Uniform boundedness of Toeplitz matrices with variable coefficients (Q930452)

Consider the \((N+1) \times (N+1)\) variable-coefficient Toeplitz matrix \(A_N(a)=(\hat{a}_{j-k}(j/N,k/N))^N_{j,k=0}\) generated by \(a(x,y,t)=\sum_{n=-\infty}^{\infty} \hat{a}_n(x,y)t^n\), where \(\hat{a}_n(x,y)= \int_{\mathbb{T}}a(x,y,t)t^{-n}\frac{| dt| }{2\pi}\). In this paper, the authors provide sufficient conditions on \(a\) that guarantee the uniform boundedness of the spectral norms \(\| A_N(a)\| _{\infty}\) as \(N \to \infty\).