuous in lambda-variation is discussed in detail. Open problems are described.

The following sections are included:

• AUTHOR INDEX