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Orthogonality 31
From (2.3.8), we get for instance
(2.4.4) Πw,n(xun) = − σn+1
ρn+1
un − τn+1
un−1, n ≥ 1.
ρn+1
Taking a truncation of the sum in (2.3.10) or (2.3.12) in order that k + j − 2m ≤ n, we
obtain an expression for Πw,n(ukuj), k,j ∈ N. Therefore, by the linearity of Πw,n,
one obtains the coefficients of Πw,np 2 in terms of the coefficients of p.
The main properties of the projection operator are investigated in section 6.2.
There, under suitable hypotheses, we shall see that limn→+∞ Πw,nf = f, where the
limiting process must be correctly interpreted.
2.5 The maximum norm
Let us assume that I is a bounded interval. In the linear space of continuous functions
in Ī, we define the norm
(2.5.1) u∞ := max
x∈ Ī
|u(x)|, ∀u ∈ C 0 ( Ī).
One easily verifies that ·∞ actually satisfies all the properties (2.1.3)-(2.1.5). We call
u∞ the maximum norm of u in Ī. It can be proven that it is not possible to define
an inner product that generates the maximum norm through a relation like (2.1.6).
Two celebrated theorems give a bound to the maximum norm of the derivative
of polynomials. The first one is due to A.Markoff and is successively generalized to
higher order derivatives by his brother (see markoff(1889) and markoff(1892)). The
second one is due to S.Bernstein (bernstein(1912a)). Other references can be found in
jackson(1930), cheney(1966) as well in zygmund(1988), Vol.2. Further refinements
are analyzed in section 3.9. Applications are illustrated in chapter six.
Theorem 2.5.1 (Markoff) - Let Ī = [−1,1], then for any n ∈ N we have
(2.5.2) p ′ ∞ ≤ n 2 p∞, ∀p ∈ Pn.