Regularity of random Fourier series

Question

The following two statements appear to be true (but do correct me if I am wrong):

1. The coefficients of a $C^k$ function on the torus $T^n$ decay at least as fast as $x^{-k}$ (where $x$ is some norm on $\mathbb{Z}^n.)$

2. If the coefficients of a Fourier series decay at least as fast as $x^{-k-n},$ then the Fourier series represents a $C^k$ function on $T^n.$

It appears that 2. is not quite a converse to 1. Now, the question is: if I take a random Fourier series whose coefficients decay as $x^{-k},$ what is its degree of regularity? Is it $C^k?, C^{k-n}?$ none of the above? (the case of $n=1$ would already be of great interest).

Answer 1

I consider the case of independent Gaussian (or any light-tailed, for that matter) coefficients with variances decaying like $x^{-2k}$ - note that in this case the coefficients themselves decay essentially like $x^{-k}$, up to a logarithmic correction. For this random series the answer will be $C^{k-n/2}$, again up to a logarithmic correction.

To prove this first note that the covariance function of this (stationary) process is in $C^{2k-n-\varepsilon}$ because Fourier series (of the covariance) decays like $x^{-2k}$. Then use the multidimensional version of Kolmogorov's continuity criterion, as formulated in, say, Lemma 2.1 of Scheutzow, (recall that for the Gaussians all $L^p$ norms are equivalent to the $L^2$ norm and use high moments there). It will follow that a Gaussian process with $C^{\alpha}$ covariance has $C^{\alpha/2-\varepsilon}$ sample paths, hence the result.

Answer 2

On $T^1$ the first assertion is essentially elementary, by integrating by parts, and in fact we find that the decay is like $o(|x|^{-k})$ in Landau's little-oh notation. For $n>1$ and odd $k$ there are (not-too-meaningful) complications in that.

For coefficients decaying like $|\xi|^{-k-n}$, the function is in the $L^2$ Sobolev space $H^{k-{n\over 2}-\epsilon}$ for every $\epsilon$, while Sobolev imbedding implies that $H^s\subset C^k$ for $s>k+{n\over 2}$. So your decay condition misses $C^k$ by $\epsilon$ in this viewpoint.

Of course, $L^1$ estimate gives various convergences under somewhat weaker hypotheses, but if one wants $C^k$ and convergence of partial sums of the Fourier series to the function in the $C^k$ topology, the $L^2$ Sobolev estimates are more reliable and intelligible.