By Van Der Merwe A. J., Du Plessis J. L.

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J, The characters of one-dimensional hypergroups often arise as the eigenfunctions of a Sturm-Liouville equation under certain boundary conditions. 2) 5 oo, where w E c( 1) (O,a) is a positive "weight" function and q E C(O,a) a "potential" function. Up to limiting cases, the differential equation is singular at the origin so that it is natural to pose the conditions u(O) = 1, u'(O) = 0. In view of a close relationship between the convolution * and the data of the Sturm-Liouville problem the corresponding hypergroup is referred to as a Sturm-Liouville hypergroup (cf.

There are many proofs of this result in the classical setting; that in Siebert(l978) is the most elegant and holds also Bloom some and Heyer(l982). straightforward We sketch the relationships details for hypergroups; see below but first consider between weak convergence of a net of bounded measures and the convergence of their Fourier transforms. 1 Theorem (a) f E p< 5 ) (KA) T satisfies If For r satisfying f 1 f a: = ~a Isupp we have " r - lim ~ I s upp - then f = and " . ~ "' (f"' ) C p<â€˘l T (KA) If (b) - ~ v lim f co .

7 the c Now consider the general case without the assumption that ~n E M8 (K) . For each f E G+(K) we have ~ * (f~) = (~ * f)~ , and taking Fourier c A An 1\ n transforms [ (~ * f)~] (D) = ~ (D)(f~) (D) for all D E K" . The assumption on th: convergence ofn (~ ) gives n lim<[(~ * f)~J"(D)u,v> = <[(~ * f)~]"(D)u,v> n for all that norm r w r DE K" , u,v E H(D) . 6 can continuity theorem transform. In ~ on KA . of a be considered inasmuch general It is as the restricted version of the Levy as it is assumed that the limit is a Fourier one has the convergence to an unspecified function not ~ is always the Fourier transform known whether bounded measure even in the case where K is a group.