J. Van Deun

Orthogonal rational functions and quadrature on an interval

Rational functions with real poles and poles in the complex lower half-plane, orthogonal on the real line, are well known. Quadrature formulas similar to the Gauss formulas for orthogonal polynomials have been studied. We generalize to the case of arbitrary complex poles and study orthogonality on a finite interval. The zeros of the orthogonal rational functions are shown to satisfy a quadratic eigenvalue problem. In the case of real poles, these zeros are used as nodes in the quadrature formulas.

Evaluation of a morphing based method to estimate muscle attachment sites of the lower extremity

To generate subject-specific musculoskeletal models for clinical use, the location of muscle attachment sites needs to be estimated with accurate, fast and preferably automated tools. For this purpose, an automatic method was used to estimate the muscle attachment sites of the lower extremity, based on the assumption of a relation between the bone geometry and the location of muscle attachment sites. The aim of this study was to evaluate the accuracy of this morphing based method. Two cadaver dissections were performed to measure the contours of 72 muscle attachment sites on the pelvis, femur, tibia and calcaneus. The geometry of the bones including the muscle attachment sites was morphed from one cadaver to the other and vice versa. For 69% of the muscle attachment sites, the mean distance between the measured and morphed muscle attachment sites was smaller than 15 mm. Furthermore, the muscle attachment sites that had relatively large distances had shown low sensitivity to these deviations. Therefore, this morphing based method is a promising tool for estimating subject-specific muscle attachment sites in the lower extremity in a fast and automated manner.

An interpolation algorithm for orthogonal rational functions

Let A={α1, α2...} be a sequence of numbers on the extended real line R^=R∪{∞} and µ a positive bounded Borel measure with support in (a subset of) R^. We introduce rational functions φn with poles {α1,...αn} that are orthogonal with respect to µ (if all poles are at infinity, we recover the polynomial situation). It is well known that under certain conditions on the location of the poles, the system {φn} is regular such that the orthogonal functions satisfy a three-term recurrence relation similar to the one for orthogonal polynomials.To compute the recurrence coefficients one can use explicit formulas involving inner products. We present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the Riesz-Herglotz-Nevanlinna transform Ωµ of the measure µ. Error bounds are derived and some examples serve as illustration.

Computing orthogonal rational functions with poles near the boundary

Computing orthogonal rational functions is a far from trivial problem, especially for poles close to the boundary of the support of the orthogonality measure. In this paper we analyze some of the difficulties involved and present two different approaches for solving this problem.