Bernstein-Jackson-type inequalities and Besov spaces associated with unbounded operators
© Dmytryshyn and Lopushansky; licensee Springer. 2014
Received: 1 January 2013
Accepted: 20 February 2014
Published: 4 March 2014
Besov-type interpolation spaces and appropriate Bernstein-Jackson inequalities, generated by unbounded linear operators in a Banach space, are considered. In the case of the operator of differentiation these spaces and inequalities exactly coincide with the classical ones. Inequalities are applied to a best approximation problem in a Banach space, particularly, to spectral approximations of regular elliptic operators.
KeywordsBesov-type spaces Bernstein-Jackson-type inequalities
1 Introduction and preliminaries
The classical Jackson and Bernstein inequalities express a relation between smoothness modules of functions and properties of their best approximations by polynomials or entire functions of exponential type that can be characterized with the help of Besov norms [, Sections 1.5, 7.2]. These results are extended to approximations of smooth functions by wavelets (see e.g. [2–4]), and to approximations of linear operators in Banach spaces by operators with finite ranks , and other similar approximations.
The motivation of our work is to extend the Bernstein-Jackson inequalities to cases of best spectral approximations in a Banach space. An analog of Bernstein-Jackson inequalities in the case of approximations in the space on a Lie group G by spectral subspaces of the group sublaplacian , where is its spectral resolution, is established in [6, 7]. Spectral subspaces are analogous subspaces of entire functions of exponential type. The appropriate Besov space is characterized by the functional of best approximation .
This approach is a prototype of our generalizations. We consider a closed operator A in a Banach space instead of and replace the spectral subspaces by invariant subspaces of exponential type entire vectors of A. Note that similar subspaces of exponential type entire vectors have appeared in [8–11].
Our goal is to investigate a best approximation problem by invariant subspaces of exponential type entire vectors of an arbitrary unbounded closed linear operator A in a Banach space . As a basic tool, we use an analog of approximate Bernstein and Jackson inequalities and an abstract quasi-normed Besov-type interpolation space , associated with exponential type entire vectors of A, which sharply characterizes the behavior of the best spectral approximation.
Using the quasi-norm of , the main result is formulated in Theorem 5 as two inequalities, estimating the minimal distance from a given element to a subspace of exponential type vectors with fixed indices. In the case of the operator of differentiation in , the spaces coincide with the classical Besov-type spaces (Theorem 7) and the estimations reduce to the known Bernstein and Jackson inequalities (Theorem 8). A new application to spectral approximations of elliptic operators is shown in Section 6 (see also Theorem 6).
exponential type entire vectors of A, where the constant is independent on and is the unit operator on . Clearly, every exponential type entire vector also is an analytic vector of A in the well-known Nelson sense.
If () and is the differentiation operator on ℝ then is the space of entire functions of exponential type, belonging to . In this case the inequality reduces to the Bernstein inequality. If the spectrum of an operator A is discrete then the subspace exactly coincides with the linear span of all its spectral subspaces in (see ).
where is called a K-functional [, Section 3.11]. Clearly, .
2 Scales of invariant subspaces
The contractive inclusions with hold.
Every space is A-invariant and the restriction is a bounded operator over with the norm .
The spectrum of A has the property .
Every space is complete.
The inequalities and yield the contractive inclusions and , respectively. If then and for all . It follows that . Therefore, for any the series is convergent. As a result, . Moreover, for all and .
Using , we obtain and when .
For any and the equality holds. It follows that for all . Hence, λ belongs to the resolvent set .
Let us use the inequality with , . It follows that if is a Cauchy sequence in the space then and are Cauchy sequences in the space for all . The completeness of implies that there exist such that and by norm of . The graph of is closed in , therefore and . It is true for all , so and by norm of for all .
Hence, . Moreover, , where in this inequality all sequences by k belong to . We obtain , . So, is complete. □
For every p () the embedding with and the equality hold.
The function (2) is a quasi-norm satisfying the inequality for all . Moreover, the contractive embedding is true.
Proof (i) Let . We reason similarly to the above. For every we have . So . It follows that . Therefore, for every the series is convergent, i.e. . Hence, .
The constant c in the definition is independent on the index . It yields the equality . Hence, . Therefore, the embedding from Theorem 2(i) yields the embedding for any index p. The inverse embedding follows from Theorem 1(i).
with hold. It follows that . Since ε is arbitrary, for all . Evidently, for all . So is a quasi-norm. The contractility of is a direct consequence of (2). □
3 Besov-type scales of approximation spaces
where by [, Lemma 7.1.6] the function is a quasi-norm on .
We can call the space endowed with the quasi-norm an abstract Besov-type space, determined by an operator A. The following properties of are deduced from well-known interpolation theorems.
The spaces are complete.
- (iii)If , , with then(4)
- (iv)If then the following continuous embedding holds:(7)
To prove the completeness of , we equip the sum (which is equal to , because ) with the norm with and . Since , we have . Hence, the space with the norm is complete. Consequently, every series with such that is convergent to an element . Using the inequality , we obtain . So is complete. The isomorphism (3) implies that the space is complete. Thus, is complete as well.
- (iii)Applying the reiteration property of the real interpolation [, Theorem 3.11.5] for the indices with (), , and , we obtain(8)
- (iv)For every there exists such that
Hence, the embedding is continuous. Finally using (4), we obtain (7). □
4 Bernstein-Jackson-type inequalities
To investigate this problem, we will use spaces defined for pair indices or .
holds. Taking in (13) and using (14), we obtain (11). □
Proof In  it is proven that for operators A, having discrete spectra, the equality holds. Hence, the inequality (11) directly implies the estimation (15) for the distance from an element to the spectral subspace . □
5 Connections with classical results
Let us put , where is the closure in () of the operator of differentiation. In the considered case we have , where (). Thus, .
where is a support of the Fourier-image Fu of a function .
For any pair or and we define the classical Besov space with the norm (see e.g. [, Section 6.2]). Let us show a relationship between the spaces and .
Hence, if then .
for any function such that for all . Hence, for all . The above inequality implies that if than .
with and . □
Hence, the quasi-norms and are equal on . Now the above claims is a consequence of Theorems 5, 7. □
Note that the equalities (17) and (18) exactly coincide with the well-known Bernstein and Jackson inequalities in the form given in [, Section 7.2].
6 An application to regular elliptic operators
is regular elliptic (see e.g. [, Section 5.2.1]). Denote , where and for all , .
endowed with the seminorms , .
where and is the Besov space.
for all and , where the constant is independent of . Hence, u has an entire analytic extension onto of exponential type.
where denotes the support of the Fourier-image Fv of a function .
It follows that . Using (25) and (26), we obtain the required equality (20). □
where is the complex linear span of root subspaces of the operator (19).
Proof From the inequality (27)-(28) and the Paley-Wiener theorem it follows that we have the quasi-norm equivalence on . It remains to apply Theorems 5, 6, and 9. □
The second author was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge at the University of Rzeszów.
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