J. Morais2 Freiberg University of Mining and Technology, Institute of Applied Analysis, Prueferstr. 9, 09596 Freiberg, Germany Received 26 November, 2010; accepted in revised form 28 July, 2011 Abstract: During the past few years considerable attention has been given to the role played by monogenic functions in approximation theory. The main goal of the present paper is to construct a complete orthogonal system of monogenic polynomials as solutions of the Riesz system over prolate spheroids in R3 . This will be done in the spaces of square integrable functions over R. As a first step towards is that the orthogonality of the polynomials in question does not depend on the shape of the spheroids, but only on the location of the foci of the ellipse generating the spheroid. Some important properties of the system are briefly discussed.
E. Mainar and Juan Manuel Pena Departamento de Matematica Aplicada, Universidad de Zaragoza, Zaragoza, Spain Received 15 December, 2009; accepted in revised form 18 October, 2011 Abstract: We analyze some properties of bivariate tensor product bases. In particular, we obtain conditions so that a general bivariate tensor product basis is optimally stable for evaluation. Finally, we apply our results to prove the optimal stability of tensor product normalized B-bases.
C. Dagnino , V. Demichelis Department of Mathematics, Faculty of Scienze Matematiche Fisiche e Naturali, University of Torino, 10100 Torino, Italy Received 30 January, 2009; accepted in revised form 13 October, 2011 Abstract: We propose a new quadrature rule for Cauchy principal value integrals based on quadratic spline quasi-interpolants which have an optimal approximation order and satisfy boundary interpolation conditions. In virtue of these spline properties, we can prove uniform convergence for sequences of such quadratures and provide uniform error bounds. A computational scheme for the quadrature weights is given. Some numerical results and comparisons with other spline methods are presented.
Luigi Brugnano and Alessandra Sestini Dipartimento di Matematica “U. Dini” Viale Morgagni 67/A, 50134 Firenze, Italy Dedicated to Prof. D. Trigiante on the occasion of his 65th birthday. Received 16 December, 2009; accepted in revised form 24 September, 2011 Abstract: We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the numerical solution of free-surface problems. In particular, we here study their application to the numerical solution of both the (linear) parabolic obstacle problem and the obstacle problem. We propose a class of effective semi-iterative Newton-type methods to find the exact solution of such piecewise linear systems. We prove that the semi-iterative Newton-type methods have a global monotonic convergence property, i.e., the iterates converge monotonically to the exact solution in a finite number of steps. Numerical examples are presented to demonstrate the effectiveness of the proposed methods.