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Mat. Sb. (N.S.), 1977, Volume 102(144), Number 2, Pages 195–215 (Mi msb2648)  

This article is cited in 13 scientific papers (total in 13 papers)

Imbedding theorems and inequalities in various metrics for best approximations

V. I. Kolyada

Abstract: Let $1\leqslant p<\infty$, and let $\lambda=\{\lambda_n\}$ be a sequence of positive numbers with $\lambda_n\downarrow0$. Denote by $E_p(\lambda)$ the class of all functions $f\in L^p(0,2\pi)$ for which the best approximation by trigonometric polynomials satisfies the condition $E_n^{(p)}(f)=O(\lambda_n)$.
In this paper the relation between best approximations in different metrics is studied. Necessary and sufficient conditions are found for the imbedding $E_p(\lambda)\subset E_q(\mu)$ ($1<p<q<\infty$), where $\{\lambda_n\}$ and $\{\mu_n\}$ are positive sequences with $\lambda_n\downarrow0$ and $\mu_n\downarrow0$.
Furthermore, it is proved that the condition of P. L. Ul'yanov
$$ \sum_{n=1}^\infty n^{q/p-2}\lambda_n^q<\infty\qquad(1\leqslant p<q<\infty) $$
is not only sufficient but is also necessary for the imbedding $E_p(\lambda)\subset L^q(0,2\pi)$.
The question of imbedding $E_p(\lambda)$ in the space of continuous functions is also considered.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 31:2, 171–189

Bibliographic databases:

UDC: 517.5
MSC: Primary 42A08, 41A50, 46E35; Secondary 26A86
Received: 31.12.1975

Citation: V. I. Kolyada, “Imbedding theorems and inequalities in various metrics for best approximations”, Mat. Sb. (N.S.), 102(144):2 (1977), 195–215; Math. USSR-Sb., 31:2 (1977), 171–189

Citation in format AMSBIB
\by V.~I.~Kolyada
\paper Imbedding theorems and inequalities in various metrics for best approximations
\jour Mat. Sb. (N.S.)
\yr 1977
\vol 102(144)
\issue 2
\pages 195--215
\jour Math. USSR-Sb.
\yr 1977
\vol 31
\issue 2
\pages 171--189

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    This publication is cited in the following articles:
    1. Oswald P., “Spline Approximation in the Lp-Metric, 0 Less-Than-Or-Equal-to P Less-Than-Or-Equal-to 1”, Math. Nachr., 94 (1980), 69–96  crossref  mathscinet  zmath  isi
    2. È. A. Storozhenko, “On a problem of Hardy-Littlewood”, Math. USSR-Sb., 47:2 (1984), 557–577  mathnet  crossref  mathscinet  zmath
    3. V. I. Kolyada, “On embedding $H_p^{\omega_1,…,\omega_\nu}$ classes”, Math. USSR-Sb., 55:2 (1986), 351–381  mathnet  crossref  mathscinet  zmath
    4. V. N. Temlyakov, “Approximation of periodic functions of several variables by trigonometric polynomials, and widths of some classes of functions”, Math. USSR-Izv., 27:2 (1986), 285–322  mathnet  crossref  mathscinet  zmath
    5. V. I. Kolyada, “Rearrangements of functions and embedding theorems”, Russian Math. Surveys, 44:5 (1989), 73–117  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. N. A. Il'yasov, “On the Order of Approximation in the Uniform Metric by the Fejér–Zygmund Means on the Classes $E_p[\varepsilon]$”, Math. Notes, 69:5 (2001), 625–633  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. G. A. Akishev, “Obobschennaya sistema Khaara i teoremy vlozheniya v simmetrichnye prostranstva”, Fundament. i prikl. matem., 8:2 (2002), 319–334  mathnet  mathscinet  zmath
    8. N. A. Il'yasov, “On the Order of Decrease of Uniform Moduli of Smoothness for the Classes of Functions $E_{p,m}[\epsilon]$”, Math. Notes, 78:4 (2005), 481–497  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Kolyada V., Marcellan F., “Kernels and Best Approximations Related to the System of Ultraspherical Polynomials”, J. Approx. Theory, 133:2 (2005), 173–194  crossref  mathscinet  zmath  isi
    10. N. A. Ilyasov, “Skorostnaya $L_p$-versiya kriteriya M. Rissa absolyutnoi skhodimosti trigonometricheskikh ryadov Fure”, Tr. IMM UrO RAN, 16, no. 4, 2010, 193–202  mathnet  elib
    11. E. S. Smailov, A. I. Takuadina, “O neuluchshaemosti predelnoi teoremy vlozheniya raznykh metrik v prostranstvakh Lorentsa s vesom Ermitta”, Ufimsk. matem. zhurn., 3:3 (2011), 140–151  mathnet  zmath
    12. N. A. Ilyasov, “O poryadke ravnomernoi skhodimosti chastnykh kubicheskikh summ kratnykh trigonometricheskikh ryadov Fure na klassakh funktsii $H_{1,m}^{l}[\omega]$”, Tr. IMM UrO RAN, 21, no. 4, 2015, 161–177  mathnet  mathscinet  elib
    13. M. E. Turova, “Otsenki nailuchshikh priblizhenii funktsii spektrom iz giperbolicheskikh krestov”, Mezhdunar. nauch.-issled. zhurn., 2015, no. 5-1(36), 29–31  mathnet  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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