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Diameters of certain sets in function spaces

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Abstract

The precise values of the diameters (according to A. N. Kolmogorov) of the class H ωC in the space C[a, b] and lower bounds for the diameters of the class H ωp in the space∼Lp(0, 2π) (1≤p≤∞), for any modulus of continuityω(δ), are obtained. The latter bounds give the exact values of the odd-numbered diameters of the class\(H_2^{1/2} = H_2^{\delta _{1/2} }\) and the exact order of decay of the diameters of the class H ω1 .

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Literature cited

  1. A. Kolmogorov (Kolmogoroff), “Uber die beste Annäherung von Funktionen einer gegebenen Funktionenklasse,” Ann. of Math.,37, 107–110 (1936).

    Google Scholar 

  2. S. M. Nikol'skii, “Fourier series of functions with given modulus of continuity,” Dokl. Akad. Nauk SSSR,52, No. 3, 191–194 (1946).

    Google Scholar 

  3. A. L. Brown, “Best n-dimensional approximation to sets of functions,” Proc. London Math. Soc.,14, No. 56, 577–594 (1964).

    Google Scholar 

  4. N. I. Chernykh, “Best approximation to periodic functions by trigonometric polynomials,” Matem. Zametki,2, No. 5, 513–522 (1967).

    Google Scholar 

  5. S. B. Stechkin, “Remarks on a theorem of Jackson” Trudy Matem. Inst. Akad. Nauk SSSR,88, 17–19 (1967).

    Google Scholar 

  6. A. L. Garkavi, “Best nets and best cross sections of sets in normed spaces” Izv. Akad. Nauk SSSR, Ser. Matem.,26, No. 1, 87–106 (1962).

    Google Scholar 

  7. Yu. I. Makovoz, Best Approximation Theorems and Diameters in Banach Spaces, Abstract of Candidate's Dissertation, Minsk (1969).

  8. V. M. Tikhomirov, “Diameters of sets in function spaces and the theory of best approximation,” Uspekhi Matem. Nauk,15, No. 3, 81–120 (1960).

    Google Scholar 

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Authors and Affiliations

  1. Erevan Polytechnic Institute, USSR

    Yu. I. Grigoryan

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  1. Yu. I. Grigoryan
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Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 637–646, May, 1973.

In conclusion, the author wishes to thank A. L. Garkavi, A. V. Efimov, and S. B. stechkin for their help.

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Grigoryan, Y.I. Diameters of certain sets in function spaces. Mathematical Notes of the Academy of Sciences of the USSR 13, 383–388 (1973). https://doi.org/10.1007/BF01147464

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  • DOI: https://doi.org/10.1007/BF01147464

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