Linear projections which implement balayage in Fourier transforms

Author:
George S. Shapiro

Journal:
Proc. Amer. Math. Soc. **61** (1976), 295-299

MSC:
Primary 43A25

MathSciNet review:
0427957

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a closed and discrete or compact subset of a second countable group and a subset of the dual group. Balayage is said to be possible for if for every finite measure on there is some measure on whose Fourier transform, , agrees on with .

If balayage is assumed possible just when is a point measure (with the norms of all the measures bounded by some constant), then there is a bounded linear projection, , from the measures on onto those on with on . An application is made to balayage in product groups.

**[1]**A. Beurling,*On balayage of measures in Fourier transforms*, Seminar notes, Institute for Advanced Study, Princeton, N.J., 1959-1960 (unpublished).**[2]**Arne Beurling,*Local harmonic analysis with some applications to differential operators*, Some Recent Advances in the Basic Sciences, Vol. 1 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1962–1964) Belfer Graduate School of Science, Yeshiva Univ., New York, 1966, pp. 109–125. MR**0427956****[3]**Jacques Dixmier,*Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann)*, Cahiers scientifiques, Fascicule XXV, Gauthier-Villars, Paris, 1957 (French). MR**0094722****[4]**N. Dunford and J. T. Schwartz,*Linear operators*. Part I, Interscience, New York, 1958. MR**22**#8302.**[5]**C. A. Rogers and R. C. Willmott,*On the uniformization of sets in topological spaces*, Acta Math.**120**(1968), 1–52. MR**0237733****[6]**M.-F. Sainte-Beuve,*On the extension of von Neumann-Aumann’s theorem*, J. Functional Analysis**17**(1974), 112–129. MR**0374364****[7]**G. S. Shapiro,*Some aspects of balayage of Fourier transforms*, Dissertation, Harvard Univ., 1973.**[8]**George S. Shapiro,*Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies*, Trans. Amer. Math. Soc.**225**(1977), 183–198. MR**0425510**, 10.1090/S0002-9947-1977-0425510-4**[9]**John von Neumann,*On rings of operators. Reduction theory*, Ann. of Math. (2)**50**(1949), 401–485. MR**0029101**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
43A25

Retrieve articles in all journals with MSC: 43A25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0427957-3

Keywords:
Balayage in Fourier transforms,
point measures,
measurable choice,
analytic set,
weak- integral,
bounded linear projection,
balayage in product groups

Article copyright:
© Copyright 1976
American Mathematical Society