Now showing 1 - 10 of 25
  • Publication
    Cauchy, Goursat and Dirichlet problems for holomorphic partial differential equations
    (Springer, 2011-01)
    n this paper we survey recent results about Fischer decomposi- tions of polynomials or entire functions and their applications to holomorphic partial di erential equations. We discuss Cauchy and Goursat problems for the polyharmonic operator. Special emphasis is given to the Khavinson-Shapiro conjecture concerning polynomial solvability of the Dirichlet problem.
      660
  • Publication
    Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces
    We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0 , and a second function f 1 such that f 1 /f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C [ a,b ] → U n with strictly increasing nodes, fixing f 0 ,f 1 ∈ U n . If U n ⊂ U n +1 and U n +1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n +1 : C [ a,b ] → U n +1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 ,f 1 ,...,f n is a basis of U n such that the linear span of f 0 ,..,f k is an extended Chebyshev space over [ a,b ] for each k = 0 ,...,n , then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold B n f ≥ B n +1 f ≥ f for all ( f 0 ,f 1 )-convex functions f ∈ C [ a,b ] . Furthermore, B n f is ( f 0 ,f 1 )-convex for all ( f 0 ,f 1 )-convex functions f ∈ C [ a,b ] .
    Scopus© Citations 48  372
  • Publication
    Polyharmonic Hardy spaces on the complexified annulus and error estimates of cubature formulas
    The present paper has a twofold contribution: first, we intro- duce a new concept of Hardy spaces on a multidimensional complexified annular domain which is closely related to the annulus of the Klein-Di rac quadric important in Conformal Quantum Field Theory. Secondly, for functions in these Hardy spaces, we provide error estimate for the p oly- harmonic Gauß-Jacobi cubature formulas, which have been introduced in previous papers.
    Scopus© Citations 5  383
  • Publication
    Harmonic functions which vanish on a cylindrical surface
    (Elsevier, 2016-01-15) ;
    Suppose that a harmonic function h on a finite cylinder vanishes on the curved part of the boundary. This paper answers a question of Khavinson by showing that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. The existence of this extension is established by an analysis of the convergence properties of a double series expansion of the Green function of an infinite cylinder beyond the domain itself.
    Scopus© Citations 9  285
  • Publication
    Bernstein operators for exponential polynomials
    Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ 0 ,...,λ n . Assume that the set U n of all solutions of the equation Lf = 0 is closed under complex conjugation. If the length of the interval [ a,b ] is smaller than π/M n , where M n := max {| Im λ j | : j = 0 ,...,n } , then there exists a basis p n,k , k = 0 ,...n , of the space U n with the property that each p n,k has a zero of order k at a and a zero of order n − k at b, and each p n,k is positive on the open interval ( a,b ) . Under the additional assumption that λ 0 and λ 1 are real and distinct, our first main result states that there exist points a = t 0
    Scopus© Citations 20  335
  • Publication
    A reflection result for harmonic functions which vanish on a cylindrical surface
    Suppose that a harmonic function h on a finite cylinder U vanishes on the curved part A of the boundary. It was recently shown that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. This paper examines what can be said if the above function h is merely harmonic near A (and inside U). It is shown that h then has a harmonic extension to a larger domain formed by radial reflection.
      338Scopus© Citations 7
  • Publication
    Reproducing kernels for polyharmonic polynomials
    (Springer, 2008-10)
    The reproducing kernel of the space of all homogeneous polynomi- als of degree k and polyharmonic order m is computed explicitly, solving a question of A. Fryant and M.K. Vemuri.
      449Scopus© Citations 6
  • Publication
    On real-analytic recurrence relations for cardinal exponential B-splines
    Let LN+1 be a linear differential operator of order N + 1 with constant coefficients and real eigenvalues λ 1, ..., λ N+1, let E( N+1) be the space of all C∞-solutions of LN+1 on the real line.We show that for N 2 and n = 2, ...,N, there is a recurrence relation from suitable subspaces εn to εn+1 involving real-analytic functions, and with εN+1 = E(Λ N+1) if and only if contiguous eigenvalues are equally spaced.
    Scopus© Citations 2  279
  • Publication
    On the mixed Cauchy problem with data on singular conics
    We consider a problem of mixed Cauchy type for certain holomorphic partial differential operators with the principal part Q2p(D) essentially being the (complex) Laplace operator to a power, Δp. We provide inital data on a singular conic divisor given by P = 0, where P is a homogeneous polynomial of degree 2p. We show that this problem is uniquely solvable if the polynomial P is elliptic, in a certain sense, with respect to the principal part Q2p(D).
    Scopus© Citations 5  236
  • Publication
    Regularity of generalized Daubechies wavelets reproducing exponential polynomials with real-valued parameters
    We investigate non-stationary orthogonal wavelets based on a non-stationary interpolatory subdivision scheme reproducing a given set of exponentials with real-valued parameters. The construction is analogous to the construction of Daubechies wavelets using the subdivision scheme of Deslauriers-Dubuc. The main result is the existence and smoothness of these Daubechies type wavelets.
    Scopus© Citations 18  476