A characterization of compact multipliers

Authors:
Gregory F. Bachelis and Louis Pigno

Journal:
Trans. Amer. Math. Soc. **165** (1972), 319-322

MSC:
Primary 43A22; Secondary 43A25

MathSciNet review:
0300012

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Abstract: Let *G* be a compact abelian group and a complex-valued function defined on the dual . The main result of this paper is that is a compact multiplier of type and , if and only if it satisfies the following condition: Given there corresponds a finite set such that whenever and are trigonometric polynomials satisfying ( the conjugate index of *q*) and for . Using the above characterization we obtain the following necessary and sufficient condition for to be the Fourier transform of a continuous complex-valued function on *G*: Given there corresponds a finite set such that whenever is a trigonometric polynomial satisfying and for .

**[1]**Gregory F. Bachelis and John E. Gilbert,*Banach spaces of compact multipliers and their dual spaces*, Math. Z.**125**(1972), 285–297. MR**0338693****[2]**Raouf Doss,*Approximations and representations for Fourier transforms*, Trans. Amer. Math. Soc.**153**(1971), 211–221. MR**0268597**, 10.1090/S0002-9947-1971-0268597-9**[3]**Raouf Doss,*On the transform of a singular or an absolutely continuous measure*, Proc. Amer. Math. Soc.**19**(1968), 361–363. MR**0222569**, 10.1090/S0002-9939-1968-0222569-4**[4]**Raouf Doss,*On the Fourier-Stieltjes transforms of singular or absolutely continuous measures*, Math. Z.**97**(1967), 77–84. MR**0209769****[5]**R. E. Edwards,*Fourier series*:*A modern introduction*. II, Holt, Rinehart and Winston, New York, 1967. MR**36**#5588.**[6]**R. E. Edwards,*Criteria for Fourier transforms*, J. Austral. Math. Soc.**7**(1967), 239–246. MR**0216243****[7]**R. E. Edwards,*On factor functions*, Pacific J. Math.**5**(1955), 367–378. MR**0072433****[8]**G. I. Gaudry,*Quasimeasures and multiplier problems*, Doctoral Thesis, Australian National University, Canberra, Australia, 1966.**[9]**Henry Helson,*Isomorphisms of abelian group algebras*, Ark. Mat.**2**(1953), 475–487. MR**0058138****[10]**Walter Rudin,*Fourier analysis on groups*, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR**0152834**

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0300012-X

Article copyright:
© Copyright 1972
American Mathematical Society