Mathematics

In this paper, half inverse spectral problem for diffusion operator with jump conditions dependent on the spectral parameter and discontinuoty coeffcient is considered. The half inverse problems is studied of determining the coeffcient and two potential functions of the boundary value problem its spectrum by Hocstadt- Lieberman and Yang-Zettl methods. We show that two potential functions on the whole interval and the parameters in the boundary and jump conditions can be determined from spectrum.

A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition orthogonality. Finally, new sequences of orthogonal polynomials with respect to the weight function x α ρ 2 ν (x), ρ ν (x)=2 x ν/2 K ν (2 x − − √ ), x>0,ν≥0,α>−1 , where K ν (z) is the modified Bessel function or Macdonald function, are investigated. Differential properties, recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are obtained. The corresponding multiple orthogonal polynomials are exhibited.

We present a general approach to sparse domination based on single-scale L p -improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hytönen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of R n .

We give a negative answer to Ulam's Problem 19 from the Scottish Book asking, is a solid of uniform density which will float in water in every position a sphere? Assuming that the density of water is 1 , we show that there exists a strictly convex body of revolution K??R 3 of uniform density 1 2 , which is not a Euclidean ball, yet floats in equilibrium in every direction. We prove an analogous result in all dimensions d?? .

Recently, Brezis, Van Schaftingen and the second author established a new formula for the W ? 1,p norm of a function in C ??c ( R N ) . The formula was obtained by replacing the L p ( R 2N ) norm in the Gagliardo semi-norm for W ? s,p ( R N ) with a weak- L p ( R 2N ) quasi-norm and setting s=1 . This provides a characterization of such W ? 1,p norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula. In this paper, we obtain an analog for the case s=0 . In particular, we present a new formula for the L p norm of any function in L p ( R N ) , which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on L p ( R N ) , which complements a formula by Maz\cprime ya and Shaposhnikova. As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space F s,p 2 ( R N ) (i.e. the Bessel potential space (I?��?) ?�s/2 L p ( R N ) ), as well as its homogeneous counterpart F ? s,p 2 ( R N ) , for s??0,1) , p??1,?? .

We present a q-extension of the duality relation for the generalized hypergeometric functions established recently by the second and the third named authors which also generalizes the q -hypergeometric identity due to the third named author (jointly with Feng and Yang). This duality relation has the form of a reduction formula for a sum of products of basic hypergeometric functions r ϕ r−1 . We further derive a confluent version of our formula for t ϕ r−1 with t<r , which involves a mixture of two types of basic hypergeometric functions. Our identity is closely related to some recent results due to Yamaguchi on contiguous relations for 2 ϕ 1 basic hypergeometric function.

In this article, we generalize a theorem of Victor L. Shapiro concerning nontangential convergence of the Poisson integral of a L p -function. We introduce the notion of ? -points of a locally finite measure and consider a wide class of convolution kernels. We show that convolution integrals of a measure have nontangential limits at ? -points of the measure. We also investigate the relationship between ? -point and the notion of the strong derivative introduced by Ramey and Ullrich. In one dimension, these two notions are the same.

In this paper we study approximation theorems for L 2 -space on Damek-Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek-Ricci spaces. We also prove Nikolskii-Stechkin inequality. To prove these inequalities we use functions of bounded spectrum as a tool of approximation. Finally, as an application, we prove equivalence of the K -functional and modulus of smoothness for Damek-Ricci spaces.

In this note, we aim to provide generalizations of (i) Knuth's old sum (or Reed Dawson identity) and (ii) Riordan's identity using a hypergeometric series approach.

The aim of this note is to provide a natural extension of Gelfond's constant e π using a hypergeometric function approach. An extension is also found for the square root of this constant. A few interesting special cases are presented.