Abstract
We state an expansion theorem of continuous functions based on the multinomial measure, which is a generalization of the Schauder one, and then apply it to the study of a system of infinitely many difference equations. We also study the differentiability of the distribution function of the multinomial measure with respect to the parameters needed to define the multinomial measures.
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References
G. de Rham, Sur quelques courbes définies par des équations fonctionnelles. Rend. Sem. Mat. Torino,16 (1957), 101–113.
K. J. Falconer, Fractal Geometry. John Wiley, 1990.
M. Hata, On the structure of self-similar sets. Japan J. Appl. Math.,2 (1985), 381–414.
M. Hata and M. Yamaguti, The Takagi function and its generalization. Japan J. Appl. Math.,1 (1984), 183–199.
S. Kakutani, On equivalence of infinite product measures. Ann. of Math.,49 (1948), 214–224.
J. Kigami, Harmonic analysis on the Sierpinski gasket. Japan J. Appl. Math.,6 (1989), 259–290.
J. Kigami, Harmonic analysis on P.C.F. self-similar sets. Trans. Amer. Math. Soc.,335 (1993), 721–755.
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis.1, Graylock, Rochester, 1957, 99.
N. Kôno, On self-affine functions. Japan J. Appl. Math.,3 (1986), 259–269.
N. Kôno, On self-affine functions II. Japan J. Appl. Math.,5 (1988), 441–454.
K. Muramoto, T. Okada, T. Sekiguchi and Y. Shiota, Digital sum problems for thep-adic expansion of positive integers. In preparation.
T. Okada, T. Sekiguchi and Y. Shiota, Applications of binomial measures to power sums of digital sums. J. Number Theory,52 (1995), 256–266.
T. Okada, T. Sekiguchi and Y. Shiota, Explicit formulas of exponential sums of digital sums. Japan J. Indust. Appl. Math.,12 (1995), 425–438.
R. Salem, On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc.,53 (1943), 427–439.
T. Sekiguchi and Y. Shiota, A generalization of Hata-Yamaguti’s results on the Takagi function. Japan J. Indust. Appl. Math.,8 (1991), 203–219.
Y. Shiota, Remarks on self-similarity. Japan J. Appl. Math.,7 (1990), 171–181.
T. Takagi, A simple example of the continuous function without derivative. The Collected Papers of Teiji Takagi, Iwanami Shoten, 1973, 5–6.
M. Yamaguti, M. Hata and J. Kigami, Mathematics on Fractal. Iwanami Shoten, 1993. (In Japanese)
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Okada, T., Sekiguchi, T. & Shiota, Y. A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case. Japan J. Indust. Appl. Math. 13, 435–463 (1996). https://doi.org/10.1007/BF03167257
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DOI: https://doi.org/10.1007/BF03167257