Franz-Jürgen Delvos

Lacunary Interpolation by Antiperiodic Trigonometric Polynomials

AbstractThe problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(xk) = ak, DmT(xk) = bk, 0 ≤ k ≤ n − 1, where xk = kπ/n is a nodal set, ak and bk are prescribed complex numbers,

Zur Konstruktion von M-Splines höheren Grades

D = \frac{d}{{dx}}

Interpolation of Odd Periodic Functions on Uniform Meshes

and m ∈ N. Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular solvability is established when n as well as m are even. In this case a periodic solution does not exist.

Multivariate Boolean Trapezoidal Rules

ZusammenfassungDie vorliegende Arbeit liefert eine systematische Anwendung der von Gordon [4], [5] entwickelten Methode der Booleschen Approximation von Funktionen mehrerer Veränderlicher in der zweidimensionalen Lagrange-Interpolation. Es werden Interpolationsmethoden untersucht, deren Interpolationsprojektoren sich als K-fache (K ∈ ℕ) Boolesche Summe von Tensorprodukt-Lagrange-Interpolationsprojektoren darstellen lassen. Unter Benutzung bestimmter Eigenschaften dieser Booleschen Lagrange-Interpolationsprojektoren werden explizite Darstellungsformeln hergeleitet. Nachdem die klassische Biermann-Interpolation auf Dreieckgittern als ein Spezialfall der Booleschen Lagrange-Interpolation charakterisiert worden ist, wird eine Methode zur Konstruktion von Serendipity-Elementen beliebiger Ordnung angegeben, welche die von Gordon-Hall [6] vorgeschlagene Konstruktion spezieller Serendipity-Elemente systematisch erweitert. Ferner wird eine explizite Restgliedformel des Projektors der Booleschen Lagrange-Interpolation hergeleitet. Am Ende folgt eine Liste von verallgemeinerten Serendipity-Elementen der Ordnung N-1 (2≦N≦8).AbstractIn this paper the Boolean method for approximation of multivariate functions developed by Gordon [4], [5] is systematically applied to bivariate Lagrange interpolation. Interpolation methods are considered whose interpolation projectors can be characterized by K-times (K ∈ ℕ) Boolean sums of tensor product Lagrange interpolation projectors. Using certain properties of Boolean Lagrange interpolation projectors we derive explicit representation formulas for the interpolants. After showing that the classical Biermann interpolation on a triangular mesh is a special case of Boolean Lagrange interpolation a method for the construction of Serendipity elements of arbitrary order is presented. This method provides a systematic generalization of the construction of special Serendipity elements proposed by Gordon-Hall [6].Furthermore, we derive an explicit remainder representation formula for Boolean Lagrange interpolation. Finally, a list of generalized Serendipity elements of order N-1 (2≦N≦8) is presented.

On the convergence of periodic splines of arbitrary degree

ZusammenfassungEs wird eine rekursive bzw. explizite Konstruktion von M-Splines höheren Grades hergeleitet, die das von einem Differentialoperator gerader Ordnung mit konstanten Koeffizienten herrührende Skalarprodukt minimieren. Dieses Vorgehen nutzt die Kenntnis der Greenschen Funktion eines zum Operator gehörenden Randwertproblems aus. Da sich diese als der reproduzierende Kern des Energieraums des Operators erweist, lassen sich die Ergebnisse der optimalen Interpolation in Hilberträumen mit reproduzierendem Kern anwenden.AbstractIn this paper the recursive and explicit construction respectively of certain higher degree M-Splines corresponding to the inner product resulting from a differential operator of even order with constant coefficients is performed. This procedure makes essential use of the Greens function of a boundary value problem related to the operator. The Greens function turns out to be the reproducing kernel of the energy-space of the operator. Thus the results of optimal interpolation in Hilbert spaces having a reproducing kernel are applicable.

Periodic Area Matching Interpolation

Interpolation methods of Hermite type in translation invariant spaces of trigonometric polynomials for any position of interpolation points and any number of derivatives are constructed. For the case of an odd number of interpolation conditionsπ-periodic trigonometric polynomials of minimum order are chosen as interpolation functions while for the case of an even number of interpolation conditionsπ-antiperiodic trigonometric polynomials of minimum order are appropriate.

Bernoulli functions and periodic B-splines

Let g ϵ C2π have an absolutely convergent Fourier series. For n ϵ ℕ we define the uniform mesh tk = 2πk/n, k ϵ ℤ, and the translates gk = g(·−tk), 0 ≦ k < n, of g. Locher [4] presented a method of interpolation of periodic functions f at the uniform mesh tk, k ϵ ℤ, by functions h from the linear space Vn (g) = lin { g0,g1,...gn−1} of translates of g. Locher’s method is only applicable if Bk (0) ≠ 0 for k = 0,1,...,n−1 where the functions Bk, k = 0,1,...,n−1, are defined by \({B_k}(t)\;\, = \;\,\sum\limits_{j = 0}^{n - 1} {\;\,g(t - {t_j})\;\exp (ik{t_j})}\). In [1] we derived a modified method of interpolation by translation which is applicable under the hypothesis Bk(0) ≠ 0 for k=1,...,n−1. It is the objective this paper to develop a related method of interpolation of odd periodic functions which works under the assumption Bk (0) ≠ 0, 0 < k < m = n/2.

Remarks on Reduced Hermite Interpolation

Boolean methods of interpolation have been applied to the construction of bivariate and trivariate numerical integration formulas3,4. These formulas are comparable with lattice rules of multivariate numerical integration5,6. In this paper we will construct Boolean trapezoidal rules for multivariate numerical integration in arbitrary dimensions which are based on the ideas of multivariate Boolean interpolation2 and which extend bi- and trivariate results3’4. A detailed error investigation is presented using Boolean remainder formulas1.