Geno Nikolov

On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions

Gauss-Lobatto quadrature formulae associated with symmetric weight functions are considered. The kernel of the remainder term for classes of analytic functions is investigated on elliptical contours. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with either the real or the imaginary axis. The results obtained here are an analogue of some recent results of T. Schira concerning Gaussian quadratures.

Sharp bounds for the extreme zeros of classical orthogonal polynomials

Bounds for the extreme zeros of the classical orthogonal polynomials are obtained by a surprisingly simple method. Nevertheless, it turns out that, in most cases, the estimates obtained in this note are better than the best limits known in the literature.

Inequalities of Duffin--Schaeffer Type

We prove here that if an algebraic polynomial f of degree at most n has smaller absolute values than Tn (the nth Chebyshev polynomial of the first kind) at arbitrary n+1 points in [-1,1], which interlace with the zeros of Tn , then the uniform norm of

Gaussian quadrature of Chebyshev polynomials

f^{\prime}

On certain definite quadrature formulae

in [-1,1] is smaller than n2 . This is an extension of a classical result obtained by Duffin and Schaeffer.

Bivariate Polynomials of Least Deviation from Zero

Abstract We investigate the behaviour of the maximum error in applying Gaussian quadrature to the Chebyshev polynomials T m . This quantity has applications in determining error bounds for Gaussian quadrature of analytic functions.

On the L 2 Markov Inequality with Laguerre Weight

Gauss and Lobatto quadrature formulae related to spaces of cubic splines with double and equidistant knots are constructed. Such quadrature formulae are known as asymptotically optimal definite formulae of order 4. Some monotonicity results concerning the associated quadrature processes are established.

Numerical Methods and Applications

Bivariate polynomials with a fixed leading term xm yn, which deviate least from zero in the uniform or L 2 -norm on the unit disk D (resp. a triangle) are given explicitly. A similar problem in L p , 1 ≤ p ≤ ∞, is studied on D in the set of products of linear polynomials.

On the monotonicity of sequences of quadrature formulae

Let w α (t) = t α e−t, α > −1, be the Laguerre weight function, and \(\Vert \cdot \Vert _{w_{\alpha }}\) denote the associated L2-norm, i.e.,