- Aldaz, J. M.

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# Aldaz, J. M.

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Aldaz, J. M.

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Aldaz, J. M.

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- PublicationBernstein operators for exponential polynomialsLet L be a linear differential operator with constant coefficients of order n and complex eigenvalues Î» 0 ,...,Î» 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 Ï€/M n , where M n := max {| Im Î» 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 âˆ’ k at b, and each p n,k is positive on the open interval ( a,b ) . Under the additional assumption that Î» 0 and Î» 1 are real and distinct, our first main result states that there exist points a = t 0
249ScopusÂ© Citations 20 - PublicationOn real-analytic recurrence relations for cardinal exponential B-splinesLet LN+1 be a linear differential operator of order N + 1 with constant coefficients and real eigenvalues Î» 1, ..., Î» N+1, let E( N+1) be the space of all Câˆž-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 Îµn to Îµn+1 involving real-analytic functions, and with ÎµN+1 = E(Î› N+1) if and only if contiguous eigenvalues are equally spaced.
235ScopusÂ© Citations 2 - PublicationShape preserving properties of generalized Bernstein operators on extended Chebyshev spacesWe 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 ] â†’ U n with strictly increasing nodes, fixing f 0 ,f 1 âˆˆ U n . If U n âŠ‚ 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 ] â†’ 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 â‰¥ B n +1 f â‰¥ f for all ( f 0 ,f 1 )-convex functions f âˆˆ C [ a,b ] . Furthermore, B n f is ( f 0 ,f 1 )-convex for all ( f 0 ,f 1 )-convex functions f âˆˆ C [ a,b ] .
306ScopusÂ© Citations 40