Louis Pigno

Fourier-Stieltjes Transforms which vanish at infinity off certain sets

In this paper G is a nondiscrete compact abelian group with character group Г and M ( G ) the usual convolution algebra of Borel measures on G . We designate the following subspaces of M ( G ) employing the customary notations: M a ( G ) those measures which are absolutely continuous with respect to Haar measure; M S ( G ) the space of measures concentrated on sets of Haar measure zero and M d ( G ) the discrete measures.

Approximations to the norm of the singular part of a measure

Let G be a non-discrete LCA group with dual group Γ. Denote by M(G) the usual convolution algebra of bounded Borel measures on G and M a ( G ) those μ ∈ M(G) which are absolutely continuous with respect to m G —the Haar measure on G .

A remark on the F. and M. Riesz theorem

Let ,u be a measure of analytic type on the unit circle. We give a short direct proof that L2 is absolutely continuous with respect to Lebesgue measure; our method also gives a convolution product version of some related several variable results. Let T be the circle group, Z the integers and M(T) the complex-valued regular Borel measures on T. For , E M(T) and n E Z define ji(n) = fTe-in djI(O). Denote by Ma(T) those , E M(T) which are absolutely continuous with respect to Lebesgue measure on T. We now cite the theorem of F. and M. Riesz. THEOREM 1. Let p E M(T) such that p(n) = O for all n 0, N>O,~~~~~~~~~~~~~O Let dT = td,L where t is a trigonometric polynomial on T. An easy consequence of (1) is that limN, . IfITB(. -N = 0; this implies that if co E M(T) and c 0 be given. We gather from (2) that there is a measure /Ae and a positive integer N(e) satisfying II tLeII N(e). (3) Notice that because of (1) and the second part of (3), ,u * (p) is a trigonometric polynomial. Inasmuch as Ma(T) is closed, we conclude from (3) that lim{, * 0,-ye)} = U2 E Ma(T). Received by the editors August 3, 1979. 1980 Mathematics Subject Classification. Primary 43A75. i 1980 American Mathematical Society 0002-9939/80/0000-0367/

Transforms which almost vanish at infinity

0 1.50

On multipliers of Fourier transforms

Let be the circle group, M ( ) the set of bounded Borel measures on and ℤ the additive group of integers. If μ ∈ M ( ) and n ∈ ℤ, define A well-known result of Rajchman states that The following quantitative generalization of this result has been given in (2) by K. de Leeuw and Y. Katznelson.

Hardy's inequality and the L1 norm of exponential sums

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, L p ( G ) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C ( G ) the set of all bounded continuous complex-valued functions on G , and C 0 ( G ) the set of all f ∈ C ( G ) which vanish at infinity.