## 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.)

### Keywords:

spline operator of Wiener space; Jackson operators
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