Spectral synthesis in hypergroups

Spectral synthesis in products and quotients of hypergroups

“…can be defined as fΛx^ = I f(x l9 x 2 )y 2 (x 2 )dm(x 2 ) J m being the Haar The following results can be deduced from Theorem 3.11 and 4.5 above and ( [9], 3.7 and 3.8). 5.5.…”

“…The motivation for all this has been the fact that hypergroups arise in a natural way as a double coset space, the space of conjugacy classes of a compact group and harmonic analysis on them is closely related to that of the groups, a survey of this can be found in ( [18] and [6]). Our main reference for the basic theory of hypergroups will be [13] and most of the further notation and terminology is as in ([12], [9] and [7]). Throughout this paper K will denote a locally compact hypergroup (same as 'convo' in [13]) possessing a Haar measure m, L\K) -L\m) the convolution algebra and H a compact subhypergroup of K. In § 2 we discuss the quotient hypergroup K//H of double cosets and the Weil's formula for L\K//H), also we briefly describe the situation for the left coset space K/H as well.…”

“…By ( [13], 3.2G), ί fdp x *σ = ί fdp x *σ and now modify 14.IB of [13] using compactness of (x*H)l) (y*H) and ( [9], 2.5).…”

“…Malliavim proved that every nondiscrete locally compact abelian group G contains a nonspectral set for A(G). It is well known that Malliavin's theorem is not true for hypergroups because every closed subset in K is even strong Ditkin for the hypergroup K = Z+ related to p-adic numbers ([11], [9] and [7] There exists f ί2 eL\K^) such that f(y l9 τ 2 ) = Λ 2 (7i) for each 7χ ε K λ .…”

“…The basic lemmas of this section, ( [9], 2.5) and the techniques of ( [22], p. 419) can be further used to modify the proof of ( [14], 3.8) and hence have the following generalizations to hypergroups for p = 1* see also ([3], 2.5.5 (f)) ( [1 Proof. Let f in L\R n ) and ψ, in L~ (R n ) be the same as /, f in ( [15], § 2, Theorem 1) and g λ be a rotationally invariant function in L\R n ) such that g x > 0.…”