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

Ka-Sing Lau; Jian-Rong Wang; Cho-Ho Chu

Studia Mathematica (1995)

- Volume: 117, Issue: 1, page 1-28
- ISSN: 0039-3223

## Access Full Article

top## Abstract

top## How to cite

topLau, Ka-Sing, Wang, Jian-Rong, and Chu, Cho-Ho. "Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures." Studia Mathematica 117.1 (1995): 1-28. <http://eudml.org/doc/216238>.

@article{Lau1995,

abstract = {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.},

author = {Lau, Ka-Sing, Wang, Jian-Rong, Chu, Cho-Ho},

journal = {Studia Mathematica},

keywords = {Choquet-Deny theorem; convolution; exponential function; matrices; renewal equation; self-similar measures; -dimension; -density; locally compact abelian groups; vector-valued measures; convolution powers; Markov chains; theorem of Deny},

language = {eng},

number = {1},

pages = {1-28},

title = {Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures},

url = {http://eudml.org/doc/216238},

volume = {117},

year = {1995},

}

TY - JOUR

AU - Lau, Ka-Sing

AU - Wang, Jian-Rong

AU - Chu, Cho-Ho

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

JO - Studia Mathematica

PY - 1995

VL - 117

IS - 1

SP - 1

EP - 28

AB - 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.

LA - eng

KW - Choquet-Deny theorem; convolution; exponential function; matrices; renewal equation; self-similar measures; -dimension; -density; locally compact abelian groups; vector-valued measures; convolution powers; Markov chains; theorem of Deny

UR - http://eudml.org/doc/216238

ER -

## References

top- [B] M. Barnsley, Fractals Everywhere, Academic Press, 1988. Zbl0691.58001
- [BW] Y. Benyamini and Y. Weit, Harmonic analysis of spherical functions on SU(1,1), Ann. Inst. Fourier (Grenoble) 42 (3) (1992), 671-694. Zbl0763.43006
- [CM] R. Cawley and R. Mauldin, Multifractal decompositions of Moran fractals, Adv. in Math. 92 (1992), 196-236. Zbl0763.58018
- [CD] G. Choquet et J. Deny, Sur l'équation de convolution μ = μ ⁎ σ, C. R. Acad. Sci. Paris 250 (1960), 799-801. Zbl0093.12802
- [CL] C. H. Chu and K. S. Lau, Operator-valued solutions of the integrated Cauchy functional equation, J. Operator Theory 32 (1994), 157-183. Zbl0847.43001
- [Ch] K. Chung, A Course in Probability Theory, 2nd ed., Academic Press, 1974.
- [Ç] E. Çinlar, Introduction to Stochastic Processes, Prentice-Hall, 1975. Zbl0341.60019
- [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. Zbl0617.60016
- [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. Zbl0764.28007
- [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. Zbl0192.12704
- [H] J. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. Zbl0598.28011
- [K] J.-P. Kahane, Lectures on Mean Periodic Functions, Tata Inst., Bombay, 1959. Zbl0099.32301
- [La] S. Lalley, The packing and covering function of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), 699-709. Zbl0665.28005
- [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. Zbl0584.62019
- [LW] K. S. Lau and J. R. Wang, Mean quadratic variations and Fourier asymptotics of self-similar measures, Monatsh. Math. 115 (1993), 99-132. Zbl0778.28005
- [LZ] K. S. Lau and W. B. Zeng, The convolution equation of Choquet and Deny on semigroups, Studia Math. 97 (1990), 115-135. Zbl0719.43002
- [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. Zbl0706.28007
- [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. Zbl0607.62005
- [Sc] A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115. Zbl0807.28005
- [Sch] L. Schwartz, Théorie générale des fonctions moyennes-périodiques, Ann. of Math. 48 (1947), 857-929. Zbl0030.15004
- [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. Zbl0765.28007
- [Str3] R. Strichartz, Self-similar measures and their Fourier transformations III, Indiana Univ. Math. J. 42 (1993), 367-411. Zbl0790.28003
- [W] J. L. Wang, Topics in fractal geometry, Ph.D. Thesis, North Texas University, 1994.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.