### Abstract

Let L_{2} be the space of 2π-periodic square-summable functions and E(f, X)_{2} be the best approximation of f by the space X in L_{2}. For n ∈ ℕ and B ∈ L_{2}, let S_{B} _{,} _{n} be the space of functions s of the form s(x)=∑j=02n−1βjB(x−jπn). This paper describes all spaces S_{B} _{,} _{n} that satisfy the exact inequality E(f,SB,n)2≤1nr∥f(r)∥2. (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

Original language | English |
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Pages (from-to) | 15-22 |

Number of pages | 8 |

Journal | Vestnik St. Petersburg University: Mathematics |

Volume | 51 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

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### Scopus subject areas

- Mathematics(all)

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*Vestnik St. Petersburg University: Mathematics*, vol. 51, no. 1, pp. 15-22. https://doi.org/10.3103/S1063454118010120

**Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces.** / Vinogradov, O. L.; Ulitskaya, A. Yu.

Research output

TY - JOUR

T1 - Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces

AU - Vinogradov, O. L.

AU - Ulitskaya, A. Yu

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ℕ and B ∈ L2, let SB , n be the space of functions s of the form s(x)=∑j=02n−1βjB(x−jπn). This paper describes all spaces SB , n that satisfy the exact inequality E(f,SB,n)2≤1nr∥f(r)∥2. (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

AB - Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ℕ and B ∈ L2, let SB , n be the space of functions s of the form s(x)=∑j=02n−1βjB(x−jπn). This paper describes all spaces SB , n that satisfy the exact inequality E(f,SB,n)2≤1nr∥f(r)∥2. (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

KW - best approximation

KW - sharp constants

KW - shift spaces

UR - http://www.scopus.com/inward/record.url?scp=85045068984&partnerID=8YFLogxK

U2 - 10.3103/S1063454118010120

DO - 10.3103/S1063454118010120

M3 - Article

AN - SCOPUS:85045068984

VL - 51

SP - 15

EP - 22

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -