Resumen de Approximation of functions of several variables by continuous linear splines on rectangular grids
Elena E. Berdysheva, S.N. Mehmonzoda, M. Sh. Shabozov
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.
