Authors:El Abdalaoui EL HOUCEİN Abstract: We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the RR-action which assert that for any family of maps (Tt)t∈R(Tt)t∈R acting onthe Lebesgue measure space (Ω,A,μ)(Ω,A,μ), where μμ is a probability measure and for any t∈Rt∈R, TtTt is measure-preserving transformation on measure space (Ω,A,μ)(Ω,A,μ) with Tt∘Ts=Tt+sTt∘Ts=Tt+s, for any t,s∈Rt,s∈R. Then, for any f∈L1(μ)f∈L1(μ), there is a single null set off which $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t} dt$limT→+∞1T∫0Tf(Ttω)e2iπθtdtexists for all θ∈θ∈\RRR. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss. PubDate: Mon, 13 Dec 2021 00:00:00 +030
Authors:Anthony SOFO; Necdet BATİR Abstract: It is commonly known that integrals containing log-polylog integrands admit representations in termsof special functions such as the Dirichlet eta and Dirichlet beta functions. We investigate two parameterized familiesof such integrals and in a particular case demonstrate a connection with the Herglotz function. In the process of theinvestigation we recover some known Euler sum equalities and discover some new identities. PubDate: Mon, 13 Dec 2021 00:00:00 +030
Authors:Valdir MENEGATTO; Claudemir OLİVEİRA Abstract: We introduce a method to construct general multivariate positive definite kernels on a nonempty set XX that employs a prescribed bounded completely monotone functionand special multivariate functions on XX. The method is consistent with a generalized version of Aitken's integral formula for Gaussians. In the case in which XX is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation. In addition, it can be interpreted as an abstract multivariate version of the well-established Gneiting's model for constructing space-time covariances commonly highly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method. PubDate: Mon, 13 Dec 2021 00:00:00 +030
Authors:Jan-christoph SCHLAGE-PUCHTA Abstract: Hilberdink showed that a continuous prime system for which there exists a constant $A$ such that the function $N(x)-Ax$ is periodic satisfies $N(x)=c(x-1)+1$. He further showed that there exists a constant $c_0>2$, such that there exists a continuous prime system of this form if and only if $c\leq c_0$. Here we determine $c_0$ numerically to be $1.25479\cdot 10^{19}\pm2\cdot 10^{14}$. To do so we compute a representation for a twisted exponential function as a sum over the roots of the Riemann zeta function. We then give explicit bounds for the error obtained when restricting the occurring sum to a finite number of zeros. PubDate: Mon, 13 Dec 2021 00:00:00 +030
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