Yu. N. Subbotin

Diameter of class WrL in L(0, 2π)and spline function approximation

The (2n−1)-dimensional diameter of the class WrL is found in the metric of the space L(0, 2π). The rate of convergence is also studied of the interpolating spline functions Sr(x, h) with equidistant nodes to a function F(x) which has a uniformly continuous k-th derivative (r ≥ k ≥ 0) on the entire axis.

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations Em,n(f)p with undetermined n nodes in the metric of the space Lp(0, 1) (1≤p≤∞) is studied in a class of functionsf(x) for which ∥fm+1 (x)∥Lq(0, 1)≤1(1≤q≤t8) or var {f(m) (x); 0, 1}≤1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n → ∞, and asymptotic formulas are obtained for p=∞ and 1≤q≤∞ in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.

Best approximation of a differentiation operator in L2-space

This paper contains the magnitude of the best approximation in the L2-sense of a k-th order differentiation operator of a bounded linear operator A(f) which acts on the class of functions which are differentiable n times.

Extremal functional interpolation and approximation by splines

This article is the authors abstract of his dissertation for the degree Doctor of Physico-mathematical Science. The dissertation was defended on April 22, 1974 at the meeting of the Academic Council for the award of higher degrees in mathematics and mechanics at the Novosibirsk State University. Official opponents were: Academician N. N. Yanenko, Corresponding Member of the Academy of Sciences of the Ukrainian SSR, Doctor of Physicomathematical Sciences; Professor N. P. Korenchuk, Doctor of Physicomathematical Sciences; Professor G. Sh. Rubinshtein.

Best approximation of a class of functions by another class

Let B be the space C=C(I) or L=L1(I), where I=(−∞, α). By ∥ϕ(n)(x)∥c we will mean the upper bound of the modulus of the values of the derivative of the function <p(n-l)(x). For any integers, k and n, 0<k<n, a lower bound for is obtained and the exact value is given for 0<k<n≤5. [6 references are cited.]

Approximation by splines and smooth bases in C(0, 2π)

An estimate of the deviation of the splines interpolating on a uniform net a function continuous on the whole axis by means of the kth module of continuity. These results are applied for the construction of smooth bases in C(0, 2π).

A relation between spline approximation and the problem of the approximation of one class by another

An investigation of the approximation in Lq(−∞, ∞) of differentiable functions whose k-th derivatives belong to Lp(−∞, ∞), by splines Sm (x) with nonfixed nodes, under the extra assumption that the norms in Ls(−∞, ∞) of theirl-th derivatives have a common bound. A relation is established with the problem of approximating functions of one class by functions of another class.

Linear method for the approximation of differentiable functions

This article is devoted to the problem of the approximation of functions in the metric of space Lp(0, 1), the s-th derivative of the functions being continuous and the (s + l)-th derivative belonging to the space Lq(0, 1), with p, q ≥1, by spline functions of order s with fixed “almost” uniform nodes.