Two problems in multidimensional prediction theory (Q1199263)

Given a positive Borel measure \(\mu\) on the \(n\)-dimensional torus \(T^ n\) and a vector \(x_ 0\in{\mathbf R}^ n\) such that the set \(\{k\in{\mathbf Z}^ n, \langle x_ 0,k\rangle=0\}\neq\{0\}\), the problem of finding an explicit expression in terms of \(\mu\) for the quantity \(\inf_ P\int_{T^ n}| 1+P(\theta)|^ 2d\mu(\theta)\), is considered. Here, the infimum is taken over the family of trigonometric polynomials \(P\) on \(T^ n\) with Fourier coefficients \(\hat P(k)\) identically zero outside of a half-space of lattice points of the form \(\{k\in{\mathbf Z}^ n,\langle x_ 0,k\rangle\geq 0, k\neq 0\}\) or of the form \(\{k\in{\mathbf Z}^ n, \langle x_ 0,k\rangle>0\}\). The results obtained generalize a famous theorem of Szegö [see \textit{U. Grenander} and \textit{G. Szegö}, Toeplitz forms and their applications (1958; Zbl 0080.095)] in prediction theory corresponding to the one-dimensional case as well as subsequent results due to \textit{H. Helson} and \textit{D. Lowdenslager} [Acta Math. 99, 165-202 (1958; Zbl 0082.282)] who studied the problem stated above under the assumption that the vector \(x_ 0\) satisfies \(x_ 0\neq 0\) and \(\{k\in{\mathbf Z}^ n, \langle x_ 0,k\rangle=0\}=\{0\}\).