Fourier Transforms of Certain Classes of Integrable Functions

Citation

Ryan, Robert Dean (1960) Fourier Transforms of Certain Classes of Integrable Functions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GHJR-RD61. https://resolver.caltech.edu/CaltechETD:etd-06152006-085338

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Let G be a locally compact Abelian group with character group [...]. M(G) will denote the class of all bounded Radon measures on G and P(G) will denote the class of all continuous positive definite functions on G. For [...] we write [...] = [...] and for [...] we write [...] = [...]. [...] will denote the linear space spanned by [...]. We find necessary and sufficient conditions on [...] in order that [...] for [...]. Theorem 5, Chapter II: [...] for [...] if and only if there exists a constant K > 0 such that [...] for all [...] where [...]. Theorem 6, Chapter II: [...] for [...] if and only if [...] for all [...]. Theorems 3 and 4, Chapter III: [...] if and only if there exists some p, [...], such that for each [...] > 0 there exists a [...] > 0 with the property that [...] whenever [...] and [...]. By taking G to be the unit circle and p = 2 in Theorems 3 and 4, Chapter III, we obtain a generalization of a theorem by R. Salem (Comptes Rendus Vol. 192 (1931)). Taking G to be the additive group of reals and p = 1 gives a generalization of a theorem by A. Berry (Annals of Math. (2) Vol. 32 (1931)).

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