A case of distinction between Fourier integrals and Fourier series

1927 ◽  
Vol 23 (7) ◽  
pp. 755-767
Author(s):  
Margaret Eleanor Grimshaw
Keyword(s):  
Fourier Series ◽  
Finite Type ◽  
Generating Function ◽  
Fourier Integral ◽  
Finite Range ◽  
Fourier Integrals

A Fourier integral is said to be of finite type if its generating function vanishes for all sufficiently large values of ¦x¦. Because the coefficient functions are defined by integrals over a finite range, the behaviour of such a Fourier integral usually resembles closely that of the corresponding series.

The Summation of a Fourier Integral of Finite Type

1926 ◽  
Vol 23 (4) ◽  
pp. 373-382 ◽  
Author(s):  
S. Pollard
Keyword(s):  
Finite Type ◽  
Generating Function ◽  
Finite Interval ◽  
Fourier Integral ◽  
Image Position

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.

Summation of the integral conjugate to the Fourier integral of finite type

1927 ◽  
Vol 23 (8) ◽  
pp. 871-881
Author(s):  
Margaret Eleanor Grimshaw
Keyword(s):  
Finite Type ◽  
Generating Function ◽  
Finite Interval ◽  
Fourier Integral ◽  
Image Position

1. The complete Denjoy-Fourier integral is of the formwherethe generating function ƒ(t) being integrable in the general Denjoy sense in every finite interval, and the integrals (1·2) being interpreted as principal values or in some other suitable way.

On uniform asymptotic expansions of finite Laplace and Fourier integrals

1980 ◽  
Vol 85 (3-4) ◽  
pp. 299-305 ◽  
Author(s):  
Kusum Soni
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Fourier Integral ◽  
Uniform Asymptotic Expansion ◽  
Laplace Integrals ◽  
Uniform Asymptotic Expansions ◽  
Fourier Integrals

SynopsisA uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.

scholarly journals Homeomorphisms of S1 and Factorization

2019 ◽  
Author(s):  
Mark Dalthorp ◽  
Doug Pickrell
Keyword(s):  
Fourier Series ◽  
Finite Type ◽  
Parameter Family ◽  
Complex Parameter ◽  
Triangular Factorization ◽  
Linear Fractional Transformations ◽  
Root Subgroup ◽  
Decay Properties ◽  
Group Of Homeomorphisms ◽  
Verblunsky Coefficients

Abstract For each $n>0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations “conjugated by $z \to z^n$”. We show that these families are free of relations, which determines the structure of “the group of homeomorphisms of finite type”. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.

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