Approximation of functions of several variables by continuous linear splines on rectangular grids

• Autores: Elena E. Berdysheva, S.N. Mehmonzoda, M. Sh. Shabozov
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 12, Nº. 0, 2020-2021, págs. 1-23
• Idioma: inglés
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• Resumen

  • We consider approximation of functions of several variables by continuous linear splines interpolating the given function in the knots of a rectilinear lattice. For function classes defined in terms of a modulus of continuity, we give an exact estimate for the error of approximation. In the particular case when the modulus of continuity is concave and the distance between points in R ͩ is measured in the ˡp-norm with 1 ≤ p ≤ 3, we calculate an explicit value of the exact approximation error on the class. Surprisingly, the behavior changes dramatically if p > 3. We show that the our estimate is no longer true, in general, when p > 3. We also consider approximation of a first derivative of a function by the corresponding derivative of the linear continuous spline and obtain an upper estimate for the error of approximation for an arbitrary modulus of continuity, all 1 ≤ p ≤ ∞, and triangulations of the staircase type.