Calculation of Improper Integrals Using (nα)-Sequences

A convergent series representation for the density of the supremum of a stable process

“…Results in [3] give us information about these products in the case when y ∈ Q and x / ∈ L, but we were not able to find any general results in the existing literature. In any case, even if one were able to prove convergence of the series (14) for a more general class of parameter α, it would not imply that the series converges to p(x), see the discussion on page 7.…”

On Weyl products and uniform distribution modulo one

“…For the case for some best approximation denominator q the product already was considered in [9, 39], and in much more general form in [3] (see also [37]). In particular, it follows from the results given there thatwhen q runs through the sequence of best approximation denominators.…”

Calculation of improper integrals using uniformly distributed sequences

“…It is the purpose of the present paper to describe generalizations of Theorem 1.1 without this assumption. Comparison with the paper's precursor [1] will show the reader that we have reused many ideas and Theorem 1.1 is contained in our results as a special case. However, our generalizations are far from straightforward and finding them required a careful analysis and further developments of the proofs given in [1].…”

“…In the case of sequences of shape ({kα}) k≥1 this condition is equivalent to the boundedness of the continued fraction expansion of α.) In a joint paper with J. Schoißengeier [1] we proved the following generalization of Oskolkov's result: In Theorem 1.1 we assumed that the singularities of the function f are all at rational points β and we made crucial use of this assumption in the proof of the implication (1)⇒(3). It is the purpose of the present paper to describe generalizations of Theorem 1.1 without this assumption.…”