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32 Polynomial Approximation of Differential Equations
Theorem 2.5.2 (Bernstein) - Let Ī = [−1,1], then for any n ∈ N we have
(2.5.3) |p ′ (x)| ≤
n
√ 1 − x 2 p∞, ∀x ∈ I, ∀p ∈ Pn.
Recalling definition (2.2.17), we finally state one of the main results characterizing
Chebyshev polynomials (compare with theorem 2.2.3).
Theorem 2.5.3 - For any n ∈ N and for any polynomial p in
such that p(x) = x n + {lower degree terms}, we have
(2.5.4) ˜ Tn∞ ≤ p∞.
The proof is given for instance in rivlin(1974). Note that
(2.5.5) ˜ Tn∞ =
2.6 Basis transformations
1 if n = 0,
2 1−n if n ≥ 1.
Ī = [−1,1] of degree n
An interesting question is how to transform the Fourier coefficients of a given polynomial
corresponding to an assigned orthogonal basis, into the coefficients of another basis
orthogonal with respect to a different weight function. The goal is to determine the
so-called connection coefficients of the expansion of any element of the first basis in
terms of the elements of the second basis. For a survey of the major results, we suggest
that the interested reader consult lecture 7 from the book of askey(1975). Here we