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Results in Approximation Theory 123
6.8 Refinements
The aim of the previous sections was to present an overview of the fundamental results
on the approximation of functions by orthogonal polynomials. Further developments can
be taken into account. In general, improvements require a certain skill in dealing with
functional spaces and their properties. We therefore confine ourselves to the discussion
of some basic facts.
In the book of timan(1963), p.262, the inequality (6.1.7) is generalized as follows:
(6.8.1) |f − Ψ∞,n(f)|(x) ≤ C [hn(x)] k
where hn(x) := 1
n
sup
|x1−x2|