Theory of Functions. Part I: Foundations of the General Theory of Analytic Functions

Given a set of functions {p k {z)}, necessary and sufficient conditions are known under which the basic series ∑ (k=0) ∞ II k f (0)p k (z) will represent all functions f ( z ) in certain classes. The various cases are included in a general theory given in part II. Questions of uniqueness are discussed, and an attempt is made to initiate a theory of representation by series of the form ∑ (k=0) ∞ α k p k (z) which are not necessarily basic. Topological methods are used, and part I is devoted largely to preliminaries. In part III is discussed the relationship between given sets and various associated sets such as the inverse and product sets.

Download Full-text

XIV. On the convergence of infinite series of analytic functions

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character ◽

10.1098/rsta.1905.0014 ◽

1905 ◽

Vol 204 (372-386) ◽

pp. 481-497

Keyword(s):

Analytic Function ◽

General Theory ◽

Infinite Series ◽

Analytic Functions ◽

Hypergeometric Functions ◽

Point Of View ◽

General Point ◽

Associated Functions ◽

Linear Differential ◽

Series Of Functions

In the first section of the following work an attempt is made to deal with the convergence of infinite series of functions defined by linear differential equations of the second order from the most general point of view. Functions of Lamé Bessel and Legendre are considered as examples. In the second section the results obtained are applied to the expansion of an arbitrary uniform analytic function of an arbitrary uniform analytic function of z in a series of hypergeometric functions, and the expansion is shown to be valid if the function is regular within a certain ellipse in the z -plane. An expansion in a series of Legendre’s associated functions is deduced by a transformation. The method has been applied by the writer to other cases, but the foregoing offer adequate illustration of the general theory.

Download Full-text

A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions.

American Mathematical Monthly ◽

10.2307/2972291 ◽

1921 ◽

Vol 28 (4) ◽

pp. 176 ◽

Cited By ~ 1

Author(s):

E. T. Whittaker ◽

G. N. Watson

Keyword(s):

General Theory ◽

Analytic Functions ◽

Modern Analysis ◽

Transcendental Functions

Download Full-text

Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

Compositio Mathematica ◽

10.1112/s0010437x11005392 ◽

2011 ◽

Vol 147 (5) ◽

pp. 1581-1607 ◽

Cited By ~ 4

Author(s):

Heiko Gimperlein ◽

Bernhard Krötz ◽

Henrik Schlichtkrull

Keyword(s):

General Theory ◽

Representation Theory ◽

Lie Groups ◽

Lie Group ◽

Analytic Functions ◽

General Framework ◽

Analytic Representation ◽

Reductive Groups ◽

Topological Properties ◽

Rapid Decay

AbstractIn this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

Download Full-text