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Books > Science & Mathematics > Mathematics > Algebra > Groups & group theory
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Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions (Paperback, Softcover reprint of the original 1st ed. 1984)
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Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions (Paperback, Softcover reprint of the original 1st ed. 1984)
Series: Graduate Texts in Mathematics, 100
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The Fourier transform and the Laplace transform of a positive
measure share, together with its moment sequence, a positive
definiteness property which under certain regularity assumptions is
characteristic for such expressions. This is formulated in exact
terms in the famous theorems of Bochner, Bernstein-Widder and
Hamburger. All three theorems can be viewed as special cases of a
general theorem about functions qJ on abelian semigroups with
involution (S, +, *) which are positive definite in the sense that
the matrix (qJ(sJ + Sk" is positive definite for all finite choices
of elements St, . . . , Sn from S. The three basic results
mentioned above correspond to (~, +, x* = -x), ([0, 00[, +, x* = x)
and (No, +, n* = n). The purpose of this book is to provide a
treatment of these positive definite functions on abelian
semigroups with involution. In doing so we also discuss related
topics such as negative definite functions, completely mono tone
functions and Hoeffding-type inequalities. We view these subjects
as important ingredients of harmonic analysis on semigroups. It has
been our aim, simultaneously, to write a book which can serve as a
textbook for an advanced graduate course, because we feel that the
notion of positive definiteness is an important and basic notion
which occurs in mathematics as often as the notion of a Hilbert
space.
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