Strong approximation of lacunary series with random gaps

We investigate the asymptotic behavior of sums (Formula presented.), where f is a mean zero, smooth periodic function on (Formula presented.) and (Formula presented.) is a random sequence such that the gaps (Formula presented.) are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps. © 2017 Springer-Verlag Wie

Channelopathies in Cav1.1, Cav1.3, and Cav1.4 voltage-gated L-type Ca2+ channels

Voltage-gated Ca2+ channels couple membrane depolarization to Ca2+-dependent intracellular signaling events. This is achieved by mediating Ca2+ ion influx or by direct conformational coupling to intracellular Ca2+ release channels. The family of Cav1 channels, also termed L-type Ca2+ channels (LTCCs), is uniquely sensitive to organic Ca2+ channel blockers and expressed in many electrically excitable tissues. In this review, we summarize the role of LTCCs for human diseases caused by genetic Ca2+ channel defects (channelopathies). LTCC dysfunction can result from structural aberrations within their pore-forming α1 subunits causing hypokalemic periodic paralysis and malignant hyperthermia sensitivity (Cav1.1 α1), incomplete congenital stationary night blindness (CSNB2; Cav1.4 α1), and Timothy syndrome (Cav1.2 α1; reviewed separately in this issue). Cav1.3 α1 mutations have not been reported yet in humans, but channel loss of function would likely affect sinoatrial node function and hearing. Studies in mice revealed that LTCCs indirectly also contribute to neurological symptoms in Ca2+ channelopathies affecting non-LTCCs, such as Cav2.1 α1 in tottering mice. Ca2+ channelopathies provide exciting disease-related molecular detail that led to important novel insight not only into disease pathophysiology but also to mechanisms of channel function

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

A summability process on Baskakov-type approximation

The summability process introduced by Bell (Proc Am Math Soc 38: 548-552, 1973) is a more general and also weaker method than ordinary convergence. Recent studies have demonstrated that using this convergence in classical approximation theory provides many advantages. In this paper, we study the summability process to approximate a function and its derivatives by means of a wider class of linear operators than a family of positive linear operators. Our results improve not only Baskakov's idea in (Mat Zametki 13: 785-794, 1973) but also the Korovkin theory based on positive linear operators. In order to verify this we display a specific sequence of approximating operators by plotting their graphs