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94 Polynomial Approximation of Differential Equations
which leads to
(6.1.3) f(x) − pn(x) =
n
j=0
j n
f(x) − f x
n j
j (1 − x) n−j , x ∈ [0,1].
The proof that this error tends to zero uniformly is now a technical exercise. One argues
as follows. For any x ∈ [0,1], the sum in (6.1.2) is decomposed into two parts. In the
first part, the indices j are such that |x − j/n| is suitably small. Therefore, from
the uniform continuity of f, the term |f(x) − f(j/n)| will also be small. To handle
the remaining part of the summation, we observe that the function n j n−j
j x (1 − x)
achieves its maximum for x = j/n, and far away from this value it decays quite fast.
By appropriately applying the above information, one can, after some manipulation,
conclude the proof.
At this point, the following question arises. Among all the polynomials of degree
less or equal to a fixed integer n, find the one which best approximates, uniformly in
Ī, a given continuous function f. In practice, with the notations of section 2.5, for any
n ∈ N, we would like to study the existence of Ψ∞,n(f) ∈ Pn such that
(6.1.4) f − Ψ∞,n(f)∞ = inf
ψ∈Pn
f − ψ∞.
This problem admits a unique solution, though the proof is very involved. An extensive
and general treatise on this subject is given for instance in timan(1963). The n-degree
polynomial Ψ∞,n(f) is called the polynomial of best uniform approximation of f in Ī.
As a consequence of theorem 6.1.1, one immediately obtains
(6.1.5) lim
n→+∞ f − Ψ∞,n(f)∞ = 0.
One can try to characterize the polynomial of best approximation of a certain degree.
Results in this direction are considered in timan(1963) and rivlin(1969) where Cheby-
shev polynomials play a fundamental role. As an example, theorem 2.5.3 provides the
polynomial that best approximates the zero function in [−1,1], in the subset of Pn given
by the polynomials of the form x n + {lower degree terms}.