Now showing 1 - 10 of 25
  • Publication
    A Characterization of the Khavinson-Shapiro Conjecture Via Fischer Operators
    (Springer, 2016-10)
    The Khavinson-Shapiro conjecture states that ellipsoids are the only bounded domains in euclidean space satisfying the following property (KS): the solution of the Dirichlet problem for polynomial data is polynomial. In this paper we show that property (KS) for a domain Ω is equivalent to the surjectivity of a Fischer operator associated to the domain Ω.
      236Scopus© Citations 6
  • Publication
    Extension results for harmonic functions which vanish on cylindrical surfaces
    The Schwarz reflection principle applies to a harmonic function which continuously vanishes on a relatively open subset of a planar or spherical boundary surface. It yields a harmonic extension to a predefined larger domain and provides a simple formula for this extension. Although such a point-to-point reflection law is unavailable for other types of surface in higher dimensions, it is natural to investigate whether similar harmonic extension results still hold. This article describes recent progress on such results for the particular case of cylindrical surfaces, and concludes with several open questions.
      478Scopus© Citations 2
  • 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 ] .
      376Scopus© Citations 48
  • Publication
    Harmonic divisors and rationality of zeros of Jacobi polynomials
    (Springer, 2013-08)
    Let Pn (α,β ) ( x ) be the Jacobi polynomial of degree n with parameters αβ The main result of the paper states the following: If b≠ 1 ; 3 and c are non-zero rel- atively prime natural numbers then P ( k +( d 3) = 2 ;k +( d 3) = 2) n p b=c 6 ≠ 0 for all natural numbers d;n and k 2 N 0 : Moreover, under the above assumption, the polynomial Q ( x ) = b c x 2 1 + ::: + x 2 d 1 + b c 1 x 2 d is not a harmonic divisor, and the Dirichlet problem for the cone f Q ( x ) < 0 g has polynomial harmonic solutions for polynomial data functions.
      400
  • 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.
      282Scopus© Citations 2
  • Publication
    The Goursat problem for a generalized Helmholtz operator in the plane
    (Springer, 2008-09) ;
    We consider the Goursat problem in the plane for partial differential operators whose principal part is the pth power of the standard Laplace operator. The data is posed on a union of 2p distinct lines through the origin. We show that the solvability of this Goursat problem depends on Diophantine properties of the geometry of lines on which the data is posed.
      424Scopus© Citations 6
  • Publication
    Polyharmonic functions of infinite order on annular regions
    (Tohoku University. Mathematical Institute., 2013-06) ;
    Polyharmonic functions f of in nite order and type on annular regions are systematically studied. The rst main result states that the Fourier-Laplace coefficients fk;l (r) of a polyharmonic function f of in nite order and type 0 can be extended to analytic functions on the complex plane cut along the negative semiaxis. The second main result gives a constructive procedure via Fourier-Laplace series for the analytic extension of a polyharmonic function on annular region A(r0; r1) of in nite order and type less than 1=2r1 to the kernel of the harmonicity hull of the annular region. The methods of proof depend on an extensive investigation of Taylor series with respect to linear differential operators with constant coefficients.
      313
  • Publication
    The Khavinson-Shapiro conjecture and polynomial decompositions
    (Elsevier, 2011-04-15) ;
    The main result of the paper states the following: Let ψ be a polynomial in n variables of degree t: Suppose that there exists a constant C > 0 such that any polynomial f has a polynomial decomposition f = ψ qf + hf with khf = 0 and deg qf deg f + C: Then deg ψ 2k. Here ∆k is the kth iterate of the Laplace operator ∆ : As an application, new classes of domains in Rn are identi ed for which the Khavinson-Shapiro conjecture holds.
      346Scopus© Citations 10
  • 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.
      469Scopus© Citations 6
  • 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.
      345Scopus© Citations 7