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Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions (Hardcover, 1984 ed.): C. Vandenberg,... Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions (Hardcover, 1984 ed.)
C. Vandenberg, J.P.R. Christensen, P. Ressel
R1,698 Discovery Miles 16 980 Ships in 10 - 15 working days

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.

Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions (Paperback, Softcover reprint of the... Harmonic Analysis on Semigroups - Theory of Positive Definite and Related Functions (Paperback, Softcover reprint of the original 1st ed. 1984)
C. Vandenberg, J.P.R. Christensen, P. Ressel
R1,525 Discovery Miles 15 250 Ships in 10 - 15 working days

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.

Potential Theory on Locally Compact Abelian Groups (Paperback, Softcover reprint of the original 1st ed. 1975): C. Vandenberg,... Potential Theory on Locally Compact Abelian Groups (Paperback, Softcover reprint of the original 1st ed. 1975)
C. Vandenberg, G Forst
R1,506 Discovery Miles 15 060 Ships in 10 - 15 working days

Classical potential theory can be roughly characterized as the study of Newtonian potentials and the Laplace operator on the Euclidean space JR3. It was discovered around 1930 that there is a profound connection between classical potential 3 theory and the theory of Brownian motion in JR . The Brownian motion is determined by its semigroup of transition probabilities, the Brownian semigroup, and the connection between classical potential theory and the theory of Brownian motion can be described analytically in the following way: The Laplace operator is the infinitesimal generator for the Brownian semigroup and the Newtonian potential kernel is the" integral" of the Brownian semigroup with respect to time. This connection between classical potential theory and the theory of Brownian motion led Hunt (cf. Hunt 2]) to consider general "potential theories" defined in terms of certain stochastic processes or equivalently in terms of certain semi groups of operators on spaces of functions. The purpose of the present exposition is to study such general potential theories where the following aspects of classical potential theory are preserved: (i) The theory is defined on a locally compact abelian group. (ii) The theory is translation invariant in the sense that any translate of a potential or a harmonic function is again a potential, respectively a harmonic function; this property of classical potential theory can also be expressed by saying that the Laplace operator is a differential operator with constant co efficients."

Potential Theory: Copenhagen 1979 - Proceedings of a Colloquium Held in Copenhagen, May 14-18, 1979 (English, French,... Potential Theory: Copenhagen 1979 - Proceedings of a Colloquium Held in Copenhagen, May 14-18, 1979 (English, French, Paperback, 1980 ed.)
C. Vandenberg, G Forst, B. Fuglede
R1,301 Discovery Miles 13 010 Ships in 10 - 15 working days
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