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Glasgow Mathematical Journal | 1978

Fourier-Stieltjes Transforms which vanish at infinity off certain sets

Louis Pigno

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.


Proceedings of the Edinburgh Mathematical Society | 1976

Approximations to the norm of the singular part of a measure

Louis Pigno

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 .


Proceedings of the American Mathematical Society | 1980

A remark on the F. and M. Riesz theorem

Louis Pigno; Brent Smith

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/


Mathematical Proceedings of the Cambridge Philosophical Society | 1980

Transforms which almost vanish at infinity

Louis Pigno

0 1.50


Mathematical Proceedings of the Cambridge Philosophical Society | 1972

On multipliers of Fourier transforms

Louis Pigno

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.


Annals of Mathematics | 1981

Hardy's inequality and the L1 norm of exponential sums

O. Carruth McGehee; Louis Pigno; Brent Smith

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.


Mathematische Zeitschrift | 1975

Fourier-Stieltjes transforms which vanish at infinity

Louis Pigno; Sadahiro Saeki


Duke Mathematical Journal | 1974

Rosenthal sets and Riesz sets

Robert E. Dressler; Louis Pigno


Archiv der Mathematik | 1979

The two sides of a Fourier-Stieltjes transform

Patrick Gardner; Louis Pigno


Colloquium Mathematicum | 1974

On strong Riesz sets

Robert E. Dressler; Louis Pigno

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Brent Smith

University of Kentucky

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Sadahiro Saeki

Tokyo Metropolitan University

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