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Remarks on some representation theorems for convolution transforms

Dany Leviatan — 1973

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

L'Autore dà due criteri necessari e sufficienti perché una funzione f ( x ) possa essere rappresentata da una trasformazione di convoluzione f ( x ) = - G ( x - t ) 𝑑 α ( t ) , - < x < , nel caso che il nucleo G ( x ) sia della classe 1 o della classe 3 ed esista un x 0 tale che - G ( x 0 - t ) | d α ( t ) | M < , essendo α ( t ) una funzione localmente a variazione limitata.

Nearly Coconvex Approximation

Leviatan, D.Shevchuk, I. — 2002

Serdica Mathematical Journal

* Part of this work was done while the second author was on a visit at Tel Aviv University in March 2001 Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are nearly coconvex with it, namely, polynomials and piecewise polynomials that preserve the convexity of f except perhaps in some small neighborhoods of the points where f changes its convexity....

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