On Riesz product measures ; mutual absolute continuity and singularity

Shelby J. Kilmer; Sadahiro Saeki

Annales de l'institut Fourier (1988)

  • Volume: 38, Issue: 2, page 63-93
  • ISSN: 0373-0956

Abstract

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We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz product measures. We shall establish a certain probabilistic dichotomy for such measures.

How to cite

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Kilmer, Shelby J., and Saeki, Sadahiro. "On Riesz product measures ; mutual absolute continuity and singularity." Annales de l'institut Fourier 38.2 (1988): 63-93. <http://eudml.org/doc/74803>.

@article{Kilmer1988,
abstract = {We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz product measures. We shall establish a certain probabilistic dichotomy for such measures.},
author = {Kilmer, Shelby J., Saeki, Sadahiro},
journal = {Annales de l'institut Fourier},
keywords = {absolute continuity; singularity; Riesz product measures; locally compact abelian groups; random Riesz product measures},
language = {eng},
number = {2},
pages = {63-93},
publisher = {Association des Annales de l'Institut Fourier},
title = {On Riesz product measures ; mutual absolute continuity and singularity},
url = {http://eudml.org/doc/74803},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Kilmer, Shelby J.
AU - Saeki, Sadahiro
TI - On Riesz product measures ; mutual absolute continuity and singularity
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 2
SP - 63
EP - 93
AB - We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz product measures. We shall establish a certain probabilistic dichotomy for such measures.
LA - eng
KW - absolute continuity; singularity; Riesz product measures; locally compact abelian groups; random Riesz product measures
UR - http://eudml.org/doc/74803
ER -

References

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  1. [1] G. BROWN and W. MORAN, On orthogonality for Riesz products, Proc. Cambridge Philos. Soc., 76 (1974), 173-181. Zbl0282.43001MR50 #2812
  2. [2] C. C. GRAHAM and O. C. MCGEHEE, Essays in Commutative Harmonic Analysis, Springer-Verlag (1979), New York, Heidelberg-Berlin. Zbl0439.43001MR81d:43001
  3. [3] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis I, Springer-Verlag (1963), New York, Heidelberg-Berlin. Zbl0115.10603
  4. [4] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis II, Springer-Verlag (1970), New York, Heidelberg-Berlin. Zbl0213.40103
  5. [5] E. HEWITT and H. S. ZUCKERMAN, Singular measures with absolutely continuous convolution squares, Proc. Cambridge Philos. Soc., 62 (1966), 399-420 ; Corrigendum, ibid., 63 (1967), 367-368. Zbl0148.38101
  6. [6] S. KAKUTANI, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224. Zbl0030.02303MR9,340e
  7. [7] M. LOÈVE, Probability Theory I, 4th ed., Springer-Verlag (1977), New York, Heidelberg-Berlin. Zbl0359.60001MR58 #31324a
  8. [8] J. M. LÓPEZ and K. A. ROSS, Sidon Sets, Marcel Dekker (1975), New York. Zbl0351.43008MR55 #13173
  9. [9] J. PEYRIÈRE, Sur les produits de Riesz, C. R. Acad. Sci., Paris Ser., A-B 276 (1973), 1417-1419. Zbl0258.43002MR47 #5512
  10. [10] L. PIGNO and S. SAEKI, Constructions of singular measures with remarkable convolution properties, Studia Math., 69 (1980), 133-141. Zbl0477.43003MR82h:43001
  11. [11] G. RITTER, Unendliche Produkte unkorrelierte Funktionen auf kompakten Abelschen Gruppen, Math. Scand., 42 (1978), 251-270. Zbl0398.42013MR80a:43001
  12. [12] G. RITTER, On Kakutani's theorem for infinite products of not necessarily independent functions, Math. Ann., 239 (1979), 35-53. Zbl0377.28004MR80g:60042
  13. [13] W. RUDIN, Fourier Analysis on Groups. Interscience Tract n° 12, Wiley (1962). New York. Zbl0107.09603MR27 #2808
  14. [14] J. L. TAYLOR, Measure Algebras. Regional Conference Series in Math., n° 16, Providence, R. I. Amer. Math. Soc. (1972). Zbl0269.28001
  15. [15] A. ZYGMUND, Trigonometric Series, 2 vols. Cambridge University Press (1959). Zbl0085.05601
  16. [16] A. PEYRIÈRE, Étude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier, Grenoble, 25-2 (1975), 127-169. Zbl0302.43003MR53 #8771

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