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.

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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.

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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.

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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.

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