Yasunobu Shiota

A generalization of Hata-Yamaguti’s results on the Takagi function

We shall generalize Hata and Yamaguti’s results on a system of infinitely many difference equations and on the connection between the Takagi function and Lebesgue’s singular function.

An explicit formula of the exponential sums of digital sums

We give a simple explicit formula of the exponential sum of digital sums by use of the distribution function of the binomial measure. We also apply our formula to the study of the power sums of digital sums.

A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case

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.

Heat Kernels on infinite graph networks and deformed Sierpinski gaskets

We shall calculate the local asymptotic decay order of heat kernels on general infinite graph networks. We shall also study deformed Sierpinski gaskets by introducing a new dimension closely related to the spectral dimension.

Hausdorff dimension of graphs of some Rademacher series

Our study is of the Hausdorff dimension and the packing dimension of graphs of Rademacher series whose coefficients form geometric progression.

Remarks on self-similarity

We will generalize Hata’s theorems on the self-similarity and show that the graph of the generalized Takagi function is a typical example of our result. We will also investigate the relation between self-affine functions and self-similar sets.