Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the entries of $f$ as functions of the entries of $d$, but it's quite cumbersome to analyze.

My question is: do you know of a nice upper bound on $|f_{i}|$ in terms of relatively simple function of the entries of $d$?

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the entries of $f$ as functions of the entries of $d$, but it's quite cumbersome to analyze.

My question is: do you know of a nice upper bound on $|f_{i}|$ in terms of relatively simple function of the entries of $d$?

UPDT: A sample $d$: $[17 \ 0 -1 -2 -3 -2 -1 \ 0]$.