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On the Positive Definiteness of Certain Functions

✍ Scribed by Toreien Maack; Zoltán Sasvári

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
679 KB
Volume
186
Category
Article
ISSN
0025-584X

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✦ Synopsis

For a coinmutative senugoup (S, +, *) with involution and a function f : S 4 [O, m), the set S ( f ) of those p 2 0 such that f* is a positive definite function on S is a closed subsemigroup of [O, 00) containing 0. For S = (Hi, +, G* = -G) it may happen that S(f) = { kd : k E No } for some d>O,anditmayhappenthatS(f)={O}u[d,m)forsomed>O. I f a > 2 a n d i f S = ( Z , + , n * = -n ) and f ( n ) = e-lnlo or S = (NO, +,no = n ) and f ( n ) = e n U , then S(f) n (0, c) = 0 and [d, 00) C S(f) for some d 2 c > 0 . Although (with c maximal and d minimal) we have not been able to show c = d in all cases, this equality does hold if .S = z and a 2 3.4. In the last section we give sinipler proofs of previously known results concerning the positive definiteness of Ge-llzllo on normed spaces.

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