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

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Not an answer, but a remark: since the algebra of n-by-n circulant matrices is isomorphic to the algebra of polynomials modulo the ideal generated by $X^n-1$, one might try to use norm estimates for polynomials (something $L^2$ flavoured would be my instinctive starting point) – Yemon Choi Dec 2 '12 at 5:47
Well, just to see that the matrix is invertible requires checking the values of the polynomial with coefficients given by $d$ at all roots of unity, so that part is unavoidable. After that you have just the standard inverse discrete Fourier transform, so there is hardly anything to simplify in that second part. Do you have some particualr restrictions on $d$ that you think might help? – fedja Dec 4 '12 at 3:03

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