Circle Of Convergence Research Articles

Summability of power series on continuous arcs outside the circle of convergence

Soit f(z) une fonction analytique dans le domaine (z) < R'(R' > 1) à l'exception des points de l'axe réel de 1 à R' ; soit f(z) bornée en ces points sauf peut-être en z = 1, et f(z) = 0(| 1 — z|-a) quand z 1, où a < 1. Des conditions sont posées pour qu'un procédé de sommation A somme la série taylorienne de fm{z) à l'intérieur d'un cercle Do et en chaque point d'une ligne droite L partant de l'origine. Si le seul point singulier de f(z) est un pôle simple en z — 1, les conditions relatives à A peuvent être modifiées de façon que la série taylorienne de fp{z) soit A-sommable dans un ensemble D -f-{zT}, où D est un domaine ouvert, simplement connexe et contenant l'origine, et {zT} peut être un ensemble arbitraire de nombres complexes. On donne un exemple d'une méthode régulière qui somme toutes les séries binômes à l'intérieur du cercle de convergence et en tout point de l'axe réel négatif. D'autres exemples illustrent des méthodes qui somment la série géométrique sur des courbes ou sections de courbes isolées à l'extérieur du cercle unitaire.

Summability of power series at unbounded sets of isolated points

Regular matrices are constructed, each of which sums every binomial series at a countable infinity of isolated points outside the circle of convergence. Some of these matrices have the same property for the Taylor series of a class of meromorphic functions.

A converse theorem on overconvergence of sequences of partial sums

A number of authors ((2), (4), (5)) have recently considered problems relating to the convergence of specified sequences of partial sums of a gap Taylor series on its circle of convergence. For convenience in stating such results we shall standardize the Taylor series as Σanzn with the unit circle as circle of convergence, and denote the partial sums and the sum function by sn(z) and f(z) respectively. Following the notation of ((4)) and (5) we shall define φ(x) for 0 ≤ x < ∞ to be the least concave majorant of log (|an| + 2). One of the recent results in this field ((5), Theorem 1, and see also (2)) may be stated asTHEOREM A. Suppose that f(z) is regular on an arc α < arg z < β. Suppose also that there are sequences of integers nk, Nk → ∞ such that Nk – nk → ∞ and am = 0 if nk < m < Nk.

A theorem on power series whose coefficients have given signs

This theorem answers in the negative the question: Is there a universal scrambling {I ek}I Ei,k = ?1, turning every power series EakZk with positive coefficients into a power series >.-kakZk having the circle of convergence as natural boundary? This problem was raised by Mrs. Turan, and I am indebted to Dr. P. Erdos for communicating it to me. An example (?4) shows that the semi-circle in the statement of the theorem can not be replaced by a larger arc. A question which remains open is to find a corresponding theorem for the case that {11} is a given sequence of complex numbers of absolute value one.

Integrations on the circle of convergence and the divergence of interpolations, I

Integrations on the circle of convergence and the divergence of interpolations, I

On power series diverging everywhere on the circle of convergence.

On power series diverging everywhere on the circle of convergence.

The Fredholm Theory of theSMatrix

The equation for the interaction representation transformation operator is put into integral form and the Fredholm theory of integral equations is used to give an explicit expression for the solution. No extension of the circle of convergence of the $S$ matrix seems possible in the general case.

Values of entire functions represented by gap Dirichlet series

Introduction. In two previous notes [9],1 I stated some results which, in a general way, may be expressed as follows: If the Taylor series which represents an entire function satisfies a certain gap condition (which depends only on the order of F(z)), the zeros of F(z) -f(z) are not exceptional with respect to the proximate order of F(z) by any meromorphic function f(z) 3 oX of lower order. On the other hand, Mandelbrojt (see for instance [3, Theorem XXV]) proves that an entire function represented by a Dirichlet series takes each value a 0 oc, except at most one, in any horizontal strip of width greater than a quantity which depends only on the order (R) and on the upper density of the sequence of exponents of the series. In the present note we shall prove some results closely related to those of my above mentioned notes and to that of Mandelbrojt just quoted. These results may briefly be stated as follows: If an entire function F(s) can be represented by a Dirichlet series satisfying certain gap conditions, the zeros of F(s) -a cannot be exceptional with respect to the proximate order (R) of F(s) in any strip of width greater than a quantity, determined by the order, for every value of a $ oX without exception. In ?1 we shall deal with functions of finite order (R); in ?2, we shall give results concerning functions of infinite order (R). I think it will be of interest to remark that the theorems of the type of the well known Hadamard's theorem (which asserts that a Taylor series satisfying a specific gap condition cannot be continued analytically outside its circle of convergence) together with the results concerning relationship between gap properties and the position of the Julia lines, and again together with the results of my above mentioned notes and those contained in one of my papers [10], enable us to state the following general principle (without pretending that it holds for every case). By means of gap conditions in the Taylor series the disappearance of the possibility of the existence of the exceptional cases can be affirmed. Therefore, the content of this paper may be regarded as a link

Note on γ-Matrices Efficient at an Isolated Point

In a recent paper1 γ-matrices were constructed summing the binomial series at isolated points to its right value. It is assumed that the reader is able to refer to this paper and is familiar with its notation and terminology.The object of this note is to give the construction of γ-matrices which sum the binomial series at an isolated point to an arbitrary given value, while inside the circle of convergence the generalised sum is necessarily the right value.

On the summability of taylor series at isolated points outside the circle of convergence

On the summability of taylor series at isolated points outside the circle of convergence

A Note on the Summability of a Power Series on Its Circle of Convergence

A Note on the Summability of a Power Series on Its Circle of Convergence

Functions with Dominant Polar Singularities on the Circle of Convergence

1. Introduction. In an earlier paper1 the functionwas considered, having on its circle of convergence, taken to be |z|=1 only isolated essential points of finite exponential order, situated at the points eiav (v = 1, 2, .., k). It was there proved,2 in the cases k = 2, k = 3, that the upper density of small coefficients is positive only if the points eiav are situated at some of the vertices of a regular polygon inscribed in the circle of convergence and if the singularities are virtually identical or linearly related. In this case the sequence of small coefficients possesses a density. The coefficients cn can be interpolated in the formwhere the Gv(z) are integral functions of order less than 1, so that the result stated may be regarded as expressing an arithmetical property of a certain type of integral function at positive integer points.

Note on the Taylor Coefficients of a Function with Algebraic-Logarithmic Singularities on its Circle of Convergence

Note on the Taylor Coefficients of a Function with Algebraic-Logarithmic Singularities on its Circle of Convergence

The application of γ-matrices to Taylor series

In a recent paper some general properties of γ-matrices were proved and Dienes' theorem on regular γ-matrices extended to semiregular γ-matrices and the binomial series. In section 2 of this paper the previous results will be extended to certain classes of Taylor series. Section 3 gives some new results on Borel's exponential summation, and section 4 introduces matrices efficient for Taylor series on the circle of convergence and others efficient for Dirichlet series on the line of convergence. A knowledge of the definitions and results of the paper mentioned above is assumed.

Mocninné řady na obvodě konvergenčního kruhu

Mocninné řady na obvodě konvergenčního kruhu

A Power Series that Converges and Diverges at Everywhere Dense Sets of Points on its Circle of Convergence

It is known that the behaviour of a power series on the circumference of its circle of convergence can be very complicated.

Coefficient Density and the Distribution of Singular Points on the Circle of Convergence

Coefficient Density and the Distribution of Singular Points on the Circle of Convergence

On the Convergence and Summability of Power Series on the Circle of Convergence (II)

Proceedings of the London Mathematical SocietyVolume s2-47, Issue 1 p. 326-350 Articles On the Convergence and Summability of Power Series on the Circle of Convergence (II) A. Zygmund, A. Zygmund Mathematical Seminar of the University, Wilno, PolandSearch for more papers by this author A. Zygmund, A. Zygmund Mathematical Seminar of the University, Wilno, PolandSearch for more papers by this author First published: 1942 https://doi.org/10.1112/plms/s2-47.1.326Citations: 24AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-47, Issue11942Pages 326-350 RelatedInformation

On the determination of the singular point on the half-line through the center of the convergence circle

On the determination of the singular point on the half-line through the center of the convergence circle

On the convergence and summability of power series on the circle of convergence (I)

On the convergence and summability of power series on the circle of convergence (I)