UBC Theses and Dissertations
Power series properties invariant under various permutation semi-groups Wick, Darrell Arne
A global analytic function is uniquely determined by one of its function elements. This function element is in turn completely determined by the coefficients of the Taylor series expansion about some point. Therefore, we should be able to determine all of the properties of the function from those of the coefficients and by the formal properties of Taylor series. Detection of these function properties from those of the coefficients is the central problem of the theory of Taylor series. Unfortunately this has proved to be an extremely difficult problem and, with the exception of simple cases, very little of the nature of a function is known from the properties of the coefficients. In Chapter One of this thesis, the central problem of the theory of Taylor series is approached. However, instead of a fixed sequence of coefficients, certain rearrangements of the order of the coefficients are also considered. Hopefully this relaxation will allow additional information to be detected. In particular, the P-preservation set consists of those rearrangements of the order of the coefficients which preserve a property P. This P-preservation set is maximal in the sense that each non-member 'maps' a function with property P onto a function without property P. The preservation sets (often groups—always semi-groups) for several function properties are found. This concept of property preservation also permits examination of those properties which are invariant under various permutation groups. This yields a division of the totality of all power series into subdivisions determined by subgroups of S[sub ω], the permutation group on the positive integers. In Chapter Two various summation processes, V, are used to sum the coefficients of a power series. For these V-summation processes, permutations which map V-summable series onto V-summable series are discussed. The program of Chapter One is continued, and the V-summability preservation sets are discussed. It is also shown that these V-summability preserving permutations leave the V-sum invariant. Finally, the results are collected in a dual lattice consisting of analytic function properties on one side and the corresponding preservation groups and semi-groups on the other side. This lattice certainly represents only a beginning. Hopefully it could lead to additional insight into analytic function theory or possibly even lead to information about the subgroups of S[sub ω].
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