Gradimir V. Milovanović

Topics in polynomials : extremal problems, inequalities, zeros

General concept of polynomials elementary inequalities zeros of polynomials special classes of polynomials extremal problems for polynomials inequalities connected with trigonometric sums.

Numerical Calculation of Integrals Involving Oscillatory and Singular Kernels and Some Applications of Quadratures

Numerical methods for strongly oscillatory and singular functions are given in this paper. Beside a summary of standard methods and product integration rules, we consider a class of complex integration methods. Several applications of quadrature processes in problems in telecom- munications and physics are also presented. (~) 1998 Elsevier Science Ltd. All rights reserved. integration, Oscillatory kernel, Singular kernel, Orthogonal polynomials, Product rules, Gaussian quaclratures, Error function, Bessel functions, Legendre functions.

Polynomials orthogonal on the semicircle

Generalizing previous work [2], we study complex polynomials {πk},πk(z)=zk+⋯, orthogonal with respect to a complex-valued inner product (f,g)=∫0πf(eiθ)g(eiθ)w(eiθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫0πw(eiθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {πk} in the case of the Jacobi weight functionw(z)=(1−z)α(1+z)β, α>−1, and its special case\(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπn are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπn(z). It has regular singular points atz=1, −1, ∞ (like Gegenbauers equation) and an additional regular singular point on the negative imaginary axis, which depends onn.

On the convergence order of a modified method for simultaneous finding polynomial zeros

AbstractUsing Newtons corrections and Gauss-Seidel approach, a modification of single-step method [1] for the simultaneous finding all zeros of ann-th degree polynomial is formulated in this paper. It is shown thatR-order of convergence of the presented method is at least 2(1+τn) where τn∈(1,2) is the unique positive zero of the polynomial

Quadratures with multiple nodes, power orthogonality, and moment-preserving spline approximation

\tilde f_n (\tau ) = \tau ^n - \tau - 1

S -orthogonality and construction of Gauss-Turán-type quadrature formulae

. Faster convergence of the modified method in reference to the similar methods is attained without additional calculations. Comparison is performed in the example of an algebraic equation.ZusammenfassungIn dieser Arbeit wird eine Modifikation einer Einschritt-Methode [1] zur gleichzeitigen Ermittlung aller Nullstellen eines Polynomsn-ter Ordnung unter Verwendung des Gauss-Seidel-Vorgehens und Newtonscher Korrekturen vorgestellt. Es wird gezeigt, daß dieR-Ordnung der vorgestellten Methode mindestens 2(1+τn) beträgt, wobei τn∈(1,2) die eindeutige positive Wurzel des Polynoms

Moment-preserving spline approximation on finite intervals

\tilde f_n (\tau ) = \tau ^n - \tau - 1