Nikol'skii space

A Banach space consisting of functions defined on an open set of an -dimensional Euclidean space and having certain difference-differentiability properties characterized by a vector , , , in the -metric, . The concept was introduced by S.M. Nikol'skii.

The Nikol'skii space can be described in terms of properties of the partial derivatives of order in the variable , where , is an integer, , ; if denotes the difference of order and of step with respect to of a function , then

if and only if has in generalized partial derivatives

, and if for ,

while for ,

where is the set of points that are distant by more than from the boundary of and is arbitrary.

The space is defined as the union of all for all , .

If , then for any , , , the Nikol'skii space is not empty and contains functions that do not belong to for any and any .

When , the are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a Hölder space. The concept of a Nikol'skii space generalizes to the case of functions that are defined on sufficiently smooth manifolds (see [2]).

There is a description of the Nikol'skii space in terms of properties of the differences of the partial derivatives of orders less than ; in particular, of those of sufficiently high order of the function itself.

Let be an isotropic space, that is, . If the domain is such that any function of class can be extended with preservation of the class to the whole space , that is, in such a way that the extended function belongs to (this is always the case when the boundary of the domain is sufficiently smooth), then if and only if for any non-negative integers and such that the function has partial derivatives of all orders and there is a constant such that

(1)

where and is the difference of order of with vectorial step . Condition (1) is equivalent to the analogous condition for the modulus of continuity of : There is an such that

where

If , for , denotes the infimum of all for which (1) holds for all and all partial derivatives of an admissible order , then

is a norm in and the norms obtained for distinct admissible pairs are equivalent.

A Nikol'skii space consisting of functions defined on the whole space can be characterized in terms of best approximations of the functions in this space by entire functions of exponential type. Let be the best approximation (error) in the -metric of an by entire functions of exponential type and of order in , . The following direct and inverse theorems of Bernshtein, Jackson and Zygmund type hold for Nikol'skii functions.

If , then for any ,

(2)

(the constant does not depend on ).

Conversely, if (2) holds for a function for , , , and if is an entire function of order 1 in each variable for which

(which exist for , by (2)), then

where

(3)

and the constants in (2) and in (3) do not depend on , .

If is periodic in all variables, then a similar description of a Nikol'skii space can be given by means of best approximations of the functions by trigonometric polynomials instead of entire functions of exponential type (see [1], [4]).

Nikol'skii spaces can be described by means of a Bessel–Macdonald operator applied to some class of generalized functions (see Imbedding theorems).

For the space Nikol'skii has proved transitive imbedding theorems for various dimensions and metrics (see [3] and Imbedding theorems), which were subsequently carried over to more general classes of functions. These theorems show that Nikol'skii spaces form a closed system relative to the boundary values of the functions occurring in them: The traces of functions in Nikol'skii spaces on smooth manifolds can in a certain sense be completely described in terms of Nikol'skii spaces.

The properties of Nikol'skii spaces make it possible to obtain necessary and sufficient conditions for the solvability of the Dirichlet problem in appropriate Nikol'skii spaces in terms of membership of the boundary function to a certain Nikol'skii space: A harmonic function belongs to the class , , where is a bounded domain in with a sufficiently smooth boundary , if and only if the boundary values belong to the class . This implies for , in particular, that if , , then the Dirichlet integral of over is finite, therefore, the Dirichlet problem can be solved by a direct variational method. From imbedding theorems for Nikol'skii spaces it follows that if the Dirichlet integral of over is finite, then (see [6]). A generalization of Nikol'skii spaces are the Besov spaces .

References

[1]	S.M. Nikol'skii, "Inequalities for entire functions of finite order and their application to the theory of differentiable functions in several variables" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 244–278 (In Russian)
[2]	S.M. Nikol'skii, "Properties of certain classes of functions of several variables on a differentiable manifold" Mat. Sb. , 33 : 2 (1953) pp. 261–326 (In Russian)
[3]	S.M. Nikol'skii, "Imbedding theorems for functions with partial derivatives, considered in differential metrics" Dokl. Akad. Nauk SSSR , 118 : 1 (1958) pp. 35–37 (In Russian)
[4]	S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[5]	O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)
[6]	S.M. Nikol'skii, "On the solution of the polyharmonic equation by a variational method" Dokl. Akad. Nauk SSSR , 88 : 3 (1953) pp. 409–411 (In Russian)