Now showing 1 - 3 of 3
  • 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
    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.
      235Scopus© 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 ] .
      306Scopus© Citations 40