# Existence of probability measure on the circle with given Fourier coefficients

We say that a Hermitian symmetric (i.e., $$f_{-n} = f_n^*$$ for any $$n \in \mathbb{Z})$$ sequence $$(f_n)_{n\in \mathbb{Z}}$$ is positive-definite if, for any $$N \geq 0$$ and any $$z_0 , \ldots, z_N \in \mathbb{C}$$, $$$$\sum_{n,m =0}^N f_{n-m} z_n z_m^* \geq 0. \tag{1}$$$$

According to the Herglotz-Bochner theorem, a Hermitian symmetric sequence $$(f_n)_{n\in \mathbb{Z}}$$ with $$f_0 = 1$$ is positive-definite if and only if there exists a probability measure $$\mu$$ in the circle $$\mathbb{T} = \mathbb{R} / \mathbb{Z}$$ such that $$$$f_n = \hat{\mu}_n := \int_{\mathbb{T}} \mathrm{e}^{2\pi \mathrm{i} n x } \mathrm{d}\mu (x).$$$$

Assume now that I am given a vector $$(f_{-N_0} , \ldots, f_0 , \ldots , f_{N_0}) \in \mathbb{C}^{2N_0+1}$$ such that $$f_0 = 1$$ and $$f_{-n} = f_n^*$$ for any $$|n|\leq N_0$$ and such that (1) holds for any $$N \leq N_0$$. Is it always possible to complete the vector $$(f_n)_{|n|\leq N_0}$$ into a positive-definite sequence $$(f_n)_{n\in \mathbb{Z}}$$, or, equivalently, is there always a probability measure $$\mu$$ in $$\mathbb{T}$$ such that $$\hat{\mu}_n = f_n$$ for $$|n|\leq N_0$$?

Yes, this works. Condition (1) says that $$\int |p(e^{ix})|^2\, d\mu(x)\ge 0$$ for every polynomial $$p(z)=\sum_{n=0}^N p_n z^n$$. By the Fejer-Riesz theorem, these squares $$|p|^2$$ range exactly over the trigonometric polynomials $$f=\sum_{|n|\le N} f_n z^n$$ with $$f\ge 0$$ on $$|z|=1$$.
So we have a positive linear functional on this vector space $$\{ f = \sum_{|n|\le N} f_n z^n \}$$. This can be extended to a positive linear functional on $$C(T)$$; see here for background. This extension gives us the desired measure.
• @Goulifet: By the way, my notation in the first line is a bit sloppy, I really mean what you get if you assume that $\mu$ has the right Fourier coefficients, and then you multiply out $\int |p|^2\, d\mu$ (in other words, this depends on the $f_n$ only and not on a hypothetical $\mu$). But I leave it in this form, it seems clear from the context. Commented Aug 20, 2020 at 2:30
• @Goulifet: One more afterthought: In the analogous problem for moments (instead of Fourier coefficients), there is a rather explicit description of all such measures $\mu$ ("Nevanlinna parametrization"). There probably must be something similar here. Commented Aug 20, 2020 at 14:15