Strict 𝑆-numbers of the Volterra operator

Bakşi, Özlem; Khan, Taqseer; Lang, Jan; Musil, Vít

doi:10.1090/proc/13769

Strict $S$-numbers of the Volterra operator
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by Özlem Bakşi, Taqseer Khan, Jan Lang and Vít Musil PDF

Proc. Amer. Math. Soc. 146 (2018), 723-731 Request permission

Abstract:

For Volterra operator $V\colon L^1(0,1)\to C[0,1]$ and summation operator $\sigma \colon \ell ^1\to c$, we obtain exact values of Approximation, Gelfand, Kolmogorov and Isomorphism numbers.

References

I. A. Alam, G. Habib, P. Lefèvre, and F. Maalouf, Essential norms of Volterra and Cesàro operators on Müntz spaces, arXiv:1612.03218, 2016.
J. Bourgain and M. Gromov, Estimates of Bernstein widths of Sobolev spaces, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 176–185. MR 1008722, DOI 10.1007/BFb0090054
M. Š. Birman and M. Z. Solomjak, Piecewise polynomial approximations of functions of classes $W_{p}{}^{\alpha }$, Mat. Sb. (N.S.) 73 (115) (1967), 331–355 (Russian). MR 0217487
Bernd Carl and Irmtraud Stephani, Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, vol. 98, Cambridge University Press, Cambridge, 1990. MR 1098497, DOI 10.1017/CBO9780511897467
D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1987. Oxford Science Publications. MR 929030
David E. Edmunds and W. Desmond Evans, Hardy operators, function spaces and embeddings, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004. MR 2091115, DOI 10.1007/978-3-662-07731-3
D. E. Edmunds and J. Lang, Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case, Math. Nachr. 279 (2006), no. 7, 727–742. MR 2226408, DOI 10.1002/mana.200510389
D. E. Edmunds and J. Lang, Bernstein widths of Hardy-type operators in a non-homogeneous case, J. Math. Anal. Appl. 325 (2007), no. 2, 1060–1076. MR 2270069, DOI 10.1016/j.jmaa.2006.02.025
Jan Lang and David Edmunds, Eigenvalues, embeddings and generalised trigonometric functions, Lecture Notes in Mathematics, vol. 2016, Springer, Heidelberg, 2011. MR 2796520, DOI 10.1007/978-3-642-18429-1
A. Kolmogoroff, Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse, Ann. of Math. (2) 37 (1936), no. 1, 107–110 (German). MR 1503273, DOI 10.2307/1968691
Pascal Lefèvre, The Volterra operator is finitely strictly singular from $L^1$ to $L^\infty$, J. Approx. Theory 214 (2017), 1–8. MR 3588527, DOI 10.1016/j.jat.2016.11.001
V. I. Levin, On a class of integral inequalities, Recueil Mathématiques 4 (1938), no. 46, 309–331.
Mikhail A. Lifshits and Werner Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion, Mem. Amer. Math. Soc. 157 (2002), no. 745, viii+87. MR 1895252, DOI 10.1090/memo/0745
Albrecht Pietsch, $s$-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223. MR 361883, DOI 10.4064/sm-51-3-201-223
Albrecht Pietsch, History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2300779
Allan Pinkus, $n$-widths in approximation theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 7, Springer-Verlag, Berlin, 1985. MR 774404, DOI 10.1007/978-3-642-69894-1
Allan Pinkus, $n$-widths of Sobolev spaces in $L^p$, Constr. Approx. 1 (1985), no. 1, 15–62. MR 766094, DOI 10.1007/BF01890021
Erhard Schmidt, Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet, Math. Ann. 117 (1940), 301–326 (German). MR 3430, DOI 10.1007/BF01450021
V. M. Tihomirov and S. B. Babadžanov, Diameters of a function class in an $L_{p}$-space $(p\geq 1)$, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 11 (1967), no. 2, 24–30 (Russian, with Uzbek summary). MR 0209747
V. M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Russian Math. Surveys 15 (1960), no. 3, 75–111. MR 0117489, DOI 10.1070/RM1960v015n03ABEH004093