On $n$-widths of certain functional classes defined by linear differential operators
HTML articles powered by AMS MathViewer
- by Y. Makovoz PDF
- Proc. Amer. Math. Soc. 89 (1983), 109-112 Request permission
Abstract:
Let $A = {D^r} + \sum \nolimits _{k = 0}^{r - 1} {{a_k}(t){D^k}}$, ${a_k} \in {C^k}$, be a differential operator and let ${W_p}(A)$ be the class of functions $x(t)$ for which $\left \| {Ax} \right \| \leqslant 1$ in ${L_p}[0,1]$. We prove that the asymptotic behavior of the Kolmogorov widths ${d_n}({W_p}(A),{L_q})$, $1 \leqslant p$, $q \leqslant \infty$, when $n \to \infty$ does not depend on ${a_k}$.References
- Charles K. Chui and Philip W. Smith, Some nonlinear spline approximation problems related to $N$-widths, J. Approximation Theory 13 (1975), 421–430. MR 370023, DOI 10.1016/0021-9045(75)90025-8 C. A. Micchelli and A. Pinkus, On $n$-widths in ${L^\infty }$: Limit as $n \to \infty$, IBM Research Report RC 5573, 1975.
- Charles A. Micchelli and Allan Pinkus, On $n$-widths in $L^{\infty }$, Trans. Amer. Math. Soc. 234 (1977), no. 1, 139–174. MR 487187, DOI 10.1090/S0002-9947-1977-0487187-1 K. Nasyrova, On asymptotic behavior of certain compacts in the space ${L_2}[0,1]$, Mat. Zametki 20 (3) (1973), 331-339. (Russian)
- E. M. Stein, Functions of exponential type, Ann. of Math. (2) 65 (1957), 582–592. MR 85342, DOI 10.2307/1970066
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 109-112
- MSC: Primary 41A46
- DOI: https://doi.org/10.1090/S0002-9939-1983-0706520-4
- MathSciNet review: 706520