Yitzhak Weit

Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenbergs theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii).

Ergodic and mixing properties of measures on locally compact abelian groups

We provide new proofs of the theorems of Choquet and Deny and of Foguel concerning iterates of convolutions of a measure on a locally compact abelian group. Let G be a locally compact abelian group, F its dual group and m its Haar measure. ,u denotes a bounded, complex, Borel measure on G with Fourier-Stieltjes transform Ai on 1F. The ergodic and mixing properties of probability measures ,u were characterized by Choquet and Deny [1] and by Foguel [2]. In this note we provide new proofs which are transparent from the point of view of harmonic analysis. In the concluding remark, the proof for the theorem of Choquet and Deny is extended naturally to compact nonabelian groups. Let Io denote {f E L1 (G): f(e) = 0}, where e is the identity of F and 1ttn is the n-times convolution of ,u. THEOREM 1. Let 1u be a bounded, complex Borel measure on G such that I,unI1 < c for all n. Then limnoo II,un * fj1j = 0 for f E Io if and only if jA(-y)j < 1 for all REMARK 1. If , is a probability measure, then the condition that jft(-y)j < 1 for e F \{e} is equivalent to Foguels condition that the support of , is not contained in any set of the form {x E G: (x, ^y) c} for any constant c and for ey E F\{e}. PROOF. Let I ={ fE Io: Iljn * fjJ1 0 as n -oo}. I is clearly a closed ideal in L1 (G). Because points of F obey synthesis, to prove I = Io it suffices to prove that Io is the only regular maximal ideal of L1 (G) that contains I [3, 39.29]. Stated more simply, for each ey E F\{e} it suffices to produce f E I such that f(^y) # O. Let ey E F\{e}. Choose K, a compact neighborhood of ^y excluding e. Choose h E L1(G) such that 0 < h < 1 hIK = 1, and the support of h is a compact set excluding e. Let f be any function in L1 (G) such that f is supported in K and f(ty) ? 0. Then (1) jjjt1 * fll = 1,tn * hn * flll ? IflflfjjI( * h)njI1. By the spectral radius theorem lim 11(,u * h)n II (l/n) SUpft I(A)h(y) C e F}. This last is less than 1 and therefore limn,oo jj(jt * h)jnj1 0. By (1) this implies that f E I, which completes the proof. Received by the editors November 23, 1983. 1980 Mathematics Subiect Classification. Primary 43A25, 60J15. (?1984 American Mathematical Society 0002-9939/84

On furstenberg’s characterization of harmonic functions on symmetric spaces

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Mean values and classes of harmonic functions

.25 per page 519 This content downloaded from 207.46.13.52 on Fri, 21 Oct 2016 05:43:18 UTC All use subject to http://about.jstor.org/terms 520 THOMAS RAMSEY AND YITZHAK WEIT For the ergodic properties we have the following simple proof of the theorem of Choquet and Deny. THEOREM 2. Let ,u be a bounded, complex, Borel measure on G. Then the following are equivalent: (i) for f E L.o(G), ,t * f = f implies f = constant. (ii) #(-i) i 1 for -y E F\{e}. REMARK 2. For ,tu a probability measure, the condition ,A(-y) i 1 for -y E F\{e} is equivalent to the fact that the support of 1u generates a dense subgroup of G; i.e. that 1u is adapted. PROOF. Let W = {f E Loo(G): ,u * f = f }. W is a w*-closed, translation invariant subspace of Loc(G). For x E F we have ,u * x = A(x)x = x. Therefore, x E F n W if and only if ,A(x) 1. Suppose that W consists only of constant functions (W Ce or W = {O}). Then there is at most one continuous character in W, the identity e. For all other elements of F we have --/(-) i 1. Conversely, suppose that (u(-y) i 1 for -y i e, -y E F. Then W can contain at most one continuous character, namely e. If e is not in W we may apply an L,,(G)-form of Wieners Theorem to conclude that W = {0} [3, 40.7]. If e is in W we must use again the fact that e, as a point of F, obeys synthesis. By [3, 40.23(b)], since W n r is exactly {e}, W = Ce. REMARK 3. Using the Peter-Weyl Theorem for compact (nonabelian) groups G, one may simply characterize minimal one-sided translation-invariant, w*-closed subspaces of Loc(G). By modifying the proof of Theorem 2 one can show that adapted probability measures are ergodic [4].

On Closed Ideals in the Motion Group Algebra

We provide a necessary and sufficient condition on a radial probability measureμ on a symmetric space for whichf =f *μ, f bounded, implies thatf is harmonic. In particular, we obtain a short and elementary proof of a theorem of Furstenberg which says that iff is a bounded function on a symmetric space which satisfiesf =f *μ for some radialabsolutely continuous probability measureμ, thenf is harmonic.

On a generalized asymptotic mean value property

Let μ be a finite complex Borel measure supported on the unit circle . In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form . and for all ξ ∈ , | ξ | = R for some fixed R > 0.

On Spectral Analysis in Locally Compact Motion Groups

A right closed ideal is said to be right-primary if it is contained in precisely one right maximal closed ideal. A set E C d//where E = h(k(E)), is said to be a set of spectral synthesis if I = k(E) is the only right closed ideal such that h(1)= E. For the Euclidean motion group M(2) we have proved in [4] that 0 is not a set of spectral synthesis. That is, the one-sided Wieners Tauberian Theorem does not hold for M(2). Our main purpose is to generalize this result by showing that for M(2) many subsets of ~ fail to be sets of spectral synthesis. In particular, every member of J// contains infinitely many right-primary ideals, whereas in the abelian case every closed primary ideal is maximal [3, p. 326]. Using the result in [4] and his methods in [-1, 2], Leptin (in a letter to the author) obtained the same results for generalized Ll-algebras (see Appendix).

Affine Invariant Subspaces of C( C )

SummaryIn this paper we study the spaces of continuous functionsf on ℝ2 satisfying(4)

Asymptotic Behavior of Dirichlet Series

\int_0^{2\pi } {f(z + \xi e^{r\theta } )} d\mu (\theta ) \to f(z)