60 research outputs found
Approximation by MĂŒntz spaces on positive intervals
International audienceThe so-called Bernstein operators were introduced by S.N. Bernstein in 1912 to give a constructive proof of Weierstrass' theorem. We show how to extend his result to MuÌntz spaces on positive intervals
Kantorovich-Bernstein a-fractal function in LP spaces
Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein a-fractal operator in the Lebesgue space Lp(I), 1 = p = 8. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in Lp(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal Lp(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in Lp spaces is proven. Further, we derive the fractal analogues of some Lp(I) approximation results, for example, the fractal version of the classical MĂŒntz-Jackson theorem. The one-sided approximation by the Bernstein a-fractal function is developed
Bernstein operators for exponential polynomials
Let be a linear differential operator with constant coefficients of order
and complex eigenvalues . Assume that the set
of all solutions of the equation is closed under complex
conjugation. If the length of the interval is smaller than , where M_{n}:=\max \left\{| \text{Im}% \lambda_{j}| :j=0,...,n\right\}
, then there exists a basis %, , of the space with
the property that each has a zero of order at and a zero of
order at and each is positive on the open interval
Under the additional assumption that and
are real and distinct, our first main result states that there exist points and positive numbers %,
such that the operator \begin{equation*}
B_{n}f:=\sum_{k=0}^{n}\alpha_{k}f(t_{k}) p_{n,k}(x) \end{equation*} satisfies
, for The second main result
gives a sufficient condition guaranteeing the uniform convergence of
to for each .Comment: A very similar version is to appear in Constructive Approximatio
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