A note on approximation by trigonometric polynomials

Research output

Abstract

Let E=∪k=1n[akbk]⊂ℝ; if n > 1, then we assume that the segments [a k, b k] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by H ω + r(E) the space of functions f defined on E such that |f (r)(x 2) − f (r)(x 1)| ≤ c fω(|x 2 − x 1|), x 1, x 2 ∈ E, f (0) ≡ f. Assume that a modulus of continuity ω satisfies the condition∫0xω(t)tdt+x∫x∞ω(t)t2dt≤cω(x). We find a constructive description of the space H ω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ H ω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.

Original languageEnglish
Pages (from-to)981-984
JournalJournal of Mathematical Sciences
Volume243
Issue number6
DOIs
Publication statusPublished - 2019

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Trigonometric Polynomial
Polynomials
H-space
Modulus of Continuity
Approximation
Pairwise
Disjoint
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Scopus subject areas

  • Mathematics(all)

Cite this

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title = "A note on approximation by trigonometric polynomials",
abstract = "Let E=∪k=1n[akbk]⊂ℝ; if n > 1, then we assume that the segments [a k, b k] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by H ω + r(E) the space of functions f defined on E such that |f (r)(x 2) − f (r)(x 1)| ≤ c fω(|x 2 − x 1|), x 1, x 2 ∈ E, f (0) ≡ f. Assume that a modulus of continuity ω satisfies the condition∫0xω(t)tdt+x∫x∞ω(t)t2dt≤cω(x). We find a constructive description of the space H ω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ H ω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.",
author = "Shirokov, {N. A.}",
note = "Shirokov, N.A. A Note on Approximation by Trigonometric Polynomials. J Math Sci 243, 981–984 (2019) doi:10.1007/s10958-019-04598-y",
year = "2019",
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language = "English",
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pages = "981--984",
journal = "Journal of Mathematical Sciences",
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T1 - A note on approximation by trigonometric polynomials

AU - Shirokov, N. A.

N1 - Shirokov, N.A. A Note on Approximation by Trigonometric Polynomials. J Math Sci 243, 981–984 (2019) doi:10.1007/s10958-019-04598-y

PY - 2019

Y1 - 2019

N2 - Let E=∪k=1n[akbk]⊂ℝ; if n > 1, then we assume that the segments [a k, b k] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by H ω + r(E) the space of functions f defined on E such that |f (r)(x 2) − f (r)(x 1)| ≤ c fω(|x 2 − x 1|), x 1, x 2 ∈ E, f (0) ≡ f. Assume that a modulus of continuity ω satisfies the condition∫0xω(t)tdt+x∫x∞ω(t)t2dt≤cω(x). We find a constructive description of the space H ω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ H ω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.

AB - Let E=∪k=1n[akbk]⊂ℝ; if n > 1, then we assume that the segments [a k, b k] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by H ω + r(E) the space of functions f defined on E such that |f (r)(x 2) − f (r)(x 1)| ≤ c fω(|x 2 − x 1|), x 1, x 2 ∈ E, f (0) ≡ f. Assume that a modulus of continuity ω satisfies the condition∫0xω(t)tdt+x∫x∞ω(t)t2dt≤cω(x). We find a constructive description of the space H ω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ H ω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.

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