Abstract. We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node distributions that prevent such phenomena an...

• Asymptotics of the Lebesgue constant for Lagrange interpolation based on Lissajous-Chebyshev node points. partial sums Fourier series generated by anisotropically dilated rhombus. Formula Dirichlet kernel with frequencies in In this paper asymptotic formulas are given constants three special approximation processes related to ? 1 -partial series. particular, we consider polynomials points, rh...

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspher...

In this paper, a new algorithm for generating codebook is proposed for vector quantization (VQ) in image coding. The significant features of the training image vectors are extracted by using the proposed Orthogonal Polynomials based transformation. We propose to generate the codebook by partitioning these feature vectors into a binary tree. Each feature vector at a non-terminal node of the bina...

Though lots of research results about self-healing key management under unreliable links have been proposed, there are still some shortcomings, such as the inefficiency of broadcasts, lack of broadcast authentication, limited sessions for key issues, disastrous risks of access polynomials, and the vulnerability of collusion attacks. In this paper, we propose two modified access polynomial based...

In this paper, by means of a new recursive algorithm of non-tensor-producttyped divided differences, bivariate polynomial interpolation schemes are constructed over nonrectangular meshes firstly, which is converted into the study of scattered data interpolation. And the schemes are different as the number of scattered data is odd and even, respectively. Secondly, the corresponding error estimat...

In this paper we propose a method for conducting stochastic collocation on arbitrary sets of nodes. To accomplish this, we present the framework of least orthogonal interpolation, which allows one to construct interpolation polynomials based on arbitrarily located grids in arbitrary dimensions. These interpolation polynomials are constructed as a subspace of the family of orthogonal polynomials...

We obtain by elementary methods necessary and sufficient conditions for a k-dimensional cubature formula to hold for all polynomials of degree up to 2m− 1 when the nodes of the formula have Lagrange polynomials of degree at most m. The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m− 1 evaluated at the node plus an orthogona...

In this paper, we summarize our initial experiences in designing head-worn displays with free-form optical surfaces. Typical optical surfaces implemented in raytrace codes today are functions mapping two dimensional vectors to real numbers. The majority of optical designs to date have relied on conic sections and polynomials as the functions of choice. The choice of conic sections is justified ...