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Conversely, every polynomial (∗∗) can be written in the form (∗), using complex ak ’s and bk ’s. Thus, the complex trig polynomials of degree n form a vector space of dimension 2n+1 over C (hence of dimension 2(2n+1) when considered as a vector space over R). But, not every polynomial in z and z¯ represents a real trig polynomial. Rather, the real trig polynomials are the real parts of the complex trig polynomials. To see this, notice that (∗∗) represents a real-valued function if and only if n n ikx ck e = n ck k=−n k=−n eikx = k=−n c¯−k eikx ; that is, ck = c¯−k for each k.

Conversely, if P (x, y) is an even polynomial, show that P (cos x, sin x) can be written using only cosines. 42. Show that Tn has dimension exactly 2n + 1 (as a vector space over R). 43. We might also consider complex trig polynomials; that is, functions of the form (∗) in which we now allow the ak ’s and bk ’s to be complex numbers. (a) Show that every trig polynomial, whether real or complex, may be written as n ck eikx , (∗∗) k=−n where the ck ’s are complex. Thus, complex trig polynomials are just algebraic polynomials in z and z¯, where z = eix ∈ T.

Any trig polynomial (∗) may be written as P (cos x) + Q(cos x) sin x, where P and Q are algebraic polynomials of degree at most n and n − 1, respectively. If (∗) represents an even function, then it can be written using only cosines. Corollary. The collection T , consisting of all trig polynomials, is both a subspace and a subring of C 2π (that is, T is closed under both linear combinations and products). In other words, T is a subalgebra of C 2π . It’s not hard to see that the procedure we’ve described above can be reversed; that is, each algebraic polynomial in cos x and sin x can be written in the form (∗).