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Optimal Equiripple Design Technique 365
Let us now turn our attention to the problem statement and equation
(7.45). It is a well-known problem in approximation theory, and the
solution is given by the following important theorem.
THEOREM 1
Alternation Theorem
Let S be any closed subset of the closed interval [0,π]. Inorder that
P (ω) be the unique minimax approximation to H dr (ω) on S, itisnecessary
and sufficient that the error function E(ω) exhibit at least (L +2)“alternations”
or extremal frequencies in S; that is, there must exist (L +2)
frequencies ω i in S such that
E (ω i )=−E (ω i−1 )=± max |E (ω)| (7.47)
S
△
= ±δ, ∀ ω 0