Erik Hendriksen

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Introduction This monograph forms an introduction to the theory of orthogonal rational functions. The simplest way to see what we mean by orthogonal rational functions is to consider them as generalizations of orthogonal polynomials. There is not much confusion about the meaning of an orthogonal polynomial sequence. One says that f n g 1 n=0 is an(More)
Let R be the space of rational functions with poles among f k ; 1= k g 1 k=0 with 0 = 0 and j k j < 1, k 1. We consider a sequence fR n g 1 n=0 of nested subspaces with 1 n=0 R n = R. We continue our investigation of the convergence as n ! 1 of quadrature rules which are exact in R n. In Part II we have discussed the convergence for a particular nesting of(More)
Let R be the space of rational functions with poles among f k ; 1= k g 1 k=0 with 0 = 0 and j k j < 1, k 1. We consider a sequence fR n g 1 n=0 of nested subspaces with 1 n=0 R n = R. First we recall from part I how to nd orthogonal bases for R for a positive measure on the unit circle. These are used in the construction of interpolatory quadrature rules(More)
Tumour growth and spread of tumour cells requires angiogenesis. Incipient angiogenesis is not induced by tumour cell hypoxia but probably by proangiogenic factors. During growth tumours depend on a further induction of vascular development for adequate oxygen and nutrient supply. If the oxygen supply is insufficient, the resulting hypoxia stimulates(More)
We investigated expression of genes involved in the proteolytic process during epileptogenesis in a rat model of temporal lobe epilepsy (TLE). In a previous microarray study we found prominent activation of this process, which reached highest expression during the acute and latent phase (1 week after SE) in CA3 and entorhinal cortex (EC). Detailed analysis(More)
In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature(More)
We shall consider nested spaces L n , n = 0; 1; 2; : : : of rational functions with n prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive real measure on the unit circle. In the special case where all poles are placed at innnity, L n = n , the polynomials of degree at most(More)
We consider a positive measure on [0, ∞) and a sequence of nested spaces L 0 ⊂ L 1 ⊂ L 2 · · · of rational functions with prescribed poles in [−∞, 0]. Let {ϕ k } ∞ k=0 , with ϕ 0 ∈ L 0 and ϕ k ∈ L k \ L k−1 , k = 1, 2,. .. be the associated sequence of orthogonal rational functions. The zeros of ϕ n can be used as the nodes of a rational Gauss quadrature(More)