A Guide to the Russian Academy of Sciences - University of Texas ...

The series of investigations on the spline theory was carried out. The existence theorems
for the interpolating and interpolating in mean polynomial splines with uniformed
knots in particular for different sets of nodal points of splines and interpolating
points have been proved, the error bounds of approximations have been obtained
for different classes of functions (Iurii. N. Subbotin). Analogous results have been
achived for L-splines connected with the ordinary linear differential operator L with
constant coefficients and for periodic splines generated by convolution operator (V.
T. Shevaldin).
There has been created a new method of approximation of functions by polynomial splines
with unfixed knots and sharp estimates for degree of corresponding approximation
have been established (Iurii. N. Subbotin, N. I. Chernykh).
The existence theorems have been proved for interpolating Dm -splines within the bounded
and unbounded domains with Lipshitz's boundaries. Proper in the sense of order
estimates of error approximation into Wsq -seminorm of classes Wrp was obtained
for "almost equidistant" points of interpolation and for all permissible s,q,r,p (O.
V. Matveev).
There have been found final results on approximation by the interpolating cubic splines and
their derivatives in the presence of the local restriction on the neighboring intervals
between the knots of splines (N. L. Zmatrakov).The exact, asymptotically exact or
exact in the sense of order estimates have been established for the approximation by
means of L-spline defined by ordinary differential operator with constant
coefficients (S. I. Novikov, A. A. Sazanov).
There has been developed a theory of the best approximation of unbounded linear operators
by means of bounded ones. Connections between this and other extreme problems
have been discovered and in a several certain cases their solutions have been found.
The above mentioned S. B. Stechkin's result on the approximation of operators has to be
pointed out here as the first one.
There have been established the relations between the continuity modulus of unbounded
operator on a class within Banach space on the one hand and the best approximation
of the operator by the linear bounded ones on the other as well as the connection of
these problems with the problem of the operator's values recovery on inexactly
known elements of the class. The proper connections between first two problem
and the problem of the best approximation of one class of elements by another was
made clear . The problems of the best approximation of an invariant operator with
respect to transference on an invariant class of element was investigated in details. It
had been shown, that the error of the best approximation in Lp -spaces on the real
axis of the k-order derivative operator on the class of functions with Lq -bounded
derivations of order n could be represented by means of the best constant into
inequality between the norm of derivatives in Lr,s -spaces, which conjugate is the
space of multiplicators from Lr into Ls (V. V. Arestov).
The necessary and sufficient conditions have been found after which the boundness Lq -
norm on of a n-variable function f and its Sobolev's Wlr (Rn )-seminorm
guarantees the boundness of its intermediate derivatives in Lp -metric and the
corresponding inequality has been written out. In several new cases the best
constant in such inequalities have been found. On the problem of optimal recovery
of derivatives of a n-variable function from Sobolev's class on the basis of its
inexact values the necessary and sufficient conditions have been found for the
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