Bochner’s theorem on Fourier-Stieltjes integrals in the framework of quaternion analysis

2012 ◽  
Author(s):  
S. Georgiev ◽  
J. Morais
Keyword(s):  
Bochner’S Theorem ◽  
Bochner's Theorem

Bochner's Theorem and the Hausdorff Moment Theorem on foundation Topological Semigroups

1985 ◽  
Vol 37 (5) ◽  
pp. 785-809 ◽  
Author(s):  
M. Lashkarizadeh Bami
Keyword(s):  
Harmonic Analysis ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Commutative Semigroups ◽  
Bochner’S Theorem ◽  
Topological Semigroups ◽  
Bochner's Theorem ◽  
Extensive Class

One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.

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