Value indefinite observables are almost everywhere Alastair A. Abbott, 2, ∗ Cristian S. Calude, † and Karl Svozil 1, ‡ Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand Centre Cavaillès, CIRPHLES, École Normale Supérieure, 29 rue d’Ulm, 75005 Paris, France Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10...

We establish a connection between the L norm of sums of dilated functions whose jth Fourier coefficients are O(j−α) for some α ∈ (1/2, 1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L and for the almost everywhere convergence of series of dilated functions.

This is a brief and concise account of the basic concepts and results on statistical convergence, strong Cesàro summability and Walsh-Fourier series. To emphasize the significance of statistical convergence, for example we mention the fact that the one-dimensional Walsh-Fourier series of an integrable (in Lebesgue’s sense) function may be divergent almost everywhere, but it is statistically con...

In this paper we continue studying the smallest universal integral IS having S as the underlying semicopula. We present convergence theorems for IS-integral sequences including monotone, almost everywhere, almost uniform, in measure and in mean converging sequences of measurable functions, respectively. It emerges that these convergences characterize the underlying measure properties such as nu...

In this paper we prove that the maximal operator of the subsequence of logarithmic means of Walsh-Fourier series is weak type (1,1). Moreover, the logarithmic means tmn (f) of the function f ∈ L converge a.e. to f as n →∞. In the literature, it is known the notion of the Riesz’s logarithmic means of a Fourier series. The n-th mean of the Fourier series of the integrable function f is defined by...

Let be introduced the Sobolev-type inner product (f, g) = 1 2 Z 1 −1 f(x)g(x)dx + M [f ′(1)g′(1) + f ′(−1)g′(−1)], where M ≥ 0. In this paper we will prove that for 1 ≤ p ≤ 4 3 there are functions f ∈ L([−1, 1]) whose Fourier expansion in terms of the orthonormal polynomials with respect to the above Sobolev inner product are divergent almost everywhere on [−1, 1]. We also show that, for some v...

We introduce a new notion of algorithmic stability, which we call training stability. We show that training stability is sufficient for good bounds on generalization error. These bounds hold even when the learner has infinite VC dimension. In the PAC setting, training stability gives necessary and sufficient conditions for exponential convergence, and thus serves as a distribution-dependent ana...

We consider the differentiation of integrals of functions in Besov spaces with respect to the basis of arbitrarily oriented rectangular parallelepipeds in R. We study almost everywhere convergence with respect to Bessel capacities. These outer measures are more sensitive than n-dimensional Lebesgue measure, and therefore we improve the positive results in [4].