Sturm Liouville Problem

Here we present Hilfer-Polya, ψ-Hilfer Ostrowski and ψ-Hilfer-Hilbert-Pachpatte types fractional inequalities. They are univariate inequalities involving left and right Hilfer and ψ-Hilfer fractional derivatives. All estimates are with respect to norms ·p, 1≤p≤∞. At the end we provide applications.

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

In this paper we obtain quantitative weighted $$L^p$$ L p -inequalities for some operators involving Bessel convolutions. We consider maximal operators, Littlewood-Paley functions and variational operators. We obtain $$L^p(w)$$ L p ( w ) -operator norms in terms of the $$A_p$$ A p -characteristic of the weight w. In order to do this we show that the operators under consideration are dominated by a suitable family of sparse operators in the space of homogeneous type $$((0,\infty ),|\cdot |,x^{2\lambda }dx)$$ ( ( 0 , ∞ ) , | · | , x 2 λ d x ) .

Abstract. The measurement signals of the quadrature homodyne interfe rometers (say x and y, usually called Sine/Cosine signals and/or quadrature sig nals) typically exhibit offsets, unequal amplitudes and a phase difference that is not exactly 90 degree as would be expected in the ideal/theoretical case. Moreover, frequently there is a ignificant component of the measurement noise which is common to both signals (caused, e.g., by the amplitude noise of the laser), and as such, it results in a non vanishing correlatio n of the measured signals. Here we present a method for estimation of the unknown correlation co efficient from the observed data and suggest its implementation into the algorithm for demod ulation and evaluation of the amplitude noise related uncertainty contribution of correla ted quadrature interferometer signals, originally proposed by Köning, Wimmer and Witkovský in [ 2] for uncorrelated interferometer signals.

Theta Functions are not an easy subject of mathematics because new researches about this wide topic are still being published regularly. The findings about Theta Functions lead directly to many applications in different fields of applied mathematics and help also to develop old interesting mathematical topics such as Poisson summation and Riemann Zeta Function. This work proves by using a previously demonstrated useful approximation that the rational inputs of Theta-3 Functions give only irrationals as outputs of these functions. The proof is very simple and I invite consequently even the students of mathematics to understand the demonstrations of this article.

In the paper we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine–Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non–singular ones. We consider well–known classes of singular differential equations with Bessel operators, such as classical and generalized Euler–Poisson–Darboux equation and the generalized radiation problem of A.Weinstein. Methods of this paper are applied to more general linear partial differential equations with Bessel operators, such as multivariate Bessel–type equations, GASPT (Generalized Axially...