The first Szego limit theorem on multi-dimensional torus

In this paper, we consider the first Szego limit theorems on d-torus T-d for 1 <= d <= +infinity. Given phi is an element of L-1 (T-d) and a finite subset sigma = {xi(1) center dot center dot center dot, xi(n)} of the dual group Z(d) of T-d, the truncated Toeplitz matrix with respect to sigma is T-sigma phi = {(phi) over cap(xi(j) - xi(i))}(1 <= i, j <= n) . For any Folner sequence {sigma(N)} of Z(d) and phi is an element of L-+(1) (T-d), it is shown that lim(N -> 8) (det T-sigma N phi)(1/vertical bar sigma N vertical bar) = exp (integral(Td) log phi dm(d)). In the case d =+infinity, we are associated with multiplicative Toeplitz matrix T-phi = {(phi) over cap (j/i)}(i,j is an element of N) and the most relevant non-Folner truncation, that is, T-N phi={(phi)over cap>(j/i)}(1 <= i,j <= N), where sigma(N) = {1,..., N}. It is shown that for each phi is an element of L-R(infinity)(T-infinity) and f is an element of C[ess-inf phi, ess-sup phi], the limit lim(N ->infinity) 1/NTrf (T-N phi) exists. Moreover, it is proven that the limit lim(N ->infinity)(det T-N phi)(1/N) exists for any phi is an element of L-+(1)(T-infinity) with strictly positive essential infimum. These results are directly related to two problems posed by Nikolski and Pushnitski in [30]. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.