Integral representation of bounded and absolutely integrable functions.
MSC 2010: Primary 33C45, 40A30; Secondary 26D07, 40C10In this article we establish a double definite integral representation, and two other indefinite integral expressions for a functional series and its derivative with members containing Jacobi polynomials.
Mathematical Subject Classification 2010: 35R11, 42A38, 26A33, 33E12.The method of integral transforms based on using a fractional generalization of the Fourier transform and the classical Laplace transform is applied for solving Cauchy-type problem for the time-space fractional diffusion equation expressed in terms of the Caputo time-fractional derivative and a generalized Riemann-Liouville space-fractional derivative.
Let denote the real-valued functions continuous on the extended real line and vanishing at . Let denote the functions that are left continuous, have a right limit at each point and vanish at . Define to be the space of tempered distributions that are the th distributional derivative of a unique function in . Similarly with from . A type of integral is defined on distributions in and . The multipliers are iterated integrals of functions of bounded variation. For each , the spaces...
Let be a natural number. Let and be real polynomials such that is not a square and has imaginary roots, if it is not linear. Effective methods for the integration of are exhibited.
This paper studies the attainable set at time T>0 for the control system showing that, under suitable assumptions on f, such a set satisfies a uniform interior sphere condition. The interior sphere property is then applied to recover a semiconcavity result for the value function of time optimal control problems with a general target, and to deduce C1,1-regularity for boundaries of attainable sets.
We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the...