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 Mat. Sb. (N.S.), 1983, Volume 122(164), Number 4(12), Pages 481–510 (Mi msb2310)

A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications

V. V. Peller

Abstract: The main result is the following description of Hankel operators in the Schatten-von Neumann class $\mathfrak{S}_p$ when $0<p<1$:
$$\Gamma_\varphi\in\mathfrak S_p\Leftrightarrow\varphi\in B_p^{1/p},$$
where $\Gamma_\varphi$ is the Hankel operator with symbol $\varphi$, and $B_p^{1/p}$ is the Besov class. This result extends results obtained earlier for $1\leqslant p<+\infty$ by the author to the case $0<p<1$. Also described are the Hankel operators in the Schatten–Lorentz classes $\mathfrak S_{pq}$, $0<p<+\infty$, $0<q\leqslant\infty$.
Precise descriptions of classes of functions defined in terms of rational approximation in the bounded mean oscillation norm are given as an application, along with a complete investigation of the case where the decrease is of power order, and some precise results on rational approximation in the $L^\infty$-norm. Certain other applications are also considered.
Bibliography: 57 titles.

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English version:
Mathematics of the USSR-Sbornik, 1985, 50:2, 465–494

Bibliographic databases:

UDC: 517.98+517.5
MSC: Primary 41A20, 41A25, 47B10, 47G05; Secondary 46E30, 46E35, 47B35, 47B05, 60G10, 60G15

Citation: V. V. Peller, “A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications”, Mat. Sb. (N.S.), 122(164):4(12) (1983), 481–510; Math. USSR-Sb., 50:2 (1985), 465–494

Citation in format AMSBIB
\Bibitem{Pel83} \by V.~V.~Peller \paper A description of Hankel operators of class $\mathfrak S_p$ for $p>0$, an investigation of the rate of rational approximation, and other applications \jour Mat. Sb. (N.S.) \yr 1983 \vol 122(164) \issue 4(12) \pages 481--510 \mathnet{http://mi.mathnet.ru/msb2310} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=725454} \zmath{https://zbmath.org/?q=an:0561.47022} \transl \jour Math. USSR-Sb. \yr 1985 \vol 50 \issue 2 \pages 465--494 \crossref{https://doi.org/10.1070/SM1985v050n02ABEH002840} 

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This publication is cited in the following articles:
1. A. A. Pekarskii, “Classes of analytic functions determined by best rational approximations in $H_p$”, Math. USSR-Sb., 55:1 (1986), 1–18
2. A. A. Pekarskii, “Tchebycheff rational approximation in the disk, on the circle, and on a closed interval”, Math. USSR-Sb., 61:1 (1988), 87–102
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5. Devore R. Popov V., “Interpolation Spaces and Non-Linear Approximation”, Lect. Notes Math., 1302 (1988), 191–205
6. Peller V., “Smoothness of Schmidt Functions of Smooth Hankel-Operators”, Lect. Notes Math., 1302 (1988), 337–346
7. Peetre J. Karlsson J., “Rational Approximation-Analysis of the Work of Pekarskii”, Rocky Mt. J. Math., 19:1 (1989), 313–333
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9. V. A. Prokhorov, “Rational approximation of analytic functions”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 139–164
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11. A. P. Petukhov, “Convergence of Fourier series for functions in the classes of Besov–Lizorkin–Triebel”, Math. Notes, 56:1 (1994), 694–698
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18. M. Putinar, “On a diagonal Padé approximation in two complex variables”, Numer Math, 93:1 (2002), 131
19. A. A. Pekarskii, “New Proof of the Semmes Inequality for the Derivative of the Rational Function”, Math. Notes, 72:2 (2002), 230–236
20. Prokhorov V., “On l-P-Generalization of a Theorem of Adamyan, Arov, and Krein”, J. Approx. Theory, 116:2 (2002), 380–396
21. Aleksandrov A., Peller V., “Distorted Hankel Integral Operators”, Indiana Univ. Math. J., 53:4 (2004), 925–940
22. V. L. Kreptogorskii, “Interpolation of Rational Approximation Spaces Belonging to the Besov Class”, Math. Notes, 77:6 (2005), 809–816
23. Joaquim Ortega-Cerdà, Jordi Saludes, “Marcinkiewicz–Zygmund inequalities”, Journal of Approximation Theory, 145:2 (2007), 237
24. L. Baratchart, M. L. Yattselev, “Meromorphic approximants to complex Cauchy transforms with polar singularities”, Sb. Math., 200:9 (2009), 1261–1297
25. Yu. S. Kolomoitsev, “On approximation of functions by trigonometric polynomials with incomplete spectrum in $L_p$, $0<p<1$”, J. Math. Sci. (N. Y.), 165:4 (2010), 463–472
26. Opmeer M.R., “Decay of Hankel Singular Values of Analytic Control Systems”, Syst. Control Lett., 59:10 (2010), 635–638
27. Opmeer M.R., “Model Reduction for Distributed Parameter Systems: a Functional Analytic View”, 2012 American Control Conference (Acc), Proceedings of the American Control Conference, IEEE Computer Soc, 2012, 1418–1423
28. Isralowitz J., “Schatten P Class Commutators on the Weighted Bergman Space l-a(2)(B-N,B-, D Nu Gamma) for 2N/(N+1+Gamma) < P < Infinity”, Indiana Univ. Math. J., 62:1 (2013), 201–233
29. T. S. Mardvilko, A. A. Pekarskii, “Conjugate Functions on the Closed Interval and Their Relationship with Uniform Rational and Piecewise Polynomial Approximations”, Math. Notes, 99:2 (2016), 272–283
30. A. B. Aleksandrov, V. V. Peller, “Operator Lipschitz functions”, Russian Math. Surveys, 71:4 (2016), 605–702
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