I'm dealing with a Fourier series which is given as:
$$ f(t,s) = \sum_{k \geq 1} \frac{ \left( \sin( n \pi s) - \sin( n \pi t \right) )^2 }{ n^{2\alpha} } $$
for some $\alpha \in (1/2,1), (s,t) \in [0,1]^2$. Or:
$$ g(s) = \sum_{n \geq 1} \frac{ \sin^2(n \pi s) }{ n^{2\alpha} } \quad h(s,t) = \sum_{n \geq 1} \frac{ \sin(n \pi s) \sin( n \pi t) }{ n^{2\alpha} }$$
For my purpose, I happen to need a lower bound on $f$ in terms of $t,s$ or $|t-s|$, up to some fractional powers. I have a feeling that it should be more or less standard result, but I don't have a lot of experience in Fourier analysis.
Using interpolation, it is relatively easy to get an upper bound of type $$f(t,s) \leq C |t-s|^{2\beta}$$
for some constant $C$ depending on $\beta,\alpha$, uniform in $t,s$, for $\beta < \alpha-1/2$
Analytical expressions can be obtained for the case $\alpha = 1$, for instance (p. 7):
https://math.mit.edu/~jerison/103/handouts/brownian.13.pdf
The expression they get, for $s=0$, is $\sim t^2 - t$, which corresponds to the Holder bound that I obtained - $\alpha = 1, \beta = 1/2$.
I tried to:
- rewrite it as $\sin(\pi n (t-s)/2) \cos(\pi n (t+s)/2)$ and compare to the infinite integral, to use change of variables and pull $|t-s|$ outside. This would have worked if it wasn't for the factor of $\cos(\pi (t+s)/2)$, which I cannot control from below.
- expand using square $(a-b)^2 = a^2 + b^2 - 2ab$, but then I need an upper bound on a product. Then I have:
$$f(t,s) = g^2(t) + g^2(s) - 2h(s,t)$$
For this I need a) a lower bound on $g$ b) an upper bound on $h$
Maybe I could get something using integral representation, but I need a precise control on constants that appear and it is not very obvious to me that it would work.