By Christian Berg, Jens Peter Reus Christensen, Paul Ressel (auth.)

The Fourier remodel and the Laplace remodel of a favorable degree percentage, including its second series, a favorable definiteness estate which less than definite regularity assumptions is attribute for such expressions. this can be formulated in special phrases within the well-known theorems of Bochner, Bernstein-Widder and Hamburger. All 3 theorems might be seen as precise instances of a normal theorem approximately capabilities qJ on abelian semigroups with involution (S, +, *) that are confident convinced within the feel that the matrix (qJ(sJ + Sk» is confident convinced for all finite offerings of components St, . . . , Sn from S. the 3 simple effects pointed out above correspond to (~, +, x* = -x), ([0, 00[, +, x* = x) and (No, +, n* = n). the aim of this publication is to supply a remedy of those confident sure features on abelian semigroups with involution. In doing so we additionally speak about similar subject matters reminiscent of damaging yes features, thoroughly mono tone features and Hoeffding-type inequalities. We view those topics as very important materials of harmonic research on semigroups. it's been our goal, at the same time, to jot down a booklet that can function a textbook for a sophisticated graduate direction, simply because we consider that the suggestion of optimistic definiteness is a crucial and uncomplicated suggestion which happens in arithmetic as usually because the idea of a Hilbert space.

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**Example text**

H" ~ T(h) ~ + g(x) > = g(x) 0, = 0, g and hi + h" = h. implying T+(f) + T+(g) so that finally T+ is additive on C + (X). We put T- := T+ - T which also is additive, nonnegative and positively homogeneous on C+(X). 3 there are two Radon measures J11. J12 on X such that 41 §2. 3 we also get immediately that there is only one signed Radon measure Jl with this property. e. Jl+(B) = Jl(B n D) and Jl-(B) = Jl(B n DC) where D E ~(X) is chosen in such a way that both Jl+ and Jl- are nonnegative (see, for example, Billingsley (1979, p.

18. Theorem. , f3 E D and for each Borel set B <;;; G", n Gp. Then there is a uniquely determined Radon measure JI. ",(B) Borel set contained in G",. if B is a PROOF. Let C <;;; X be compact. We say that C = U~= 1 Ai is a decomposition of C if (Ai) is a finite family of pairwise disjoint Borel sets such that for each i = 1, ... i E D. 1' ... 17 there exist compact sets C i <;;; G"" with C = U~=I C i . Finally we put A I := C b A i := Ci\(C U··· u Ci- 1) for i = 2, ... , n. If we have two decompositions of a compact set C, C = U~= 1 Ai = 1 B j with Ai <;;; G"", Bj <;;; Gpi , then I Uj,: 31 §l.

E. we have the desired equality. ff(Y)} = *(A, B), using in the last equality once more that is a Radon bimeasure. We also see from the preceding argument that K is indeed uniquely determined from its values on products of compact sets (still assuming (X, Y) < 00). In the second step we abandon the finiteness restriction on <1>. For two compact sets K <;; X, L <;; Y we know that there is a uniquely determined KK,L E M +(K x L) such that K K, L(A x B) = (A, B) for all Borel sets A <;; K, B <;; L. *