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## Studia Mathematica

1995-1996 | 117 | 1 | 1-28
Tytuł artykułu

### Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Choquet-Deny theorem and Deny's theorem are extended to the vector-valued case. They are applied to give a simple nonprobabilistic proof of the vector-valued renewal theorem, which is used to study the $L^p$-dimension, the $L^p$-density and the Fourier transformation of vector-valued self-similar measures. The results answer some questions raised by Strichartz.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
1-28
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-01-25
poprawiono
1995-05-31
Twórcy
autor
• Department Of Mathematics, University Of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.
autor
• Houston Advanced Research Center, 4800 Research Forest Drive, The Woodlands, Texas 77381, U.S.A.
autor
• Goldsmiths' College, University Of London, London Se14, England
Bibliografia
• [B] M. Barnsley, Fractals Everywhere, Academic Press, 1988.
• [BW] Y. Benyamini and Y. Weit, Harmonic analysis of spherical functions on SU(1,1), Ann. Inst. Fourier (Grenoble) 42 (3) (1992), 671-694.
• [CM] R. Cawley and R. Mauldin, Multifractal decompositions of Moran fractals, Adv. in Math. 92 (1992), 196-236.
• [CD] G. Choquet et J. Deny, Sur l'équation de convolution μ = μ ⁎ σ, C. R. Acad. Sci. Paris 250 (1960), 799-801.
• [CL] C. H. Chu and K. S. Lau, Operator-valued solutions of the integrated Cauchy functional equation, J. Operator Theory 32 (1994), 157-183.
• [Ch] K. Chung, A Course in Probability Theory, 2nd ed., Academic Press, 1974.
• [Ç] E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, 1975.
• [DS] L. Davies and D. N. Shanbhag, A generalization of a theorem of Deny with application in characterization problems, Quart. J. Math. Oxford 38 (1987), 13-34.
• [D] J. Deny, Sur l'équation de convolution μ ⁎ σ = μ, Sém. Théor. Potent. M. Brelot, Fac. Sci. Paris 4 (1960).
• [EM] G. Edgar and R. Mauldin, Multifractal decompositions of digraph recursive fractals, Proc. London Math. Soc. 65 (1992), 196-236.
• [F] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990.
• [Fe] W. Feller, An Introduction to Probability Theory and its Applications, 3nd ed., Vol. 2, Wiley, New York, 1968.
• [Fu] H. Fürstenberg, Poisson formula for semi-simple Lie groups, Ann. of Math. 77 (1963), 335-386.
• [H] J. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
• [K] J.-P. Kahane, Lectures on Mean Periodic Functions, Tata Inst., Bombay, 1959.
• [La] S. Lalley, The packing and covering function of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), 699-709.
• [L1] K. S. Lau, Fractal measures and mean p-variations, J. Funct. Anal. 108 (1992), 421-457.
• [L2] K. S. Lau, Self-similarity, $L^p$-spectrum and multifractal formalism, preprint.
• [LR] K. S. Lau and C. R. Rao, Integrated Cauchy functional equation and characterizations of the exponential law, Sankhyā A 44 (1982), 72-90.
• [LW] K. S. Lau and J. R. Wang, Mean quadratic variations and Fourier asymptotics of self-similar measures, Monatsh. Math. 115 (1993), 99-132.
• [LZ] K. S. Lau and W. B. Zeng, The convolution equation of Choquet and Deny on semigroups, Studia Math. 97 (1990), 115-135.
• [Ma] B. Mandelbrot, The Fractal Geometry of Nature, Freeman, 1983.
• [MW] R. Mauldin and S. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811-829.
• [M] H. Minc, Nonnegative Matrices, Wiley, 1988.
• [RL] B. Ramachandran and K. S. Lau, Functional Equations in Probability Theory, Academic Press, 1991.
• [RS] C. R. Rao and D. N. Shanbhag, Recent results on characterizations of probability distributions: A unified approach through an extension of Deny's theorem, Adv. Appl. Probab. 18 (1986), 660-678.
• [Sc] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115.
• [Sch] L. Schwartz, Théorie générale des fonctions moyennes-périodiques, Ann. of Math. 48 (1947), 857-929.
• [S] E. Seneta, Nonnegative Matrices, Wiley, New York, 1973.
• [Str1] R. Strichartz, Self-similar measures and their Fourier transformations I, Indiana Univ. Math. J. 39 (1990), 797-817.
• [Str2] R. Strichartz, Self-similar measures and their Fourier transformations II, Trans. Amer. Math. Soc. 336 (1993), 335-361.
• [Str3] R. Strichartz, Self-similar measures and their Fourier transformations III, Indiana Univ. Math. J. 42 (1993), 367-411.
• [W] J. L. Wang, Topics in fractal geometry, Ph.D. Thesis, North Texas University, 1994.
Typ dokumentu
Bibliografia
Identyfikatory