Strict $S$-numbers of the Volterra operator
HTML articles powered by AMS MathViewer
- 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
Additional Information
- Özlem Bakşi
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174 — and — Department of Mathematics, Yildiz Technical University, Faculty of Art and Science, Istanbul, Turkey
- Email: baksi@yildiz.edu.tr
- Taqseer Khan
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174 — and — Aligarh Muslim University, Aligarh, U.P.-202002, India
- Email: taqi.khan91@gmail.com
- Jan Lang
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174
- MR Author ID: 367896
- Email: lang@math.osu.edu
- Vít Musil
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210-1174 — and — Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
- Email: musil@karlin.mff.cuni.cz
- Received by editor(s): February 27, 2017
- Received by editor(s) in revised form: March 31, 2017
- Published electronically: October 5, 2017
- Additional Notes: This research was partly supported by the United States-India Educational Foundation (USIEF), by the grant P201-13-14743S of the Grant Agency of the Czech Republic and by the grant SVV-2017-260455
- Communicated by: Thomas Schlumprecht
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 723-731
- MSC (2010): Primary 47B06; Secondary 47G10
- DOI: https://doi.org/10.1090/proc/13769
- MathSciNet review: 3731705