By Carothers N.L.
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Extra info for A Short course on approximation theory
41 Trig Polynomials That is, Tn ≡ span A ⊂ span B. By comparing dimensions, we have 2n + 1 = dim Tn = dim(span A) ≤ dim(span B) ≤ 2n + 1, and hence we must have span A = span B. The point here is that Tn is a finite-dimensional subspace of C 2π of dimension 2n + 1, and we may use either one of these sets of functions as a basis for Tn . , the case where we allow complex coefficients in (∗). Now it’s clear that every trig polynomial (∗), whether real or complex, can be written as n ck eikx , (∗∗) k=−n where the ck ’s are complex; that is, a trig polynomial is actually a polynomial (over C ) in z = eix and z¯ = e−ix .
Bn are real numbers. The degree of a trig polynomial is the highest frequency occurring in any representation of the form (∗); thus, (∗) has degree n provided that one of an or bn is nonzero. , the union of the Tn ’s). It is convenient to take the space of all continuous 2π-periodic functions on R as the containing space for Tn ; a space we denote by C 2π . The space C 2π has several equivalent descriptions. For one, it’s obvious that C 2π is a subspace of C(R), the space of all con- tinuous functions on R.
Tn . [n/2] n 2 Tn−2k (x); for n even, 2 T0 should be replaced by T0 . k n n C7. For n odd, 2 x = k=0 Proof. For −1 ≤ x ≤ 1, 2n xn = 2n (cos θ)n = (eiθ + e−iθ )n n i(n−4)θ n i(n−2)θ e + ··· e + 2 1 = einθ + ··· + n n e−i(n−2)θ + e−inθ e−i(n−4)θ + n−1 n−2 = 2 cos nθ + n n 2 cos(n − 4)θ + · · · 2 cos(n − 2)θ + 2 1 = 2 Tn (x) + n n 2 Tn−4 (x) + · · · , 2 Tn−2 (x) + 2 1 where, if n is even, the last term in this last sum is binomial expansion, namely (n) C8. The zeros of Tn are xk n [n/2] = n [n/2] n [n/2] T0 (since the central term in the T0 , isn’t doubled in this case).
A Short course on approximation theory by Carothers N.L.