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# Chebyshev–Laurent Polynomials and Weighted Approximation

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Chebyshev–Laurent Polynomials and Weighted Approximation book

# Chebyshev–Laurent Polynomials and Weighted Approximation

DOI link for Chebyshev–Laurent Polynomials and Weighted Approximation

Chebyshev–Laurent Polynomials and Weighted Approximation book

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## ABSTRACT

Let (a, b) ⊂ (0, ∞) and for any positive integer n, let S_{n}
be the Chebyshev space in [a, b] defined by S_{n}
:= span { x^{−n}
/^{2+k
}, k = 0,…, n }. The unique (up to a constant factor) function τ_{n}
∈ S_{n}
, which satisfies the orthogonality relation
∫
a
b
τ
n
(
x
)
q
(
x
)
(
x
(
b
−
x
)
(
x
−
a
)
)
−
1
/
2
d
x
=
0
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/5c315e12-b249-44e0-8be1-fb5a0ad8cd49/content/eq1.tif"/>
for any q ∈ S
_{
n – 1}, is said to be the orthogonal Chebyshev S_{n}
-polynomials. This paper is an attempt to exibit some interesting properties of the orthogonal Chebyshev S_{n}
-polynomials and to demonstrate their importance to the problem of approximation by S_{n}
-polynomials. A simple proof of a Jackson-type theorem is given and the Lagrange interpolation problem by functions from S_{n}
is discussed. It is shown also that τ_{n}
obeys an extremal property in L_{q}
, 1 ≤ q ≤ ∞. Natural analogues of some inequalities for algebraic polynomials, which we expect to hold for the S_{n}
-polynomials, are conjectured.