Δ h -Appell sequences and related interpolation problem

In this paper, we introduce poly-central factorial sequences and poly-central Bell polynomials arising from the polyexponential functions, reducing them to central factorials and central Bell polynomials of the second kind respectively when k = 1. We also show some relations: between poly-central factorial sequences and power of x; between poly-central Bell polynomials and power of x; between poly-central Bell polynomials and the poly-Bell polynomials; between poly-central Bell polynomials and higher order type 2 Bernoulli polynomials of second kind; recurrence formula of poly-central Bell polynomials.

The present paper contains a small survey on the principal and elementary methods for a systematic study of general polynomial sequences. For the set of polynomial sequences the algebraic structure of a group is given. Matricial forms, recurrence relations and conjugate sequences of polynomials are examined. For every element of a polynomial sequence determinant forms are determined by suitable Hessenberg matrices. Generating functions and derivation matrix are derived. Then, associated differential operator and Sheffer classification are considered. As an application, a general interpolation problem is hinted at. For a real valued function, the generalized Taylor polynomial and series are given. As an illustrative example, the shifted (with respect to the degree) Fibonacci polynomial sequence is considered.

A note on Appell sequences, Mellin transforms and Fourier series

The classical Caratheodory-Fejer and Nevanlinna-Pick interpolation problems have a long and distinguished history, appearing in a variety of applications in mathematics and electrical engineering. It is well-known that these problems can be solved in O(n<SUP>2</SUP>) operations, where n is the overall multiplicity of interpolation points. In this paper we suggest a superfast algorithm for solving the more general confluent tangential interpolation problem. The cost of the new algorithm varies from O(n log<SUP>2</SUP> n) to O(n log<SUP>3</SUP> n), depending on the multiplicity pattern of the interpolation points. The new algorithm can be used to factorize, invert, and solve a linear system of equations with confluent- Cauchy-like matrices. This class of matrices includes Hankel-like (i.e., permuted Toeplitz-like), Vandermonde-like and Cauchy-like matrices as special cases. An important ingredient of the proposed method is a new fast algorithm to compute a product of a confluent- Cauchy-like matrix by a vector.

The general rational interpolation problem in the scalar case and its Hankel vector

In this article we give necessary and sufficient conditions for logarithmic convexity of some sequences of Appell polynomials. Then we apply our results to Turan’s type polynomial inequalities. Precise upper and lower bounds for this class of polynomials are also determined and asymptotic behavior of

as well.

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.

We present a study of a specific kind of lowering operator, herein called $\Lambda$, which is defined as a finite sum of lowering operators, proving that this configuration can be altered, for instance, by the use of Stirling numbers. We characterize the polynomial sequences fulfilling an Appell relation with respect to $\Lambda$, and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are $\Lambda$-Appell, with $\Lambda$ defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also $\Lambda$-Appell, when $\Lambda$ is the lowering operator $\Lambda=a_{0}D+a_{1}DxD+a_{2}\left(Dx\right)^2D$, where $a_{0}$, $a_{1}$ and $a_{2}$ are constants and $a_{2} \neq 0$. The case where $a_{2}=0$ and $a_{1} \neq 0$ is also naturally recaptured.

A sequence of general Hermite trigonometric interpolation polynomials with equidistant interpolation points is given. Integrating these interpolation formulae a sequence of quadrature formulae for the integration of periodic functions is obtained. Derivative-free remainders are stated for these interpolation and quadrature formulae.

Using properties of the de Branges-Rovnyak spaces we include the classical lifting problem into the general scheme of the abstract interpolation problem.

We study a version of the classical Nevanlinna-Pick interpolation problem for Blaschke products where all interpolation points are located on the boundary of the unit disc. It turns out that all problems fall into three classes which are distinguished by the minimal degree of the interpolating function. The properties of the classes are studied in some detail. In particular it is shown that exactly one class consists of well-posed problems. An algorithm for classifying a given boundary interpolation problem is developed and a procedure for finding a solution with minimal degree for the generic class of regular problems is described.

To date, researchers usually use spectral and pseudospectral methods for only numerical approximation of ordinary and partial differential equations and also based on polynomial basis. But the principal importance of this paper is to develop the expansion approach based on general basis functions (in particular case polynomial basis) for solving general operator equations, wherein the particular cases of our development are integral equations, ordinary differential equations, difference equations, partial differential equations, and fractional differential equations. In other words, this paper presents the expansion approach for solving general operator equations in the form Lu+Nu=g(x),x∈Γ, with respect to boundary condition Bu=λ, where L, N and B are linear, nonlinear, and boundary operators, respectively, related to a suitable Hilbert space, Γ is the domain of approximation, λ is an arbitrary constant, and g(x)∈L2(Γ) is an arbitrary function. Also the other importance of this paper is to introduce the general version of pseudospectral method based on general interpolation problem. Finally some experiments show the accuracy of our development and the error analysis is presented in L2(Γ) norm.

A general interpolation problem with operator argument is studied for functions f from sub-Bergman spaces associated with an analytic function S mapping the open unit disk ⅅ into the closed unit disk.

A new quadrature rule based on a generalized mixed interpolation formula of exponential type

On the parametrization of the suboptimal solutions in generalized interpolation