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6
RESULTS IN
APPROXIMATION THEORY
According to a theorem of Weierstrass, any continuous function in a bounded closed in-
terval can be uniformly approximated by polynomials. From this celebrated statement,
a large variety of sophisticated results have emerged. We review those that are most
relevant to the analysis of spectral methods.
6.1 The problem of best approximation
We begin with a classical theorem in approximation theory.
Theorem 6.1.1 (Weierstrass) - Let I be a bounded interval and let f ∈ C0 ( Ī). Then,
for any ǫ > 0, we can find n ∈ N and pn ∈ Pn such that
(6.1.1) |f(x) − pn(x)| < ǫ, ∀x ∈ Ī.
Hints for the proof - We just give a brief sketch of the proof in the interval
Ī = [0,1].
We follow the guideline of the original approach presented in bernstein(1912b). Other
techniques are discussed for instance in timan(1963), todd(1963) and cheney(1966).
For sufficiently large n, the approximating polynomial is required to be of the form
(6.1.2)
n
j n
pn(x) := f x
n j
j (1 − x) n−j , x ∈ [0,1],
j=0