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A Property of a Class of Functions Regular in the Unit Circle and a Theorem on Translations
Author(s) -
Edwards R. E.
Publication year - 1950
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-25.1.33
Subject(s) - mathematics , closure (psychology) , norm (philosophy) , banach space , normed vector space , class (philosophy) , pure mathematics , bounded function , discrete mathematics , unit circle , unit (ring theory) , continuous function (set theory) , space (punctuation) , function (biology) , combinatorics , mathematical analysis , linguistics , philosophy , mathematics education , artificial intelligence , evolutionary biology , political science , economics , computer science , law , market economy , biology
The symbol C is used to denote the class of functions φ( z ), ψ( z ), … of the complex variable z which are regular for | z | < 1 and bounded and continuous for | z | ⩽ 1. C is turned into a complex linear normed space by defining the norm ‖φ‖ =sup| x | ⩽ 1| φ ( z )|This paper is concerned with the problem of approximating (in the sense of the above norm) an arbitrary function of class C by the linear combinations of functions of the type φ(ζ z ), where φ( z ) is a given fixed function of class C and where the numbers ζ may be chosen froma given set E contained in the closed unit circle. The treatment is based upon some classical results of Banach on linear normed spaces. From the result (Theorem 1 below) concerning this problem is drawn a conclusion concerning the translations of an element of the space of boundary functions associated with C , and this conclusion is in turn compared with the “closure of translations” theorem for the space of continuous periodic functions.