R. P. Boas

Polynomial expansions of analytic functions

This is an excellent little book written with a commendable attention to detail. The authors take pains to indicate frequently what is going on behind the scenes and they include numerous pertinent and useful illustrations of the theory. The book contains much new material in addition to an organized treatment of known phenomena. The work is motivated by a desire to examine in some detail the fact that certain polynomial sets yield nontrivial representations of zero. Boas and Buck find that, for a specified polynomial set, (a) the nonexistence of such representations of zero, or (b) the existence and enumeration of distinct representations of zero can be deduced from properties of generating functions of certain forms. A very few misprints were noted, none of them likely to disturb the reader. Chapter I (about 20 pages) contains motivation of the study and some underlying function theoretic results. A set of pseudo Laguerre polynomials (superscript dependent upon the subscript) is used in an illustration. A fairly general class of generating relations previously studied by Boas and Buck is introduced and the corresponding polynomials are given the name generalized Appell polynomials. This class of polynomials is to play a major role in the succeeding chapters. On page 18 two problems, which have long confronted workers in the study of generating relations for polynomial sets, are pointed out to the reader. Chapter II (about 27 pages) is a study of the representation of entire functions by series of generalized Appell polynomials. There are determinations of the existence of convergent, Mittag-Leffler summable, or Borel summable expansions in such series. Numerous useful illustrations are given. These include polynomials associated with names Bernoulli, Euler, pseudo Laguerre, reversed proper Laguerre (superscript independent of the subscript), Hermite, and a rearranged form of Legendre polynomials. Additional application of much interest is made to the Sheffer polynomials (Sheffers type zero classification) which include proper Laguerre polynomials, those of Angelescu, Mittag-Leffler and Newton, the actuarial polynomials and some others associated with interpolation problems. Boas and

Partial Sums of Infinite Series, and How They Grow

Introduction. The four chapters of this article are separate informal essays on topics connected with the problem of finding out, as accurately as possible, how fast a convergent series converges and how fast a divergent series diverges. Chapter I is a general discussion with numerical examples. Chapter II is a self-contained development of the Euler-Maclaurin formula, the formula that lets us approximate the partial sums of series (if the series have sufficiently simple structure) by integrals, with errors that can be estimated. The Euler-Maclaurin formula, in its more sophisticated versions, involves the Bernoulli numbers, and Chapter III develops the properties of these numbers to the point where some of their interesting applications can be made. In Chapter IV I show how sums of some convergent series, the Euler constants for some divergent series, and good approximate formulas for partial sums can be actually calculated numerically. I hope that some of this material, which is really quite elementary, may find its way into the undergraduate curriculum, perhaps in place of some of the dull and not particularly useful material that is traditionally there. I am much indebted to John W. Wrench, Jr., who provided me with some essential numerical data, as well as advice in my first ventures into numerical analysis; and to Leonard Evens, who has helped me learn to talk to the computer without its scolding me too often. I have used portions of these essays as talks at section meetings of the Association and elsewhere, as well as in my retiring Presidential Address to the Association, and I have profited from useful comments made by members of my audiences.

Inequalities Involving a Function and Its Inverse

The paper presents a simple technique for establishing a class of inequalities, some of which arise in connection with

Entire functions of exponential type

\varepsilon

A remark on principal‐value integrals

-entropy and its applications in probability, and which include a eneralization of Young’s inequality.

Infinite Processes: Background to Analysis. By A. Gardiner

it is immaterial which value of z is used in (2). If (1) holds in a region of the s-plane, for example in an angle, ƒ(z) is said to be of exponential type c in that region. Functions of exponential type have been extensively studied, both for their own sake and for their applications. I shall discuss here a selection of their properties, chosen to illustrate how the restriction (1) on the growth of a function restricts its behavior in other ways.

The exceptional integral values of the index

Many physical problems lead to integrals that can be evaluated by contour integration, and sometimes there are poles on the contour. In practice, it usually is appropriate in such cases to express the result as a principal‐value integral. Most textbooks treat each problem involving principal values individually by considering an indented contour and taking limits. We want to call the attention of users of the calculus of residues to a theorem which, although known, seems not to be well‐known.

L P problems, 1 < p < ∞

(1983). Infinite Processes: Background to Analysis. By A. Gardiner. The American Mathematical Monthly: Vol. 90, No. 9, pp. 651-653.

More general classes of functions; conditional convergence

Whereas the theorems of §§ 3 and 4 involve the function and the coefficients symmetrically when the index γ is not an integer, and sometimes when it is, there are a number of asymmetric theorems, and some missing ones, for integral values of γ, usually at values differing by 2. In this section we shall summarize the known facts and point out some connections with other aspects of the theory of trigonometric series, mentioning also a number of results that do not involve absolute convergence.

Asymptotic formulas and Lipschitz conditions

We now consider conditions for x−γϕ(x) to belong to L P , 1 < p < ∞, where ϕ stands for f or g and λ n are its associated Fourier coefficients.