On Spectral Approximations of Unbounded Operators

We establish an improvement of Bernstein–Jackson inequalities by explicitly calculating constants on special approximation scales of analytic vectors of finite exponential types, generated by unbounded operators. Inequalities are applied to analytical estimates of spectral approximations of unbounded operators. Applications to spectral approximations of elliptic and ordinary differential boundary-value problems are shown.


Introduction
We investigate a spectral approximation problem for a linear closed unbounded operator A in a Banach space X , using the subspace E (A) ⊂ X of its analytic vectors of finite exponential types. We call this the spectral approximation because in the case of operators A with discrete spectrum the subspaces E (A) exactly coincide with the linear span of all its spectral subspaces R(A) [5]. For many other operators (see e.g., [13]), the subspace E (A) also contains all their spectral subspaces. Non-triviality of E (A), for example, in the case of generators A of strongly continuous 1-parameter groups, is checked in Proposition 1. The basic tool in our approach is the functional E (t, x; E (A), X ) (more details in [2,16]) which in our cases characterizes the shortest distance from x ∈ X to a subspace of vectors with an exponential type not larger than t > 0. We use an adaptation of approximation scales (in terms of [17]) of quasi-normed Besov spaces B s τ (A), determined by E (t, x; E (A), X ), which are connected with the approximation errors by Bernstein-Jackson type inequalities.
Earlier applications of analytic vectors of finite exponential types to approximation problems can be found in [7,8]. Looking historically, the idea of exponential type vectors comes from analytic vectors in Nelson's theorem [14]. Analyses of various estimates via approximation functionals are carried out in [1] and in other publications. In [9,10] instead of the functional E (t, x; E (A), X ), a modulus of smoothness ω(·) was used.
One of our aims is to prove the inverse and direct theorems that give an estimate of approximation errors by means of elements E (A). Namely, the inverse Theorem 2(a) which is usually identified with Bernstein's inequality (1) and the direct Theorem 2(b) which is identified with Jackson's inequality (2). Here, the main result is that in the inequalities (1)- (2) we obtain the explicit dependence of constants c s,τ , C s,τ on parameters of the Besov spaces B s τ (A). The calculated constants are exact in the sense that the limits (3) of direct and inverse sequences as τ → ∞ coincide with 1. These theorems characterize subclasses of elements from X in relation to rapidity of approximations. In our case, these subclasses are completely described by the quasi-normed Besov-type spaces B s τ (A). Theorem 2 uses the completeness of quasi-normed space E (A), which previously is proved in Theorem 1(a). The proof of completeness is based on Bernstein's compactness principle for entire analytic functions of exponential type. Under an implicit assumption of this completeness, the second part of this theorem in a somewhat different form was given in [6, Thm 3(i)].
It is important that in the case, when A is the operator of differentiation D in L p (R), the scale of Besov spaces B s τ (A), as well as, the Bernstein-Jackson inequalities fully coincide with known classical analogs (see [6,). However, in this case, the exact values of constants in these inequalities were not earlier calculated.
The last two sections are applications. It is essential that estimates of spectral approximations are given in terms of quasi-norms of classic Besov spaces what are well investigated.
If A is a regular elliptic operator with variable smooth coefficients over the space L p (Ω) on a bounded domain Ω ⊂ R n , we prove in Theorem 3 that the space B s τ (A) coincides with an appropriate subspace in the classic Besov space B s p,τ (Ω) and the Bernstein-Jackson inequalities for the last space give estimates of spectral approximations. A partial case of constant coefficients in elliptic boundary-value problems was considered earlier in [6].
Second example of analytic estimates of spectral approximations errors for some self-adjoint ordinary differential boundary-value problems is described in Theorem 4.

Approximation Scales Generated by Unbounded Operators
In what follows, let A : D(A) → X be a closed linear operator with dense domain where all integer powers A k are assumed closed. We associate with an element x ∈ D ∞ (A) the scalar functions in variable z ∈ C, interconnected by Laplace's transform.
An element x ∈ D ∞ (A) is called the vector of exponential type ν > 0 of A, if one of the following equivalent conditions holds (see, e.g. We consider the subspace of all exponential type vectors E (A) as the union

Then the continuous embedding E ν (A)
E μ (A) with μ > ν holds and each E ν (A) is A-invariant, as well as, the restriction A| E ν (A) is a bounded operator (see [6,Thm 1]).
We assume that exponential type vectors E (A) are dense in X . Such an assumption is not restrictive, since E (A) contains spectral subspaces of A (see, e.g., Propositions 1 and Remark 3 or [5, Thm 2.2]).
To investigate approximation errors, we consider a special scale of Besov-type spaces B s x ∈ X and t > 0. We will use the real interpolation method. Recall that for a pair of quasi-normed spaces (X 0 , | · | X 0 ), (X 1 , | · | X 1 ) and 0 < θ < 1, 1 ≤ q < ∞ the interpolation space generated by K -functional is defined as x 1 ∈ X 1 and t > 0 (see, e.g. [2]).
By Bernstein's compactness theorem [15, Thm 3.3.6] there exists a convergent subsequence x n i (t) exp (−tν) : i ∈ N with respect to the topology of uniform convergence in variable t ∈ [0, r ] for all r > 0. Thus, ∀ε > 0, ∃n ε ∈ N : Thus, (x n i ) is fundamental in E 2ν (A). Below, it will be proven that E 2ν (A) is complete. As a conclusion of this, there exists x 0 ∈ E 2ν (A) such that x n i → x 0 at i → ∞ and, consequently, it follows that the sequences (x n ) and (A/ν) k x n for any k ∈ Z + are fundamental in X . By completeness of X and closeness of A k there exist x, y k ∈ X such that x n → x and (A/ν) k x n → y k in X . So, The proof is completed.

Analytical Estimates of Best Approximation Errors
The analytical estimates of approximation errors are based on exact values of constants. In direct and inverse approximation theorems this problem is solved by exact values of constants in the Bernstein-Jackson inequalities, presented in the following statements.
Herewith, for a fixed s, we have By combining the previous inequalities, The following inequalities are a consequence of definitions K and K ∞ [16, Rem.

3.1],
According to the left inequality from (5), we have On the other hand, from the right inequality (5) it follows that Thus, combining the previous inequalities, we get Via [2, Lemma 7.1.2] for every v > 0 there exists t > 0 such that Since |x| θ B s ∞ (A) ≤ |x| (E (A),X ) θ,∞ , the inequalities (7) yield . By applying (4), we obtain Setting s = 1/θ − 1 and τ = θq in (8), we get the inequalities (1). b) By integration both sides of min ( respectively. As a result, Applying (6)  for all x ∈ B s ∞ (A). Thus, the inequalities (2) hold for both cases that ends the proof. The limits (3) are calculated directly.

Remark 2
The known values of constants in the Bernstein-Jackson inequalities for some particular cases can be found in [11, pp.

Conditions for Non-triviality of Approximation Scales
We give a simple criterion that, in the case of self-adjoint operators, ensures the equality of E (A) with all spectral subspaces of A.
Proposition 1 If a strongly continuous group R t → e it A on X , generated by iA, is such that lim sup |t|→∞ e it A x = M x < ∞ for all x ∈ X then the embedding E (A) X is dense.

.5]) via integration by parts, we have
Using the equality Since the X -valued function R t → e it A x − x is continuous at t = 0, for every ε > 0 there exists δ > 0 such that max |t|≤δ e it A x − x ≤ ε. Therefore, Since ε is arbitrary, (9) holds. Hence, α>0 {P α x : x ∈ X } and therefore E (A) are dense in X what ends the proof.

Remark 3 If
A is self-adjoint in a Hilbert space X then the group e it A is unitary by Stone's theorem. Thus, in this case M x ≡ 1 and the embedding E (A) X is dense. Approximation problems for cases of self-adjoint operators A was analyzed in [10] where, instead of the Besov-type quasi-norm, the smoothness modulus is used.

Applications to Elliptic Operators on Bounded Domains with Smooth Coefficients
Note that a simple case of elliptic operators with constant coefficients was considered in [6]. Now, we adapt the Bernstein-Jackson inequalities (1)(2) to the case of regular elliptic operators with variable smooth coefficients.
We will need one general result obtained earlier in [5]. Suppose that A has a discrete spectrum σ (A), i.e., its resolvent R(λ, A) = (λ − A) −1 has only isolated eigenvalues {λ j ∈ C : j ∈ N} of finite multiplicities which are poles with the limit at infinity. In particular, this guarantees the compactness of R(λ, A) (see, e.g. [12, p.187]). Let R λ j (A) = x ∈ D ∞ (A) : (λ j − A) r j x = 0 be the spectral subspace, corresponding to the eigenvalue λ j of multiplicity r j . Denote by R ν (A) the complex linear span in X of all spectral subspaces R λ j (A) such that |λ j | < ν. Let us define on R(A) := In [5,Thm 2.2] it is proved that the following equalities hold, As a consequence, in the case of operators A with discrete spectrum, Theorem 2 can be slightly strengthened. Namely (see [6,Thm 6]), the following inequalities hold, with the constants c s,τ and C s,τ from Theorem 2. Consider the space L p (Ω), (1 < p ≤ ∞) on Ω ⊂ R n . The Sobolev space W m p (Ω) has norm u W m p (Ω) = |α|≤m D α u L p (Ω) , α = (α 1 , . . . , α n ) ∈ N n , |α| = α 1 + · · · + α n , where D is differentiation. The asymptotic equality is the mth order modulus of smoothness for u ∈ L p (Ω) and Δ m h (u, t) := m k=0 (−1) m−k m k u(t + kh) is the mth difference with step h. Take m = [s] + 1 (the smallest integer larger than s), then the classic Besov space B s p,τ (Ω) can be endowed with the norm (see e.g., [4,16])

Now, let
We assume that its resolvent set ρ(A) is non empty. This is enough for the compactness of R(λ, A) for any λ ∈ ρ(A) and the closeness for all integer powers A k . Thus, the spectrum σ (A) is discrete and is independent on p [18,Sec. 5.4.4]. Let 0 ∈ ρ(A) for simplicity.

Theorem 3
The following Bernstein-Jackson inequalities hold, with the constants c s,τ and C s,τ from Theorem 2, where is denoted E(t, u; R(A), In addition, for each functions u ∈ B s p,τ,A (Ω) the following inequality holds, Proof First, we show that for any a α ∈ C ∞ (Ω) the following equality holds, It is enough to prove the equality Since (Ω) . Substituting μ = ν 2 with ν > 1, we have .
On the other hand, according to [18,Thm 5.4.3] for any k ∈ N there exists c k > 0 such that A k u L p (Ω) ≥ c k u W 2mk p (Ω) for all u ∈ D k (A). Thus, where c k+1 = c k c 1 = c k 1 by induction on k. Hence, for each k ∈ Z + and u ∈ D k (A), from which it follows that Hence, the equality (16) holds. Applying now Theorem 2 and (10), as well as, taking into account (16), we obtain the required inequalities (12)(13), while (14) directly follows from (11).