(1) f(x) = x+o(x) . The problem is discussed explicitly by Landau ([7], pp . 597-604 ; [8]), Ingham ([3]), Karamata ([5], [6]), Gordon ([1]), and is implicit in the 'Eratosthenian' summation method introduced by Wintner ([10], [11]) . In this paper we consider the analogous problem in which the sequence {n} of all positive integers is replaced by a finite or infinite sequence 1, a,, a2, . . . o...

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In this article we prove a Wiener Tauberian theorem for L(SL2(R)), 1 ≤ p < 2. Let G be the group SL2(R) and K its maximal compact subgroup SO(2,R). Let M be {±I}. We show that if the Fourier transforms of a set of functions in L(G) do not vanish simultaneously on any irreducible Lp− -tempered representation for some > 0, where they are assumed to be defined, and if for each M-type at least one ...