Margit Lenard

College Of Nyíregyháza · Mathematics and Computer Science

The regularity of a special lacunary interpolation problem is investigated, where for a given r (r≥2, r∈ℕ) the derivatives up to the (r-2)nd order together with the weighted rth derivative are prescribed at the nodes. Sufficient conditions on the nodes and the weight function, for the problem to be regular, are derived. Under these conditions a met...

The weighted (0,2)-interpolation with additional Hermite-type conditions is studied in a unified way with respect to the existence, uniqueness and representation (explicit formulae). Sufficient conditions are given on the nodes and the weight function, for the problem to be regular. Examples are presented on the zeros of the classical orthogonal po...

The main object of this paper is to construct a Birkhoff quadrature formula of the form ∫1−1f(x)dx≈∑i-1nAif′(xi)+∑j=0kCj(fj(1)+(−1)jf(j)(−1), which is exact for the polynomials of degree ⩽ 2n + 2k + 1. We construct the formula when the nodes {Xi}1n and {xi∗}1n−1 are the zeros of the ultraspherical polynomials Pn(k)(x) and Pn(k)′(x), respectively.

We consider the following Pál interpolation problem: On two sets of nodes (one consists of the zeros of a polynomial p n of degree n, while the elements of the other one are the zeros of p n ' ) different interpolation conditions are prescribed simultaneously. Weighted (0,1,⋯,r-2,r)-interpolation conditions (r≥2) are given on one of the sets of the...

The weighted (0,1,3)-interpolation with Hermite-type conditions is studied in a unified way with respect to the existence, uniqueness and representation (explicit formulae). Sufficient conditions are given on the nodes and the weight function, for the problem to be regular. Examples are presented on the zeros of the classical orthogonal polynomials...

The weighted (0,2)-interpolation is studied in a unified way with two additional interpolatory conditions. The question is how to choose the nodal points and the weight function w so that the problem is regular. We formulate sufficient conditions on the nodal points and on the weight function. In the regular cases find simple explicit forms of the...

In this paper two modified (0,2)-interpolation problems with boundary conditions are studied on the zeros of the ultraspherical polynomials. The aim of these constructions is to generalize Turán’s (0,2)-interpolation problem, in which the nodes are the zeros of the integrated Legendre polynomial. In the first paper of this series [Int. J. Appl. Mat...

In [4] A. M. Chak, A. Sharma and J. Szabados characterized the Jacobi matrices P(a,ß), (a,ß > -1) for which the (0,2)-interpolation problem is regular. It follows from their result, that if n is odd and a = ß, or if a, ß are both odd integers and n > 1 + (a + ß)/2, then the (0,2)-interpolation problem is not regular. Recently, the author proved tha...

In this paper two modified (0,2)-interpolation problems with boundary conditions are studied on the zeros of the ultraspherical polynomials. The aim of these constructions is to generalize Turán’s (0,2)-interpolation problem, in which the nodes are the zeros of the integrated Legendre polynomial. In this paper the existence and uniqueness of the in...

Let the set of knots $$ - 1 = x_{n + 1} < x_n < ... < x_1 < x_0 = 1 (n \geqq 1)$$ (n ? 1) be given on the interval [-1, 1]. Find a polynomial Qm(x) of minimal degree satisfying (0, 2)-interpolational conditions at the inner knots and boundary conditions at the endpoints, that is $$Q_m^{(s)} (x_i ) = y_i^{(s)} (s = 0,2) for i = 1,..., u_1$$ and $...

Let the set of the knots -1 = xn < xn* < xn-1 < xn-1* < ⋯ < x1 < x1* < x0 = 1 (n ≥ 2) be given on the interval [-1, 1]. Find a polynomial Qm(x) of minimal degree satisfying the (0;1) interpolation properties Qm(xi) = yi (i = 1, . . . , n-1), Q′m(xi*) = y′i (i = 1, . . ., n), with the boundary conditions Q(j)m(1) = αj (j = 0, . . ., k), Q(j)m(-1) =...

In this paper a multi-variate spline interpolational method on a rectangular grid is presented. The method is based on the use of a special continuous piecewise polynomial which is quadratic in each variable. In addition to approximation properties, the shape preserving characteristics and stability of the method have been proved.

We construct multiple quadrature formulae by integrating special spline functions in several variables. This spline construction, the reduced n-quadratic interpolation of Hermite-type is discussed by the author [J. Approximation Theory 68, No. 2, 113-135 (1992; Zbl 0757.41018)].

The present paper contains a construction of a two-variable spline function and convergence theorems concerning it. The advantage of this construction is its simplicity on the one hand, and that the sequence of spline functions and their derivatives converge uniformly “in the best way” on the other hand. It means, that the speed of convergence is j...