Nonanalytic Functions of Absolutely Convergent Fourier Series

Rudin, Walter

doi:10.1073/pnas.41.4.238

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“…D'autre part, un certain nombre de résultats « négatifs » a été découvert par Kahane et Rudin. Dans [9] et [10] Kahane construit une fonction /e 0Lr telle |/| ^=0L,.. Rudin [23] a montré indépendam-ment qu'il existe une fonction non négative/€=(9L,., telle que \/f^^r' Un autre résultat de Rudin [24] concernant les fonctions qui « opèrent » sur les coeffi-^ M. Y. KATZNELSON. cients de Fourier des fonctions sommables entraîne que les fonctions qui opèrent sont nécessairement Lipschitziennes d'ordre i. Que cette condition n'est pas suffisante résulte d'une construction de Kahane.…”

Section: /(^)Ea=»f(/(^))ea(t)unclassified

Sur le calcul symbolique dans quelques algèbres de Banach

Katznelson¹

1959

Ann. Sci. École Norm. Sup.

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Section: /(^)Ea=»f(/(^))ea(t)unclassified

Sur le calcul symbolique dans quelques algèbres de Banach

Katznelson¹

1959

Ann. Sci. École Norm. Sup.

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1990

Fourier Analysis on Groups

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Measure algebras on abelian groups

Rudin¹

1959

Bull. Amer. Math. Soc.

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The recent developments in the general field of Fourier analysis which I wish to describe, illustrate the algebraic point of view which has established itself here as well as in most other parts of analysis, and which has been most fruitful in suggesting new and interesting problems. Typically, a question will pose itself in algebraic terms, will then be transformed into a problem which can be attacked by the methods of classical analysis, and the algebraic aspect will reappear in the solution.To cite just one example now, one may ask for the set of all automorphisms of the group algebra of the circle group ; this turns out to be equivalent to the problem of finding all permutations of the set of all integers which carry Fourier series into Fourier series; the solution of this analytical problem involves arithmetic progressions, i.e., cosets of the subgroups of the additive group of the integers, and the group theoretic nature of the problem becomes apparent (the result is described in 3.3(d) below). This is a large subject, and I will restrict myself to those topics with which I have had close personal contact. This choice will, of course, result in the omission of much that is interesting. In particular, I will say nothing about what is perhaps the most intriguing of the open problems in Fourier analysis: the problem of spectral synthesis, or, equivalently, the problem of finding all closed ideals in LKG).I. Preliminaries. 1.1. Unless the contrary is explicitly stated, any group mentioned in the sequel will be abelian and locally compact, with addition as group operation. Associated with every such group G there is a measure m (not identically 0 or GO), the so-called Haar measure of G, with the following properties [6; 7; 21; 3l]: m is a non-negative completely additive set function defined for all Borel sets in G, which is regular (i.e., m(E) = sup m(K) =inf rn(V), where K ranges over all compact subsets of E and V ranges over all open supersets of £), and which is translation invariant:

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Nonanalytic Functions of Absolutely Convergent Fourier Series

Cited by 3 publications

References 0 publications

Sur le calcul symbolique dans quelques algèbres de Banach

Sur le calcul symbolique dans quelques algèbres de Banach

Bibliography

Measure algebras on abelian groups

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