2
votes
1answer
86 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
5
votes
2answers
420 views

Do all analytic and $2\pi$ periodic functions have a finite Fourier series?

Consider a function $f:\mathbb{R}\to\mathbb{R}$ which is periodic with period $2\pi$. Let us impose the condition that $f$ is analytic. Now does that imply that $f$ has a finite Fourier series? PS : ...
4
votes
1answer
448 views

Analytic functions and Fourier Series

I'm taking my first real analysis course and I'm trying to get a better feel about analytic functions. My understanding is that an analytic function is one which can be written as a power series. My ...