Consider the Dirichlet kerel:
$f(x) = 1+2\sum_{k=1}^{N}\cos(kx)$.
Now, given a diagonalizable real matrix $A$, one can consider $f(A)$, using the standard notation of matrix functions. Namely, $f(A) = I+2\sum_{k=1}^{N}\sum_{j=0}^{\infty}\frac{(-1)^j}{(2j)!}(kA)^{2j}$.
I am interested in bounding $\|f(A)\|_{\infty}$ from above. I am also assuming:
$\|A\|_{\infty} \le c$ for some $c$ that is independent of $N$, $A$ or its dimension.
The eigenvalues of $A$ are real and also bounded by $c$.
I have tried following quite a few paths. Among them, are:
1) Trivial bounds, mainly using subadditivity.
2) Using the Cauchy definition, $f(A) = \frac{1}{2\pi i}\int_{\Gamma}f(z)(zI-A)^{-1}dz$.
All of the above led me to upper bounds which are polynomial (or square root) in the dimension of $A$. However, numerical tests for random matrices consistently show very small results (specifically, if we take $N$ to be logarithmic in the dimension of $A$).
Do you have any suggestions?
Thank you.
Clarification: I am using the notation $\|B\|_{\infty} = \max_{i}\sum_{j=1}^{n}|B_{i,j}|$.
Edit: It might be the case, that if $A$ is stochastic and symmetric, $\cos(kA)$ itself has a bounded infinity norm (i.e., independent of both $N$ and $k$). It that is indeed correct, the above question is resolved. I posted this question there, but it might also be appropriate here as well.
I apologize for the duplication.
