Widths of embeddings in function spaces

Journal of Complexity

Volumn 24, Issue 4, 2008, Pages 545-570

  Vybíral, Jan ^a  

^a FRIEDRICH SCHILLER UNIVERSITY JENA   (Germany)

Author keywords

Approximation numbers; Besov spaces; Gelfand numbers; Kolmogorov numbers; Sobolev embeddings

Indexed keywords

BANACH SPACES; FUNCTIONAL ANALYSIS;

APPROXIMATION NUMBERS; BESOV SPACES; GELFAND NUMBERS; KOLMOGOROV NUMBERS; SOBOLEV EMBEDDINGS;

SOBOLEV SPACES;

EID: 48049124310     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2008.01.002     Document Type: Article

References (66)

1. Bastero J., Bernués J., and Peña A. An extension of Milman's reverse Brunn-Minkowski inequality. Geom. Funct. Anal. 5 3 (1995) 572-581
2. O. Besov,, Trudy. Mat. Inst. Steklova 60 (1961) 42-81. (Engl. transl.: Investigation of a family of function spaces in connection with theorems of imbedding and extension, Amer. Math. Soc. Transl. (2) 40 (1964) 85-126).
3. α, Math. USSR Sb. 2 (1967) 295-317)
4. Caetano A.M. About approximation numbers in function spaces. J. Approx. Theory 94 (1998) 383-395
5. Carl B. Entropy numbers, s-numbers, and eigenvalue problems. J. Funct. Anal. 41 (1981) 290-306
6. Carl B., and Stephani I. Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics vol. 98 (1990), Cambridge University Press, Cambridge
7. Z. Ciesielski, T. Figiel, Construction of Schauder bases in function spaces on smooth compact manifolds. Approximation and function spaces, in: Proceedings of the International Conference, Gdansk, 1979, 1981, pp. 217-232.
8. Cohen A. Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications vol. 32 (2003), North-Holland, Elsevier, Amsterdam
9. Cohen A., Dahmen W., and DeVore R. Multiscale decompositions on bounded domains. Trans. Amer. Math. Soc. 352 (2000) 3651-3685
10. Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 909-996
11. Daubechies I. Ten Lectures on Wavelets (1992), SIAM, Philadelphia
12. d. Trans. Amer. Math. Soc. 335 (1993) 843-864
13. Dispa S. Intrinsic characterizations of Besov spaces on Lipschitz domains. Math. Nachr. 260 (2003) 21-33
14. Dispa S. Intrinsic descriptions using means of differences for Besov spaces on Lipschitz domains. Function Spaces, Differential Operators and Nonlinear Analysis (2003), Birkhäuser, Basel, Boston, Stuttgart 279-287
15. Edmunds D.E., and Triebel H. Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics vol. 120 (1996), Cambridge University Press, Cambridge
16. Frazier M., and Jawerth B. A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93 1 (1990) 34-170
17. M. Frazier, B. Jawerth, G. Weiss, Littlewood-Paley theory and the study of function spaces, in: Regional Conference Series, vol. 79, AMS, Providence, 1991.
18. Fuchssteiner B. Verallgemeinerte Konvexitätsbegriffe und der Satz von Krein-Milman. Math. Ann. 186 (1970) 149-154
19. p-Räume. Math. Ann. 186 (1970) 171-176
20. Garnaev A.Yu., and Gluskin E.D. O. Dokl. Akad. Nauk SSSR 277 (1984) 1048-1052 (Engl. transl.: On widths of the Euclidean ball, Soviet Math. Dokl. 30 (1984) 200-204)
21. Gluskin E.D. O. Vestnik Leningrad. Univ. Seria Mat. 13 (1981) 5-10 (Engl. transl.: On some finite-dimensional problems of in the theory of widths., Vestnik Leningrad Univ. Math. 14 (1982) 163-170)
22. Gluskin E.D. u. Mat. Sb. 120 (1983) 180-189 (Engl. transl.: Norms of random matrices and widths of finite-dimensional sets, Math. USSR Sb. 48 (1984) 173-182)
23. Hernández E., and Weiss G. A First Course on Wavelets (1996), CRC Press, Boca Raton
24. Ismagilov R.S. Uspekhi Mat. Nauk 29 3 (1974) 161-178 (Engl. transl.: Diameters of sets in normed linear spaces and approximation of functions by trigonometric polynomials, Russian Math. Surveys 29(3) (1974) 169-186)
25. Kalton N.J. Compact p-convex sets. Quart. J. Math. Oxford (2) 28 (1977) 301-308
26. Kashin B.S. O. Dokl. Akad. Nauk SSSR 214 (1974) 1024-1026 (Engl. transl.: On Kolmogorov diameters of octahedra, Soviet Math. Dokl. 15 (1974) 304-307)
27. Kashin B.S. O (The diameters of octahedra). Uspekhi Mat. Nauk 30 4 (1975) 251-252
28. Kashin B.S. u. Izv. Akad. Nauk, Seria Mat. 41 (1977) 334-351 (Engl. transl.: Diameters of some finite-dimensional sets and classes of smooth functions, Math. USSR, Izv. 11 (1977) 317-333)
29. Kashin B.S. O (On the diameters of Sobolev classes of small smoothness). Vestnik Mosk. Univ. Seria Mat. 5 (1981) 50-54
30. Kolmogorov A.N. Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. of Math. 37 (1936) 107-110
31. ν. Doklady Akad. Nauk SSSR 48 (1945) 535-538
32. König H. Eigenvalue Distribution of Compact Operators (1986), Birkhäuser, Basel, Boston, Stuttgart
33. q (0, 1), 2 < q < ∞, Russ. Math. Survey 38(5) (1983) 146-147)
34. R. Linde, s-Numbers of diagonal operators and Besov embeddings, in: Proceedings of the 13th Winter School, Suppl. Rend. Circ. Mat. Palermo (1986).
35. Lorentz G.G., Golitschek M.v., and Makovoz Y. Constructive approximation. Advanced problems, Grundlehren der Mathematischen Wissenschaften vol. 304 (1996), Springer, Berlin
36. C. Lubitz, Weylzahlen von Diagonaloperatoren und Sobolev-Einbettungen, Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, 1982.
37. Mai{dotless}̌orov V.E. (Discretization of the problem of diameters). Uspekhi Mat. Nauk 30 6 (1975) 179-180
38. 2 (R). Trans. Amer. Math. Soc. 315 (1989) 69-87
39. Meyer Y. Wavelets and Operators (1992), Cambridge University Press, Cambridge
40. Peetre J. New Thoughts on Besov Spaces, Duke University Mathematics Series (1976), Duke University, Durham
41. Pietsch A. Einige neue Klassen von kompakten linearen Operatoren. Rev. Math. Pures Appl. 8 (1963) 427-447
42. Pietsch A. s-Numbers of operators in Banach spaces. Studia Math. 51 (1974) 201-223
43. A. Pietsch, Operator ideals, Deutsch. Verlag Wiss., Berlin, 1978; North-Holland, Amsterdam, London, New York, Tokyo, 1980.
44. Pietsch A. Eigenvalues and s-Numbers (1987), Cambridge University Press, Cambridge
45. Pietsch A. History of Banach Spaces and Linear Operators (2007), Birkhäuser, Boston, Basel, Berlin
46. Pinkus A. n-Widths in Approximation Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.7 (1985), Springer, Berlin
47. Pisier G. The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics vol. 94 (1989), Cambridge University Press, Cambridge
48. Rellich F. Ein Satz über mittlere Konvergenz. Nach. Wiss. Gesell. Göttingen, Math.-Phys. Kl. (1930) 30-35
49. Ropela S. Spline bases in Besov spaces. Bull. Acad. Pol. Sci., S. Sci. Math. Astron. Phys. 24 (1976) 319-325
50. Rudin W. Functional Analysis. second ed. (1991), McGraw-Hill, New York, St.Louis, San Francisco
51. Runst T., and Sickel W. Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations (1996), W. de Gruyter, Berlin
52. Rychkov V.S. On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. (2) 60 (1999) 237-257
53. Schütt C. Entropy numbers of diagonal operators between symmetric Banach spaces. J. Approx. Theory 40 (1984) 121-128
54. L. Skrzypczak, Approximation and Entropy Numbers of Compact Sobolev Embeddings, Approximation and Probability, vol. 72, Banach Center Publications, Warszawa, 2006, pp. 309-326 (Papers of the Conference Held on the Occasion of the 70th anniversary of Prof. Zbigniew Ciesielski, Bedlewo, Poland, September 20-24, 2004).
55. Sobolev S.L. Mat. Sb. 4 (1938) 471-497 (Engl. transl.: On a theorem of functional analysis, Amer. Math. Soc. Transl. (2) 34 (1963) 39-68)
56. Stechkin S.B. O (On the best approximation of given classes of functions by arbitrary polynomials). Uspekhi Mat. Nauk 9 (1954) 133-134
57. Tikhomirov V.M. u. Uspekhi Mat. Nauk 15 3 (1960) 81-120 (Engl. title: Diameters of sets in function spaces and the theory of best approximations, Russ. Math. Survey 15(3) (1960) 75-111)
58. Triebel H. Interpolation Theory, Function Spaces, Differential Operators (1978), North-Holland, Amsterdam
59. Triebel H. Theory of Function Spaces, Geest & Portig, Leipzig (1983), Birkhäuser, Basel, Boston, Berlin
60. Triebel H. Theory of Function Spaces II (1992), Birkhäuser, Basel, Boston, Berlin
61. Triebel H. Theory of Function Spaces III (2006), Birkhäuser, Basel, Boston, Berlin
62. H. Triebel, Function spaces and wavelets on domains, to appear.
63. H. Triebel, Local Means and Wavelets in Function Spaces, Banach Center Publications, to appear.
64. Wojtaszczyk P. A Mathematical Introduction to Wavelets, London Mathematical Society Student Text vol. 37 (1997), Cambridge University Press, Cambridge
65. Kühn T. A lower estimate for entropy numbers. J. Approx. Theory 110 1 (2001) 120-124
66. Donoho D.L. Compressed sensing. IEEE Trans. Inform. Theory 52 4 (2006) 1289-1306