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...

Formulas are given for n-point osculatory and hyperosculatory (as well as ordinary) polynomial interpolation for f(x), over ( —1, 1), in terms of fixi), f'(xi) and f"(x/) at the irregularly-spaced Chebyshev points x¡ = —cos {(2i — l)ir/2n}, i = 1, • • ■ , n. The advantage over corresponding formulas for Xi equally spaced is in the squaring and cubing, in the respective osculatory and hyperoscul...

properties of the hybrid of block-pulse functions and lagrange polynomials based on the legendre-gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known legendre interpolation operator. the uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. the applicabili...

In this report, we study in a detailed way higher order variances and quadrature Gauss Jacobi. Recall that the variance of order j measures the concentration of a probability close to j points x j,s with weight λ j,s which are determined by the parameters of the quadrature Gauss Jacobi. We shall study many example in which these measures specify adequately the distribution of probabilities. We ...

The problem of counting lattice points on a hyperboloid of two sheets is Gauss’ circle problem in hyperbolic geometry. The problem of counting lattice points on a hyperboloid of one sheet does not have the same geometric interpretation, and in general, the solution(s) to Gauss’ circle problem gives a lower bound, but not an upper bound. In this paper, we describe an exception. Given an ample he...

In a normed linear space X, consider a nonempty closed set K which has the property that for some r > 0 there exists a set of points xo € X\K, d(xoK) > r, which have closest points p(xo) € K and where the set of points xo — r((xo — p(xo))/\\xo — p(zo)||) is dense in X\K. If the norm has sufficiently strong differentiability properties, then the distance function d generated by K has similar dif...