# Condition number and Chebyshev systems

Suppose I have a square matrix $A$ of size $n$ with elements $a_{mn}=\phi_m(x_n)$ where $\phi_m(x)$ can be thought of as a very friendly function: orthogonal, bandlimited, bounded and analytic. Also, they form a Chebyshev system, so $det(A)>0$.

Is there a way I can bound the smallest singular value of $A$ from below with a bound that is strictly positive?

What I did was looking on $AA^*$ which ideally should be equal to $I$ (inner product $\langle \phi_m, \phi_n\rangle=\delta_{mn}$) with some error, and then I used Gresgorin on the worst case according to the numerical integration error. This leads to a lower bound that is very often negative and hence useless.

Note that when $n$ is going to infinity, this does not mean that $AA^*$ is going to $I$ because as $n$ is larger $\phi_n$ needs more and more points as well for its integral to be computed accurately. So it can be said only on an uppler-left submatrix of $AA^*$

If it helps $x_n$ can be thought as equally spaced (or any other form of distribution)

Any ideas? I would be grateful for any help!

Thanks.

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