Estimates of Kolmogorov, Gelfand and linear 𝑛-widths on compact Riemannian manifolds

Pesenson, Isaac

doi:10.1090/proc/13054

Estimates of Kolmogorov, Gelfand and linear $n$-widths on compact Riemannian manifolds
HTML articles powered by AMS MathViewer

by Isaac Z. Pesenson PDF

Proc. Amer. Math. Soc. 144 (2016), 2985-2998 Request permission

Abstract:

We determine lower and exact estimates of Kolmogorov, Gelfand and linear $n$-widths of unit balls in Sobolev norms in $L_{p}$-spaces on compact Riemannian manifolds. As it was shown by us previously these lower estimates are exact asymptotically in the case of compact homogeneous manifolds. The proofs rely on two-sides estimates for the near-diagonal localization of kernels of functions of elliptic operators.

References

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
Gavin Brown and Feng Dai, Approximation of smooth functions on compact two-point homogeneous spaces, J. Funct. Anal. 220 (2005), no. 2, 401–423. MR 2119285, DOI 10.1016/j.jfa.2004.10.005
Gavin Brown, Dai Feng, and Sun Yong Sheng, Kolmogorov width of classes of smooth functions on the sphere $\Bbb S^{d-1}$, J. Complexity 18 (2002), no. 4, 1001–1023. MR 1933699, DOI 10.1006/jcom.2002.0656
B. Bordin, A. K. Kushpel, J. Levesley, and S. A. Tozoni, Estimates of $n$-widths of Sobolev’s classes on compact globally symmetric spaces of rank one, J. Funct. Anal. 202 (2003), no. 2, 307–326. MR 1990527, DOI 10.1016/S0022-1236(02)00167-2
Daryl Geller and Azita Mayeli, Continuous wavelets on compact manifolds, Math. Z. 262 (2009), no. 4, 895–927. MR 2511756, DOI 10.1007/s00209-008-0405-7
Daryl Geller and Isaac Z. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds, J. Geom. Anal. 21 (2011), no. 2, 334–371. MR 2772076, DOI 10.1007/s12220-010-9150-3
Daryl Geller and Isaac Z. Pesenson, $n$-widths and approximation theory on compact Riemannian manifolds, Commutative and noncommutative harmonic analysis and applications, Contemp. Math., vol. 603, Amer. Math. Soc., Providence, RI, 2013, pp. 111–122. MR 3204029, DOI 10.1090/conm/603/12043
Daryl Geller and Isaac Z. Pesenson, Kolmogorov and linear widths of balls in Sobolev spaces on compact manifolds, Math. Scand. 115 (2014), no. 1, 96–122. MR 3250051, DOI 10.7146/math.scand.a-18005
E. D. Gluskin, Norms of random matrices and diameters of finite-dimensional sets, Mat. Sb. (N.S.) 120(162) (1983), no. 2, 180–189, 286 (Russian). MR 687610
Klaus Höllig, Approximationszahlen von Sobolev-Einbettungen, Math. Ann. 242 (1979), no. 3, 273–281 (German). MR 545219, DOI 10.1007/BF01420731
Lars Hörmander, The analysis of linear partial differential operators. III, Classics in Mathematics, Springer, Berlin, 2007. Pseudo-differential operators; Reprint of the 1994 edition. MR 2304165, DOI 10.1007/978-3-540-49938-1
A. I. Kamzolov, The best approximation of classes of functions $W^{\alpha }_{p}\,(S^{n})$ by polynomials in spherical harmonics, Mat. Zametki 32 (1982), no. 3, 285–293, 425 (Russian). MR 677597
A. I. Kamzolov, Kolmogorov widths of classes of smooth functions on a sphere, Uspekhi Mat. Nauk 44 (1989), no. 5(269), 161–162 (Russian); English transl., Russian Math. Surveys 44 (1989), no. 5, 196–197. MR 1040278, DOI 10.1070/RM1989v044n05ABEH002208
B. S. Kašin, The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 334–351, 478 (Russian). MR 0481792
George G. Lorentz, Manfred v. Golitschek, and Yuly Makovoz, Constructive approximation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 304, Springer-Verlag, Berlin, 1996. Advanced problems. MR 1393437, DOI 10.1007/978-3-642-60932-9
V. E. Maĭorov, Linear diameters of Sobolev classes, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1127–1130 (Russian). MR 514776
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
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463, DOI 10.1515/9781400886104
Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1