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Average error bounds of best approximation of continuous functions on the Wiener space. (English) Zbl 0816.41030

In this paper, the authors study the approximation of the identity operator and the integral operator \(T_ m\) defined as:
Let \(m \geq 0\) be an integer. Define \((T_ 0g) (x) = g(x)\) \(\forall g \in C_ 0 [0,1] : = \{f : [0,1] \to \mathbb{R},f\) continuous on \([0,1]\), \(f(0) = 0\}\); when \(m \geq 1\) \[ (T_ mg) (x) : = {1 \over (m-1)!} \int^ 1_ 0 (x-t)_ +^{m-1} g(t)dt \quad \forall g \in C_ 0 [0,1]; \] by Jackson operators \(I_ n^{(m)}\) \((m \geq 0)\), discrete Jackson operators, and spline operators, respectively, on the Wiener space and obtain average error estimation. The paper consists of four sections. Section 1 is introduction, section 2 is a preliminary section containing some auxiliary lemmas and useful results. Section 3 contains two theorems dealing with the approximation of the identity operator and the integral operator \(T_ m\) by Jackson operators. In section 4 the authors give a discretized version of the Jackson operator \(I_ n^{(m)}\) and a spline operator defined on \(C_ 0^ m [0,1] : = \{f \in C^ m [0,1] : f^{(j)} (0) = 0\), \(j = 0, \dots, m\}\), and get some analogous results for these operators.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A50 Best approximation, Chebyshev systems
68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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