Direct and inverse theorems on approximation of solution of operator equation
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| Опубліковано в: : | Нелинейные граничные задачи |
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Інститут прикладної математики і механіки НАН України
1999
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| Цитувати: | Direct and inverse theorems on approximation of solution of operator equation / M.L. Gorbachuk, V.I. Gorbachuk // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 110-116. — Бібліогр.: 8 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860191183070822400 |
|---|---|
| author | Gorbachuk, M.L. Gorbachuk, V.I. |
| author_facet | Gorbachuk, M.L. Gorbachuk, V.I. |
| citation_txt | Direct and inverse theorems on approximation of solution of operator equation / M.L. Gorbachuk, V.I. Gorbachuk // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 110-116. — Бібліогр.: 8 назв. — англ. |
| collection | DSpace DC |
| container_title | Нелинейные граничные задачи |
| first_indexed | 2025-12-07T18:06:15Z |
| format | Article |
| fulltext |
DIRECT AND INVERSE THEOREMS ON APPROXIMATION
OF SOLUTIONS OF OPERATOR EQUATIONS
c© M.L.Gorbachuk, V.I.Gorbachuk
1. In the theory of approximation of functions, the so-called direct and inverse
theorems give the relation of smoothness of a continuous function to the behavior of
its best approximation by more elementary objects. For example, in the periodic case,
trigonometric polynomials play the role of such objects. If we put for a continuous
2π-periodic function f(x) (x ∈ R1)
En(f) = inf
T∈Tn
sup
x∈[0,2π]
|f(x)− T (x)|
(Tn is the set of all (≤ n)-order polynomials), then the S. N. Bernshtein result asserts:
f ∈ C̃∞[0, 2π] ⇐⇒ ∀k ∈ N nkEn(f) → 0 as n →∞;
f ∈ Ã[0, 2π] ⇐⇒ ∃q : 0 < q < 1, ∃c > 0 : En(f) < cqn;
f ∈ Ãc[0, 2π] ⇐⇒ ∀q : 0 < q < 1,∃c = c(q) > 0 : En(f) < cqn.
Here C̃∞[0, 2π], Ã[0, 2π], and Ãc[0, 2π] denote the space of infinitely differentiable on
R1, analytic on R1, and entire 2π-periodic functions respectively. There are a number
of other direct and inverse theorems (see [1]). They belong mainly to D. Jackson and
S. N. Bernshtein. The similar situation holds in the case where a function continuous
on a closed interval is approximated by algebraic polynomials, or a continuous function
given on R1 is approximated by entire functions of exponential type.
Some of the direct theorems were often applied to finding the error estimation in ap-
proximation of the exact solutions of various equations (differential, integral, operator)
by their approximate ones. As for inverse theorems (they make possible to predetermine
the smoothness degree of a solution on the basis of the available estimates), the number
of them is considerably lesser. As far as we know, the results of such kind were obtained
for the first time by A. V. Babin [2, 3] (1976 - 1984). We formulate one of them which
concerns an abstract parabolic equation.
Consider the Cauchy problem
dy(t)
dt
+ Ay(t) = 0 (t ≥ 0), y(0) = f, f ∈ H,
Partially supported by Ukrainian Fundamental Science Grant 1.4/62, and INTAS (project 93-0249-
ext.)
where A is a nonnegative selfadjoint operator in a complex separable Hilbert space H.
As is known, the solution of this problem is of the form
y(t) = e−Atf.
It was shown that if f ∈ D(eα
√
A) for some α > 0, then
y = lim
n→∞
Pn(A)f,
where Pn(A) is a certain n-order polynomial of A, and there exist the constants σ > 0
and c > 0 such that the following exact estimate is valid:
‖y − Pn(A)f‖ ≤ c exp
(−σ(ln n)2
)
. (1)
Taking a partial differential operator with polynomial coefficients as A and a polynomial
as f (then Pn(A)f are also polynomials), A. V. Babin found, by using in this concrete
situation the S. N. Bernshtein weighted inverse theorems and the estimate (1), the
smoothness degree of the solution of a corresponding equation. But the process itself of
constructing the polynomials Pn was rather complicated.
M. L. Gorbachuk and V. V. Gorodetskii [4] revealed (1984) that for the functions
associated with some evolution equations (for the above equation, for example, this
function is e−λt), the partial sums of their Fourier-Laguerre series give (up to the factor
e−λ) the polynomials Ln(λ) such that
lim
n→∞
Ln(A)f = y.
Moreover, if f is an analytic vector of the operator A, then for every closed interval
[0, b]
∃ρ = ρ(b) : 0 < ρ < 1, ∃c > 0 : ‖y − Ln(A)f‖ ≤ cρn+1 (2)
uniformly. The polynomials Ln(λ) are of the form
Ln(λ) = µ1/2(t + µ)−1
n∑
k=0
(
− t
t + µ
)k
l0,µ,k(λ),
where
l0,µ,k(λ) = (−1)kµ1/2eµλ
(
(µλ)ke−µλ
)(k)
k!
, µ = µ(f) = const.
Conversely, if the relation (2) is fulfilled for a vector f infinitely differentiable for the
operator A, then f is an analytic vector of A.
As we can see, for the polynomial approximation method of solving the Cauchy
problem considered we have both direct and inverse theorems. The analogous theorems
were proved for some other classes of differential equations in a Hilbert space.
2. Let us turn attention to the direct and inverse theorems in the approximation of
solutions of operator equations by projective not polynomial methods. To do this we
need a brief survey of the results on approximation of vectors from a Banach space by
entire vectors of exponential type of a closed operator acting in this space.
Let A be a closed linear operator in a Banach space B, D(A) = B, {mn}n∈N0 (N0 =
{0, 1, 2, . . . }) a non-decreasing sequence of positive numbers (for the sake of simplicity
we consider m0 = 1). Denote by C∞(A) the set of infinitely differentiable vectors of A:
C∞(A) =
⋂
n∈N0
D(An).
For α > 0 we put
C{mn}(A) = ind lim
α→∞
Cα〈mn〉(A), C(mn)(A) = proj lim
α→0
Cα〈mn〉(A),
where
Cα〈mn〉(A) =
{
f ∈ C∞(A)
∣∣∃c > 0 : ‖Akf‖ ≤ cαkmk, ∀k ∈ N0
}
(‖ · ‖ is the norm in B; everywhere c means a certain constant) is a Banach space with
respect to the norm
‖f‖Cα〈mn〉(A) = sup
n∈N0
‖Anf‖
αnmn
.
The convergence in C{mn}(A) (C(mn)(A)) is that in some (any) Cα〈mn〉(A). If mn = n!
or mn ≡ 1, then we get the spaces C{n!}(A), C(n!)(A), and C{1}(A) known as the spaces
of analytic, entire, and entire of exponential type vectors of the operator A respectively.
The spaces C{nnβ}(A) and C(nnβ)(A) are called the Gevrey classes of Roumieu and
Beurling type.
E x a m p l e 1. If B = C[a, b] (−∞ < a < b < ∞), A = d
dx , D(A) = C1[a, b],
then C∞(A), C{n!}(A), C(n!)(A), C{1}(A) coincide with the spaces of usual infinitely
differentiable on [a, b], analytic on [a, b], entire and entire of exponential type functions;
C(1)(A) is the set of all polynomials; C{nnβ}(A) and C(nnβ)(A) are the usual Gevrey
classes.
E x a m p l e 2. If B = L2(R1), A is the closure of the operator A0 =
1
2
(
− d2
dx2 + x2 + 1
)
, D(A0) = C∞0 (R1) (the set of infinitely differentiable functions with
compact support), then C∞(A) = S, C{nnβ}(A) = S
β/2
β/2 , where S is the well-known
Schwartz space of slowly decreasing functions and
Sβ
α =
{
f
∣∣∃h > 0 : sup
m,n∈N0,x∈R1
|xmf (n)(x)|
hm+nmmαnnβ
< ∞}
.
So, by entire vectors of exponential type of the operator A we mean the vectors
g ∈ C{1}(A) =
⋃
α>0
Cα〈1〉(A).
The type σ(g) of a vector g ∈ C{1}(A) is defined as follows:
σ(g) = inf
α>0:g∈Cα〈1〉(A)
α.
For every f ∈ B we set
Er(f) = inf
g∈C{1}(A):σ(g)≤r
‖f − g‖.
The function Er(f) is monotonically non-increasing, and
∀f ∈ B Er(f) → 0 as r →∞⇐⇒ C{1}(A) = B. (3)
If A is a normal operator in a Hilbert space H, the equality in the right hand side of
(3) is fulfilled because
C{1}(A) = {y ∈ H
∣∣y = E∆x},
where x runs through the whole space H, ∆ is any compact set in R2, E∆ is the spectral
measure of the operator A.
Theorem 1. Let A be a normal operator in H, γ(t) (t ∈ [0,∞)) a monotonically non-
increasing positive function, and γ(t) → 0 as t →∞. Then there exists a vector f ∈ H
such that
Er(f) = γ(r).
Theorem 1 shows that in the case of a normal A the rate of the convergence Er(f) →
0 (r →∞) may be arbitrary. It depends on the smoothness degree of f with respect to
the operator A.
Denote by Hα(A) the |A|-scale of Hilbert spaces that is,
Hα(A) = D(Cα), C = I + (A∗A)1/2,
(f, g)Hα(A) = (Cαf, Cαg)
((·, ·) is the inner product in H), and H−α(A) is the dual of Hα(A), (α ≥ 0) with respect
to (·, ·).
Theorem 2. Let A be a normal operator in H and B a Banach space such that the
continuous embeddings
Hk1(A) ⊆ B ⊆ H−k2(A)
hold for some ki ∈ N0 (i = 1, 2). Assume also that the sequence {mn}n∈N0 satisfies the
condition
∃c > 0, ∃h > 1 : mn+1 ≤ chnmn.
Then the following equivalences are true:
f ∈ C∞(A) ⇐⇒ lim
r→∞
rαEBr (f) = 0, ∀α > 0,
f ∈ C{mn}(A) ⇐⇒ ∃α > 0,∃c > 0 : EBr (f) ≤ cG−1(αr),
f ∈ C(mn)(A) ⇐⇒ ∀α > 0,∃c > 0 : EBr (f) ≤ cG−1(αr),
where G(λ) = sup
n∈N0
(λn/mn), EBr (f) = inf
g∈C{1}(A):σ(g)≤r
‖f − g‖B.
(The proofs of Theorems 1 and 2 under stronger assumptions are contained in [5]).
Theorem 2 implies many well-known results of the theory of approximation of func-
tions. For instance, put
H = L2(0, 2π), Af = i
d
dx
, D(A) =
{
f ∈ W 1
2 [0, 2π] : f(0) = f(2π)
}
.
In this case the spectrum of A σ(A) = Z and
{
1√
2π
eikx
}
k∈Z
is the orthonormal basis
of eigenvectors of the operator A, C∞(A) = C̃∞[0, 2π], C{1}(A) coincides with the set
of trigonometric polynomials, σ(g) ≤ n means g ∈ Tn. In view of the estimate
c1‖f‖L2(0,2π) ≤ ‖f‖C̃[0,2π] ≤ c2‖f‖W 1
2 [0,2π],
the space B = C̃[0, 2π] of continuous 2π-periodic functions with the norm ‖f‖C̃[0,2π] =
max
x∈[0,2π]
|f(x)| satisfies the conditions of Theorem 2. So, the next assertion is valid.
Theorem 3. The following equivalences hold:
f ∈ C̃∞[0, 2π] ⇐⇒ ∀α > 0 lim
r→∞
nαEn(f) = 0 (4)
(S. N. Bernshtein),
f ∈ C̃{mn}[0, 2π] ⇐⇒ ∃α > 0, ∃c > 0 : En(f) ≤ cG−1(αn), (5)
f ∈ C̃(mn)[0, 2π] ⇐⇒ ∀α > 0, ∃c > 0 : En(f) ≤ cG−1(αn), (6)
where
C̃{mn}[0, 2π] = {f ∈ C̃∞[0, 2π] : ∃α > 0, ∃c > 0 : |f (k)(x) ≤ cαkmk}
C̃(mn)[0, 2π] = {f ∈ C̃∞[0, 2π] : ∀α > 0, ∃c > 0 : |f (k)(x) ≤ cαkmk}
(∀k ∈ N0), G(λ) = sup
n∈N0
λn
mn
, En(f) = E C̃[0,2π]
n (f) is the best uniform approximation of
f(x) by trigonometric polynomials from Tn.
The equivalences (5), (6) with mn = n! lead to the S. N. Bernshtein results from
the subsection 1 on behavior of the best approximation En(f) of an analytic or entire
function by trigonometric polynomials. Indeed, in this case C̃{mn}[0, 2π] = Ã[0, 2π],
C̃(mn)[0, 2π] = Ãc[0, 2π], G(λ) = eλ. Thus, the mentioned S. N. Bernshtein results are
a consequence of (5) and (6).
Taking the spaces L2
(
[−1, 1], 1√
1−x2
)
, L2(R1), L2
(
R1, e−x2)
as H, and the operators
generated by the expressions
l1 =
√
1− x2
d
dx
(
√
1− x2
d
dx
), l2 = i
d
dx
, l3 = ex2(− d2
dx2
+ x2
)
respectively and the corresponding boundary conditions as A, we arrive at some of
known and unknown results on the approximation of smooth functions by algebraic
polynomials on a finite interval, entire functions of exponential type and algebraic poly-
nomials on the whole real axis in the corresponding metrics.
3. Now consider the equation
Au = f, (7)
where A = A∗ is a positive definite operator in H with a discrete spectrum. The equation
(7) has a unique solution. According to the Dirichlet principle (see, e. g. [6]), finding
the solution of (7) is equivalent to that of the vector u ∈ D(A) such that the functional
F (v) = (Av, v)− 2<(f, v) (8)
given on D(A) attains its minimum at u.
Let {ek}k∈N be a complete system of linearly independent vectors in H1(A) and
Ln = span{e1, e2, . . . , en}. Denote by un ∈ Ln the vector on which the functional (8)
considered on Ln has minimum. The vector un is called the Rietz approximation of
the solution u. If {ek}k∈N is an orthonormal basis of the eigenvectors of a selfadjoint
positive definite operator B related to A (that is, D(B) = D(A)), then as is well-known,
un → u in H1/2, Aun → f in H. (9)
It turns out to be that the convergence Aun → f in (8) may be arbitrarily slow.
Theorem 4. Let {αn}n∈N be a sequence of positive numbers convergent monotonically
to 0. Under the above conditions on the operator A and the system {ek}k∈N, there exists
a vector f ∈ H such that
‖Aun − f‖ ≥ αn.
The question arises under what conditions on f ∈ H, the value Aun − f has a
preassigned order of decrease. The predecessors’ results (see, e. g. [7]) concerned only
the power decrease , and they dealt only with direct theorems. The following statement
reduces this restriction.
Theorem 5. Let A = A∗ > γI (γ > 0) be an operator with a discrete spectrum, and
{ek}k∈N be the orthonormal basis of eigenvectors of an operator B related to A. Then
u ∈ C∞(B) ⇐⇒ ∀k ∈ N ‖Aun − f‖ = o(λ−k
n (B)), (10)
u ∈ C{mn}(B) ⇐⇒ ∃α > 0, ∃c > 0 : ‖Auk − f‖ ≤ cG−1(αλk) (11)
(k ∈ N is arbitrary),
u ∈ C(mn)(B) ⇐⇒ ∀α > 0, ∃c > 0 : ‖Auk − f‖ ≤ cG−1(αλk), (12)
where λn is the eigenvalue of the operator B corresponding to en, G(λ) = sup
n∈N0
λn
mn
.
If mn = nnβ (β > 0), then G(λ) = exp(λ1/β), and the estimate in (11) and (12) takes
the form
‖Aun − f‖ ≤ c exp(−λ1/β
n ).
Even in the elementary case where H = L2(0, π), A = − d2
dx2 +q(x) (q ≥ 0 is a continuous
function on [0, π], D(A) = W 2
2 [0, π] ∩W 1
2
◦[0, π], B = − d2
dx2 , D(B) = D(A), we obtain
new results of both direct and inverse character.
Theorem 6. Let the function q(x) be infinitely differentiable, and
q(2k−1)(0) = q(2k−1)(π) = 0 (k ∈ N). (13)
Then
∀α > 0 nα‖u− un‖ → 0 (n →∞)
⇐⇒ f is infinitely differentiable on [0, π] and
f (2k)(0) = f (2k)(π) = 0 (k ∈ N). (14)
If q(x) is analytic and satisfies (13), then
∃α > 0 : eαn‖u− un‖ → 0 (n →∞)
⇐⇒ f is analytic on [0, π] and satisfies (14)
If q(x) is entire and satisfies (13), then
∀α > 0 : eαn‖u− un‖ → 0(n →∞) ⇐⇒ f is entire and satisfies (14).
We have formulated the results for the Rietz method. The similar assertions are true
for the moment method (in this case the operator A in (7) is invertible and K-positive
definite in the W.V. Petryshin sense [8]). If K = I, the latter method becomes the
Rietz one. If K = A, we have the method of least squares.
References
1. Gutler R.S., Kudryavtzev L.D., Levitan B.M., Elements of Function Theory, Moscow: Nauka, 1963.
2. Babin A.V., Solving the Cauchy problem with the help of weighted approximation of exponents by
polynomials, Funkts. Anal. and Pril. 17 (1983), no. 4, 75-76.
3. Babin A.V., Construction and investigation of solutions of differential equations by approximation
theory methods, Mat. Sb. 123 (1984), no. 2, 147-173.
4. Gorbachuk M.L., Gorodetskii V.V., On polynomial approximation of solutions of operator differen-
tial equations in a Hilbert space, Ukr. Mat. Zh. 36 (1984), no. 4, 500-502.
5. Gorbachuk M.L., Gorbachuk V.I., On approximation of smooth vectors of a closed operator by entire
vectors of exponential type, Ukr. Mat. Zh. 47 (1995), no. 5, 616-628.
6. Mikhlin S.G., Variation Methods in Mathematical Physics, Moscow: Nauka, 1970.
7. Luchka A.Yu., Luchka T.F., Origin and Development of direct methods of Mathematical Physics,
Kiev: Naukova Dumka, 1985.
8. Petryshin W.V., On a class of K.-p.d. and non-K.-p.d. operators and operator equations, J. Math.
Anal. and Appl. 10 (1965), 1-24.
Institute of Mathematics of Ukrainian National Academy of Sciences
Tereshchenkivska Str. 3
Kyiv 252601, Ukraine
Tel.: (044) 517-21-82
Fax:(044) 225-20-10
E-mail: imath@horbach.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-169280 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0236-0497 |
| language | English |
| last_indexed | 2025-12-07T18:06:15Z |
| publishDate | 1999 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Gorbachuk, M.L. Gorbachuk, V.I. 2020-06-09T16:36:08Z 2020-06-09T16:36:08Z 1999 Direct and inverse theorems on approximation of solution of operator equation / M.L. Gorbachuk, V.I. Gorbachuk // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 110-116. — Бібліогр.: 8 назв. — англ. 0236-0497 https://nasplib.isofts.kiev.ua/handle/123456789/169280 Partially supported by Ukrainian Fundamental Science Grant 1.4/62, and INTAS (project 93-0249-ext.). en Інститут прикладної математики і механіки НАН України Нелинейные граничные задачи Direct and inverse theorems on approximation of solution of operator equation Article published earlier |
| spellingShingle | Direct and inverse theorems on approximation of solution of operator equation Gorbachuk, M.L. Gorbachuk, V.I. |
| title | Direct and inverse theorems on approximation of solution of operator equation |
| title_full | Direct and inverse theorems on approximation of solution of operator equation |
| title_fullStr | Direct and inverse theorems on approximation of solution of operator equation |
| title_full_unstemmed | Direct and inverse theorems on approximation of solution of operator equation |
| title_short | Direct and inverse theorems on approximation of solution of operator equation |
| title_sort | direct and inverse theorems on approximation of solution of operator equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169280 |
| work_keys_str_mv | AT gorbachukml directandinversetheoremsonapproximationofsolutionofoperatorequation AT gorbachukvi directandinversetheoremsonapproximationofsolutionofoperatorequation |