Abstract
We describe the construction of extension operators with minimal possible norm τm from the half-line to the entire real line for the spaces \(W_2^m \) and derive the asymptotic estimate \(\tau _m \approx K_0 m\;\;({\text{as }}m \to \infty )\), where
The proof is based on the investigation of the maximum and minimum eigenvalues and the corresponding eigenvectors of some special matrices related to Vandermonde matrices and their inverses, which can be of interest in themselves.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. I. Burenkov and A. L. Gorbunov, “Sharp estimates of the norms of extension operators for Sobolev spaces,” Izv. RAN, 61, No.1, 3–44 (1997).
V. I. Burenkov and G. A. Kalyabin, “Lower estimates of the norm of extension operators for Sobolev spaces on the halfline,” Math. Nachr., 218, 19–23 (2000).
H. B. Dwight, Tables of Integral and Other Mathematical Data [in Russian], Nauka, Moscow, 1978.
A. Klinger, “The Vandermonde matrix, Amer. Math. Monthly,” 74, 571–574 (1967).
V. N. Gabushin, “Inequalities for the norms of the function and its derivatives in Lp metrics,” Mat. Zametki, 1, No.3, 291–298 (1968).
V. M. Tikhomorov, Some Questions in Approximation Theory [in Russian], Moscow University Press, Moscow, 1976.
L. V. Taikov, “Kolmogorov type inequalities and the best formulas for numerical differentiation,” Mat. Zametki, 4, No.2, 233–238 (1968).
N. I. Akhiezer, Lectures in Approximation Theory [in Russian], Nauka, Moscow, 1965; English transl.: N. I. Achieser, Theory of approximation, Dover Publications, New York, 1992.
G. A. Kalyabin, “Asymptotic formulas for smallest eigenvalues of Hilbert type matrices,” Funkts. Anal. Prilozhen., 35, No.1, 80–84 (2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kalyabin, G.A. The Best Extension Operators for Sobolev Spaces on the Half-Line. Functional Analysis and Its Applications 36, 106–113 (2002). https://doi.org/10.1023/A:1015614406023
Issue Date:
DOI: https://doi.org/10.1023/A:1015614406023