While reading some things about analytic functions earlier tonight it came to my attention that Fourier series are not necessarily analytic. I used to think one could prove that they are analytic using induction
- Let P(n) be some statement parametrized by the natural number n (in this case: the nth partial sum of the Fourier series is analytic)
- Show that P(0) is true
- Show that P(n−1)⇒P(n)
- (Invalid) conclusion: P(n) continues to be true as we take the limit n→∞*
Why exactly is the conclusion not valid here? It seems very strange that even though P(n) is true for any finite n, it ceases to be valid when I remove the explicit upper bound on n. Are there circumstances under which I can make an argument of this form?
Example of invalid proof: Define the truncated Fourier series Fn(x) as the partial sum
where Ak and Bk are the Fourier coefficients for some arbitrary function f. Using the facts that sin(t) and cos(t) are analytic, and that any linear combination of analytic functions is analytic:
- P(n) is the statement “Fn(x) is analytic”
- F0(x) is clearly analytic because it is a linear combination of sine and cosine functions
Fn(x) can be written as the linear combination
So if Fn−1(x) is analytic, Fn(x) is analytic.
- F(x)≡limn→∞Fn(x) is analytic. But F(x) is the Fourier series for f; therefore, the Fourier series for f is analytic.
*I’m assuming that P(n) is a statement about some sequence which is parametrized by n and for which taking the limit as n→∞ is meaningful
A trivial case where P(n) is true for all n∈N but P(∞) is false is the statement “n is finite”.