Mariko Yasugi

Effective properties of sets and functions in metric spaces with computability structure

We consider an abstract metric space with a computability structure and an effective separating set. In this article, we also introduce an effectively σ-compact space. The computability of real-valued functions on such a space is defined. It is shown that some of typical propositions in a metric space, namely Baire category theorem, Tietzes extension theorem and decomposition of unity, can be effectivized. It is also proved that computable functions are dense in continuous functions.

Metrization of the Uniform Space and Effective Convergence

The subject of the present article is the following fact. Consider an effective uniform space. A generally constructed metric from the uniformity has the property that a sequence from the space effectively converges with respect to the uniform topology if and only if it does with respect to the induced metric. This can be shown without assuming the computability of the metric.

Effective Fine‐convergence of Walsh‐Fourier series

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integrals such as the Bounded Convergence Theorem, the Dominated Convergence Theorem, and the Second Mean Value Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form a Euclidian computable sequence of reals which converges effectively to zero. This property of convergence is the effectivization of the Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version of Dirichlets test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Sequential Computability of a Function

We consider the real sequences in I=0,1) and real functions on I. A computability notion with respect to the uniformity {Un}, where Un(x)=k2n,k+12n) if x∈k2n,k+12n) will be called D-computability. An R-computable sequence from I will be shown to be approximated by a recursive sequence of rational numbers with a limiting recursive modulus of convergence with respect to {Un}. Using this result, we relate two extended notions of sequential computability of a function or a function sequence, one formulated in terms of limiting recursion and one in terms of {Un}.

Some Properties of the Effective Uniform Topological Space

We develop the theory of the computability structure and some notions of computable functions on a uniform topological space, and will apply the results to some real functions which are discontinuous in the Euclidean space.

Two Notions of Sequential Computability of a Function with Jumps

Abstract Given a strictly increasing computable sequence of real numbers (with respect to the Euclidean topology), one can define an effective uniform space of the real line, where the elements in the sequence are regarded as isolated. The relation between two notions of computability of real sequences, one with respect to the Euclidean space and one with respect to the uniform space as above, is discussed. As a consequence, we prove the equivalence of two notions of sequential computability of a function which is effectively uniformly continuous on the intervals between the given points and which may jump at those points.

Integral of Fine Computable functions and Walsh Fourier series

We define the effective integrability of Fine-computable functions and effectivize some fundamental limit theorems in the theory of Lebesgue integral such as Bounded Convergence Theorem and Dominated Convergence Theorem. It is also proved that the Walsh-Fourier coefficients of an effectively integrable Fine-computable function form an E-computable sequence of reals and converge effectively to zero. The latter fact is the effectivization of Walsh-Riemann-Lebesgue Theorem. The article is closed with the effective version of Dirichlets test.

Computability of Probability Distributions and Characteristic Functions

As a part of our works on effective properties of probability distributions, we deal with the corresponding characteristic functions. A sequence of probability distributions is computable if and only if the corresponding sequence of characteristic functions is computable. As for the onvergence problem, the effectivized Glivenkos theorem holds. Effectivizations of Bochners theorem and de Moivre-Laplace central limit theorem are also proved.

Fine-Continuous Functions and Fractals Defined by Infinite Systems of Contractions

Motivated by our study in [12] of the graph of some Fine-computable (hence Fine-continuous) but not locally uniformly Fine-continuous functions defined according to Brattkas idea in [2], we have developed a general theory of the fractal defined by an infinite system of contractions. In our theory, non-compact invariant sets are admitted. We note also that some of such fractals, including the graph of Brattkas function, are also characterized as graph-directed sets. Furthermore, mutual identity of graph-directed sets and Markov-self-similar sets is established.

A NOTE ON RADEMACHER FUNCTIONS AND COMPUTABILITY

We will speculate on some computational properties of the system of Rademacher functions f n g. The n-th Rademacher function n is a step function on the interval [0; 1), jumping at nitely many dyadic rationals of size 1 2 n and assuming values f1; 1g alternatingly.