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Optimal Higher Order Time Discretizations 83
Proof. Let P m,k = ψ m (R m,k ) be an optimal polynomial (Corollary 1). The
difference D = R − R m,k ∈ P k−1 takes at x>0 the value
D(x) = P (x) − P m,k(x)
x m+1 .
Since R m,k is optimal, P m,k (τ j ) ∈ [0, 4] for j =0,...,k.ThenD(τ j ) ≥ 0
(resp. D(τ j ) ≤ 0) when P (τ j ) = 4 (resp. P (τ j )=0).SinceP (τ j ), j =0,...,k,
alternates in {0, 4}, we have shown that
(−1) j (P (τ 0 ) − 2) D(τ j ) ≥ 0,
for j =0,...,k.
These inequalities tell us that D ∈ P k−1 satisfies the conditions of Lemma 4.
Therefore, D = 0 proving that R is optimal.
⊓⊔
The necessary and sufficient optimality conditions of Theorems 4 and 5
will be used to determine the optimal polynomials in Section 4. We conclude
this section with two corollaries of these optimality conditions. The first one
deals with the uniqueness of the solution. The second one provides a full
description of the optimal polynomials when m = 1, relating them to the
Chebyshev polynomials of the first kind [Che66, LT86, Wei06].
Corollary 2 (Uniqueness of the solution). The maximization problem
(20) has one and only one solution. It has no other local maximum.
Proof. Existence has been quoted in Corollary 1. Uniqueness is is actually a
by-product of the proof of Theorem 5, where it is shown that if a polynomial
P = ψ m (R), for some R ∈ P k−1 , satisfies the optimality conditions (this
is the case for any local maximum, by Theorem 4), then R is equal to an
arbitrarily fixed solution. Hence there cannot be more than one solution or
local maximum.
⊓⊔
Corollary 3 (Optimal polynomials when m =1).For k ≥ 0,
α 1,k =4(k +1) 2 (28)
and the optimal polynomial ψ 1 (R 1,k ) takes at x ∈ [0,α 1,k ] the value
(
[ψ 1 (R 1,k )](x) =2
[1 − T k+1 1 − 2x )]
, (29)
α 1,k
where T k denotes the Chebyshev polynomial of the first kind and degree k,
which verifies T k (x) = cos(k arccos x) for x ∈ [−1, 1].
Proof. Let α 1,k be defined by (28) and let ϕ be the function defined at x ∈
[0,α 1,k ] by the right-hand side of (29). The fact that ϕ ≡ ψ 1 (R 1,k ) will result
from the following observations: