In "Toeplitz Forms and Their Applications" by U.Grenader and G.Szego p. 18 in a proof of a theorem there is the following reasoning.
The expression
$T_n(r, \theta) = \sum c_{\nu-\mu}r^{|\mu-\nu|}e^{i(\mu-\nu)\theta}u_{\mu}\bar{u}_{\nu}$,
where $\mu,\nu=0,1\dots n$, $c_k$-s are complex coefficients of a Fourier series of some function f(x), $r$ is real.
Then
"$T_n(r, \theta)$ defines a harmonic function of the polar coordinates $r,\theta$ so that its nonnegativity for $r=1$ involves the same property for $r<1$"
Would somebody, please, explain why the expression in bold is correct?
The function $\phi: re^{i\theta} \longmapsto T_n(r,\theta)$ is harmonic and nonnegative on the unit circle.
As $\phi$ is harmonic on $\{|z| < 1\}$, it has no local minimum. As a consequence, its minimum on the unit disk must be reached on the unit circle. But $\phi$ is nonnegative on the unit circle: so $\phi$ must be non-negative.