Why the Fourier series are intriguing?

Question

On the $2\pi$-periodic functions on reals, it is well-known that $ \|\hat{f}\|_{\infty}\leq \|f\|_1$. Moreover the ratio $\frac{\|\hat{f}\|_{\infty}}{\|f\|_1}$ can be arbitraily small, where $\hat{f}$ is the fourier transform of $f$ whose domain is intergers.

Based on the R. E Edwards' book (Fourier series), page 32:

Were this type of phonamena (the ratio is small) absent, the theory of Fourier series would be much simpler and much less interguing than it in fact is.

How one may interpert this sentence?

Answer 1

If we had a positive lower bound for $\frac {\|f\|_1} {\|f\|_{\infty}} $ then convergence in $L^{1}, L^{2}$ and $L^{\infty}$ would all be equivalent. Since Fourier series of any $L^{2}$ function converges in $L^{2}$ this would make pointwise and uniform convergence trivial in many cases. For example, the FS of any continuous function would converge uniformly.

Answer 2

Your inequality should go the other way, and it likely requires a $2\pi $.

I have no idea what the author was thinking, but here's a possibility. Fourier series appear naturally in the context of Hilbert spaces, so with $\|\cdot\|_2$. But to make actual calculations it is often more interesting/necessary to have pointwise convergence, and in particular uniform convergence when you represent a continuous function. If the quotient $\|f\|_\infty/\|f\|_2$ were bounded, then we would be able to deal with Fourier series just at the level of Hilbert spaces, without need for all the special conditions for uniform convergence, Cesaro convergence, etc.