This post is partly inspired by Fourier Coefficients and Hölder Continuity.

Typical proofs of the nowhere differentiability of Brownian paths is by contradiction using binary expansion from real analysis. For example, Prop 11. of S.Lally's notes.

But usually one can prove a function $f$ is nowhere differentiable by examining their Fourier coefficients $\hat{f}$. For example Simple Proofs of Nowhere-differentiability Weierstrass's Function and Cases of Slow Growth

Can we prove the nowhere differentiability (w.p.1) of Brownian paths by examining its Karhunen–Loève coefficients?

Moreover, what is the relationship between Karhunen–Loève coefficients $\hat{X_t}$ and the smoothness of the path of an $L^2$ stochastic process $X_t$?

If there is such a relationship, is this relationship similar to the relationship between Fourier coefficients and original function?

In response to Christian's comment,

The Fourier coefficients don't look like the right thing to look at to me if you want to establish that Brownian paths are

nowheredifferentiable, since the FT responds to lack of smoothnesssomewhereby lack of decay.

A natural thing to ask is that *what object characterizes/detects global non-smoothness in in situation?*

nowheredifferentiable, since the FT responds to lack of smoothnesssomewhereby lack of decay. $\endgroup$ – Christian Remling Apr 22 '17 at 18:22somewhere.For me, the natural intuitive explanation of the fact is that typically $B_{t+h}\simeq h^{1/2}$. $\endgroup$ – Christian Remling Apr 25 '17 at 19:27