A Strong Converse of the Wiener-Levy Theorem

1962 ◽  
Vol 14 ◽  
pp. 694-701 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Unit Circle ◽  
Trigonometric Series ◽  
Fourier Coefficients ◽  
Complex Functions ◽  
Strong Converse ◽  
Convergent Trigonometric Series ◽  
Image Position

Let A1 denote the class of all complex functions on the unit circle which are sums of absolutely convergent trigonometric series; that is to say, the class of all f of the formSimilarly, if 1 < p < ∞, let Ap be the class of all complex functions f on the circle whose Fourier coefficientssatisfy the condition

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

The Representation of (C, k) Summable Series in Fourier Form

1978 ◽  
Vol 21 (2) ◽  
pp. 149-158 ◽  
Author(s):  
G. E. Cross
Keyword(s):  
Trigonometric Series ◽  
Point Of View ◽  
Perron Integral ◽  
Image Position

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series1.1

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

scholarly journals Radial growth and exceptional sets for Cauchy–Stieltjes integrals

1994 ◽  
Vol 37 (1) ◽  
pp. 73-89 ◽  
Author(s):  
D. J. Hallenbeck ◽  
T. H. MacGregor
Keyword(s):  
Unit Circle ◽  
Radial Growth ◽  
Borel Measure ◽  
Local Conditions ◽  
Exceptional Sets ◽  
Image Position ◽  
Complex Valued

This paper considers the radial and nontangential growth of a function f given bywhere α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

scholarly journals A Determinantal Expansion for a Glass of Definite Integral. Part 1

1953 ◽  
Vol 9 (1) ◽  
pp. 44-52 ◽  
Author(s):  
L. R. Shenton
Keyword(s):  
Weight Function ◽  
Fourier Coefficients ◽  
Finite Interval ◽  
Orthonormal System ◽  
Definite Integral ◽  
Negative Weight ◽  
Image Position

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficientsParseval's formula gives

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