Fourier Approximation of Functions

Consider a function on :

where is even and

are Fourier Series coefficients.
The de la Vallee Poussin Theorem
Carlesson’s Theorem
Convergence at Spectral Speed

Lemma

Proof

Integration by parts.

Lemma

For nice enough , decay faster than for any

Theorem

For nice enough , we have Convergence at Spectral Speed.