TY - JOUR

T1 - A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators

AU - Vinogradov, O. L.

AU - Gladkaya, A. V.

N1 - Vinogradov, O.L., Gladkaya, A.V. A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators. J Math Sci 217, 3–22 (2016). https://doi.org/10.1007/s10958-016-2950-7

PY - 2016/8/1

Y1 - 2016/8/1

N2 - Let σ > 0, m, r ∈ ℕ, m ≥ r, let Sσ,m be the space of splines of order m and minimal defect with nodes jπσ (j ∈ ℤ), and let Aσ,m(f)p be the best approximation of a function f by the set Sσ,m in the space Lp(ℝ). It is known that for p = 1,+∞, (Formula presented.) where Kr are the Favard constants. In this paper, linear operators Xσ,r,m with values in Sσ,m such that for all p ∈ [1,+∞] and f ∈ Wp (r)(ℝ),ǁf−Xσ,r,m(f)ǁp≤Kr/σrǁf(r)ǁp are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.

AB - Let σ > 0, m, r ∈ ℕ, m ≥ r, let Sσ,m be the space of splines of order m and minimal defect with nodes jπσ (j ∈ ℤ), and let Aσ,m(f)p be the best approximation of a function f by the set Sσ,m in the space Lp(ℝ). It is known that for p = 1,+∞, (Formula presented.) where Kr are the Favard constants. In this paper, linear operators Xσ,r,m with values in Sσ,m such that for all p ∈ [1,+∞] and f ∈ Wp (r)(ℝ),ǁf−Xσ,r,m(f)ǁp≤Kr/σrǁf(r)ǁp are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.

UR - http://www.scopus.com/inward/record.url?scp=84978173187&partnerID=8YFLogxK

U2 - 10.1007/s10958-016-2950-7

DO - 10.1007/s10958-016-2950-7

M3 - Article

AN - SCOPUS:84978173187

VL - 217

SP - 3

EP - 22

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 1

ER -