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  • 1
    Electronic Resource
    Electronic Resource
    Barycentric formulae for cardinal (SINC-)interpolants (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 703-718 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): Primary 65D05, 41A05 ; Secondary 42A15, 65D07 ; CR: G1.1
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present a barycentric representation of cardinal interpolants, as well as a weighted barycentric formula for their efficient evaluation. We also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and, in other cases, is even more accurate. In numerical examples, we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396489
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  • 2
    Electronic Resource
    Electronic Resource
    Barycentric Formulae for Cardinal (SINC-).Interpolants (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 747-747 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389340
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Baryzentrische Formeln zur trigonometrischen Interpolation (II) Stabilität und Anwendung auf die Fourieranalyse bei ungleichabständigen Stützstellen (1984)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 35 (1984), S. 193-205 
    Details
    ISSN: 1420-9039
    Keywords: AMS(MOS) ; 65D05 ; 65T05 ; 30C30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Résumé Dans la première partie, nous avons présenté des formules barycentriques pour l'interpolation trigonométrique. Ici, nous montrons que ces formules permettent une analyse de Fourier particuliérement simple et efficiente; leur seul inconvénient réside dans leur instabilité lorsque le nombre de noeuds croît, instabilité qui peut être évitée dans un cas particulier. Elles sont applicables à l'approximation de l'“inverse” d'une fonction périodique, par exemple de la fonction de correspondance des frontières en application conforme numérique.
    Notes: Summary In Part I we have presented barycentric formulas for trigonometric interpolation. Here we show that these formulas make it possible to calculate Fourier coefficients easily and efficiently. The only inconvenience is their instability when the number of interpolating points becomes large; this instability can be avoided in a special case. The formulas can be used to approximate the “inverse” of a periodic function, for instance of the boundary correspondence function in numerical conformal mapping.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00947932
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Baryzentrische Formeln zur Trigonometrischen Interpolation (I) (1984)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 35 (1984), S. 91-105 
    Details
    ISSN: 1420-9039
    Keywords: AMS (MOS) ; 65D05 ; 65T05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Résumé Des formules barycentriques pour l'interpolation d'une fonction périodiquef par un polynôme trigonométrique ont été données par Salzer [11] pour le cas d'un nombre impair de noeuds quelconques et par Henrici [7] dans le cas particulier de noeuds équidistants. Nous présentons ici des formules pour l'interpolation avec un nombre pair de noeuds quelconques, ainsi que des versions simplifiées pour une fonctionf paire ou impaire.
    Notes: Summary Barycentric formulas for the interpolation of a periodic functionf by a trigonometric polynomial have been given by Salzer [11] in the case of an odd number of arbitrary (interpolating) points and by Henrici [7] in the special case of equidistant points. We present here formulas for the interpolation with an even number of arbitrary points as well as simpler versions for an even or an odd functionf.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00945179
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  • 5
    Electronic Resource
    Electronic Resource
    Rational interpolation through the optimal attachment of poles to the interpolating polynomial (2000)
    Springer
    Numerical algorithms 23 (2000), S. 315-328 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; rational interpolation ; optimal interpolation ; 65D05 ; 41A05 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract After recalling some pitfalls of polynomial interpolation (in particular, slopes limited by Markov's inequality) and rational interpolation (e.g., unattainable points, poles in the interpolation interval, erratic behavior of the error for small numbers of nodes), we suggest an alternative for the case when the function to be interpolated is known everywhere, not just at the nodes. The method consists in replacing the interpolating polynomial with a rational interpolant whose poles are all prescribed, written in its barycentric form as in [4], and optimizing the placement of the poles in such a way as to minimize a chosen norm of the error.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019168504808
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  • 6
    Electronic Resource
    Electronic Resource
    A matrix for determining lower complexity barycentric representations of rational interpolants (2000)
    Springer
    Numerical algorithms 24 (2000), S. 17-29 
    Details
    ISSN: 1572-9265
    Keywords: interpolation ; rational interpolation ; barycentric representation ; barycentric weights ; complexity ; 65D05 ; 41A05 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Among the representations of rational interpolants, the barycentric form has several advantages, for example, with respect to stability of interpolation, location of unattainable points and poles, and differentiation. But it also has some drawbacks, in particular the more costly evaluation than the canonical representation. In the present work we address this difficulty by diminishing the number of interpolation nodes embedded in the barycentric form. This leads to a structured matrix, made of two (modified) Vandermonde and one Löwner, whose kernel is the set of weights of the interpolant (if the latter exists). We accordingly modify the algorithm presented in former work for computing the barycentric weights and discuss its efficiency with several examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019180807534
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  • 7
    Electronic Resource
    Electronic Resource
    A closed formula for the čebyšev barycentric weights of optimal approximation in H2 (1993)
    Springer
    Numerical algorithms 5 (1993), S. 155-163 
    Details
    ISSN: 1572-9265
    Keywords: Optimal approximation ; barycentric formula ; čebyšev points ; AMS(MOS) 65D05 ; 65D15 ; 41A20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The barycentric formula has several advantages over other means of evaluating the polynomial interpolating a function betweenn points in an interval. In particular, it is much more stable for sets of points clustered at the extremities of the interval, as are all the sets guaranteeing a good approximation forn sufficiently large. Also, it requires onlyO(n) operations for every function to be interpolated, once some weights, which depend only on the points, have been computed. Computing those weights usually requiresO(n2) operations; for čebyšev points, however,O(n) operations suffice. We show here that all the above is also true for the optimal evaluation of functionals in H2 by giving a closed formula for the corresponding weights.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02215678
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