\(N\)-widths on the classes of multivariate bandlimited functions (Q2583368)

Let be given \(p\in (1\infty)\), \(v=(v_1,\dots,v_d)\in\mathbb R^d_+\), \(m\in\mathbb N\) and \(N\in \mathbb N\). The authors deal with the subset \[ E_{v,m}=\{f\in B_{v,p}(\mathbb R^d)\mid \| x^m f(x)\|_{L_p(\mathbb R^d)}\leq 1\} \] of the seet \(V_{v,p}(\mathbb R^d)\) of bandlimited functions. By using results obtained by the first author together with \textit{J. Wang} (see [Acta Math. Appl. Sin. 19, No.~4, 481--488 (1996; Zbl 1019.62500)]) and results by \textit{S. M. Nikol'skij} [Approximation of functions of several variables and imbedding theorems (Die Grundlehren der mathematischen Wissenschaften 205, Berlin-Heidelberg-New York: Springer-Verlag) (1975; Zbl 0307.46024)] they prove that the Kolmogorov \(N\)-width \(d_N(E_{v,m};L_p(\mathbb R^d))\) of \(E_{v,m}\) in \(L_p(\mathbb R^d)\) satisfies \[ C_2\overline h^{-m} N^{-\frac md}\leq d_N(E_{v,m}; L_p(\mathbb R^d))\leq C_1h^{-m}_0 N^{-\frac md}, \] where \[ \overline h=\max\{\pi/v_1,\dots,\pi/v_d\},\quad h_0=\min\{\pi/v_1,\dots,\pi/v_d\}, \] and the constants \(C_1\) and \(C_2\) depend only on \(p\) and \(d\). Moreover, in the case \(p=2\) an exact estimate of \(d_N(E_{v,m};L_2(\mathbb R^d))\) is given without proof.