Now showing 1 - 10 of 25
  • Publication
    Padé approximation for a multivariate Markov transform
    Methods of Padè approximation are used to analyse a multivariate Markov transform which has been recently introduced by the authors. The first main result is a characterization of the rationality of the Markov transform via Hankel determinants. The second main result is a cubature formula for a special class of measure.
      239Scopus© Citations 3
  • Publication
    Positivity properties for the clamped plate boundary problem on the ellipse and strip
    The positivity preserving property for the biharmonic operator with Dirichlet boundary condition is investigated. We discuss here the case where the domain is an ellipse (that may degenerate to a strip) and the data is a polynomial function. We provide various conditions for which the positivity is preserved.
      340Scopus© Citations 1
  • 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.
      344Scopus© Citations 5
  • 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 ] .
      306Scopus© Citations 40
  • 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.
  • 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.
      247Scopus© Citations 7
  • 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
      249Scopus© Citations 20
  • 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.
      256Scopus© Citations 5
  • 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.
      409Scopus© Citations 16
  • 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.
      304Scopus© Citations 7