Direct Theorems in the Theory of Approximation of the Banach Space
  Vectors by Entire Vectors of Exponential Type
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION OF
BANACH SPACE VECTORS BY EXPONENTIAL TYPE ENTIRE
VECTORS
YA. GRUSHKA AND S. TORBA
Abstra
t. For an arbitrary operator A on a Bana
h spa
e Xwhi
h is the generator
of C0�group with 
ertain growth 
ondition at in�nity, the dire
t theorems on 
onne
-
tion between the smoothness degree of a ve
tor x ∈ X with respe
t to the operator
A, the rate of 
onvergen
e to zero of the best approximation of x by exponential type
entire ve
tors for the operator A, and the k-module of 
ontinuity are established.
The results allow to obtain Ja
kson-type inequalities in a number of 
lassi
 spa
es of
periodi
 fun
tions and weighted Lp spa
es.
1. Introdu
tion
The dire
t and inverse theorems establishing a relationship between the smoothness
degree of a fun
tion with respe
t to the di�erentiation operator and the rate of 
onver-
gen
e to zero of its best approximation by trigonometri
 polynomials are well known in
the theory of approximation of periodi
 fun
tions. Ja
kson's inequality is one among
su
h results.
N. P. Kuptsov proposed a generalized notion of the module of 
ontinuity, expanded
onto C0-groups in a Bana
h spa
e [1℄. Using this notion, N. P. Kuptsov [1℄ and A. P.
Terekhin [2℄ proved the generalized Ja
kson's inequalities for the 
ases of a bounded
group and s-regular group. Remind that the group {U(t)}t∈R is 
alled s-regular if the
resolvent of its generator A satis�es the 
ondition ∃θ ∈ R : ‖Rλ(eiθAs)‖ ≤ CImλ .
G. V. Radzievsky studied the dire
t and inverse theorems [3, 4℄, using the notion of K-
fun
tional instead of module of 
ontinuity, but it should be noted that the K-fun
tional
has two-sided estimates with regard to module of 
ontinuity at least for bounded C0-
groups.
In the papers [5, 6℄ and [7℄ the authors investigated the 
ase of a group of unitary
operators in a Hilbert spa
e and established Ja
kson-type inequalities in Hilbert spa
es
and their rigs. These inequalities are used to estimate the rate of 
onvergen
e to zero of
the best approximation of both �nite and in�nite smoothness ve
tors for the operator A
by exponential type entire ve
tors.
We 
onsider the C0-groups, generated by the so-
alled non-quasianalyti
 operators [8℄,
i.e. the groups satisfying
(1.1)
ln ‖U(t)‖
1 + t2
dt < ∞.
As was shown in [5℄, the set of exponential type entire ve
tors for the non-quasianalyti
operator A is dense in X, so the problem of approximation by exponential type entire
Re
eived by the editors 04/04/2007.
2000 Mathemati
s Subje
t Classi�
ation. Primary 41A25, 41A17, 41A65.
Key words and phrases. Dire
t and inverse theorems, modulo of 
ontinuity, Bana
h spa
e, entire
ve
tors of exponential type.
This work was partially supported by the Ukrainian State Foundation for Fundamental Resear
h
(proje
t N14.1/003).
http://arxiv.org/abs/0704.0298v3
2 YA. GRUSHKA AND S. TORBA
ve
tors is 
orre
t. On the other hand, it was shown in [9℄ that 
ondition (1.1) is 
lose to
the ne
essary one, so in the 
ase when (1.1) doesn't hold, the 
lass of entire ve
tors isn't
ne
essary dense in X, and the 
orresponding approximation problem loses its meaning.
The purpose of this work is to obtain Ja
kson-type inequalities in the 
ase where a
ve
tor of a Bana
h spa
e is approximated by exponential type entire ve
tors for a non-
quasianalyti
 operator, and, in parti
ular, Ja
kson-type inequalities in various 
lassi
al
fun
tion spa
es.
2. Preliminaries
Let A be a 
losed linear operator with dense domain of de�nition D(A) in the Bana
h
spa
e (X, ‖·‖) over the �eld of 
omplex numbers.
Let C∞(A) denotes the set of all in�nitely di�erentiable ve
tors of the operator A, i.e.
C∞(A) =
D(An), N0 = N ∪ {0}.
For a number α > 0 we set
α(A) =
x ∈ C∞(A) | ∃c = c(x) > 0 ∀k ∈ N0
∥∥Akx
∥∥ ≤ cαk
The set E
α(A) is a Bana
h spa
e with respe
t to the norm
Eα(A) = sup
‖Anx‖
Then E(A) =
α>0 E
α(A) is a linear lo
ally 
onvex spa
e with respe
t to the topology
of the indu
tive limit of the Bana
h spa
es E
α(A):
E(A) = lim ind
α(A).
Elements of the spa
e E(A) are 
alled exponential type entire ve
tors of the operator A.
The type σ(x,A) of a ve
tor x ∈ E(A) is de�ned as the number
σ(x,A) = inf {α > 0 : x ∈ Eα(A)} = lim sup
‖Anx‖
Example 2.1. Let X is one of the Lp(2π) (1 ≤ p < ∞) spa
es of integrable in p-th
degree over [0, 2π], 2π-periodi
al fun
tions or the spa
e C(2π) of 
ontinuous 2π-periodi
al
fun
tions (the norm in X is de�ned in a standard way), and let A is the di�erentiation
operator in the spa
e X (D(A) = {x ∈ X∩AC(R) : x′ ∈ X}; (Ax)(t) = dx
, where AC(R)
denotes the spa
e of absolutely 
ontinuous fun
tions over R). It 
an be proved that in
su
h 
ase the spa
e E(A) 
oin
ides with the spa
e of all trigonometri
 polynomials, and for
y ∈ E(A) σ(y,A) = deg(y), where deg(y) is the degree of the trigonometri
 polynomial
In what follows, we always assume that the operator A is the generator of the group
of linear 
ontinuous operators {U(t) : t ∈ R} of 
lass C0 on X. We re
all that belonging
of the group to the C0 
lass means that for every x ∈ X the ve
tor-fun
tion U(t)x is

ontinuous on R with respe
t to the norm of the spa
e X.
For t ∈ R+, we set
MU (t) := sup
τ∈R, |τ |≤t
‖U(τ)‖ .
The estimation ‖U(t)‖ ≤ Meωt for some M,ω ∈ R implies MU (t) < ∞ (∀t ∈ R+). It is
easy to see that the fun
tion MU (·) has the following properties:
1) MU (t) ≥ 1, t ∈ R+;
2) MU (·) is monotoni
ally non-de
reasing on R+;
3) MU (t1 + t2) ≤ MU (t1)MU (t2), t1, t2 ∈ R+.
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 3
A

ording to [1℄, for x ∈ X, t ∈ R+ and k ∈ N we set
ωk(t, x, A) = sup
0≤τ≤t
∥∥∆kτx
∥∥ , where(2.1)
∆kh = (U(h)− I)k =
(−1)k−j
U(jh), k ∈ N0, h ∈ R (∆0h ≡ 1).(2.2)
Moreover, let
(2.3) ω̃k(t, x, A) = sup
|τ |≤t
∥∥∆kτx
Remark 2.1. It is easy to see that in the 
ase of the bounded group {U(t)} (‖U(t)‖ ≤
M, t ∈ R) the quantities ωk(t, x, A) and ω̃k(t, x, A) are equivalent within 
onstant fa
tor
(ωk(t, x, A) ≤ ω̃k(t, x, A) ≤ M ωk(t, x, A)), and in the 
ase of isometri
 group (‖U(t)‖ ≡
1, t ∈ R) these quantities 
oin
ide.
It is immediate from the de�nition of ω̃k(t, x, A) that for k ∈ N:
1) ω̃k(0, x, A) = 0;
2) for �xed x the fun
tion ω̃k(t, x, A) is non-de
reasing and is 
ontinuous by the
variable t on R+;
3) ω̃k(nt, x,A) ≤
1 + (n− 1)MU ((n− 1)t)
ω̃k(t, x, A) (n ∈ N, t > 0);
4) ω̃k(µt, x,A) ≤
1 + µMU (µt)
ω̃k(t, x, A) (µ, t > 0);
5) for �xed t ∈ R+ the fun
tion ω̃k(t, x, A) is 
ontinuous in x.
For arbitrary x ∈ X we set, a

ording to [7, 6℄,
Er(x,A) = inf
y∈E(A) :σ(y,A)≤r
‖x− y‖ , r > 0,
i.e. Er(x,A) is the best approximation of the element x by exponential type entire ve
tors
y of the operator A for whi
h σ(y,A) ≤ r. For �xed x Er(x,A) does not in
rease and
Er(x,A) → 0, r → ∞ for every x ∈ X if and only if the set E(A) of exponential type
entire ve
tors is dense in X. Parti
ularly, as indi
ated above, the set E(A) is dense in X
if the group {U(t) : t ∈ R} belongs to non-quasianalyti
 
lass.
3. Abstra
t Ja
kson's inequality in a Bana
h spa
e
Theorem 3.1. Suppose that {U(t) : t ∈ R} satis�es 
ondition (1.1). Then ∀k ∈ N there
exists a 
onstant mk = mk(A) > 0, su
h that ∀x ∈ X the following inequality holds:
(3.1) Er(x,A) ≤ mk · ω̃k
, x, A
, r ≥ 1.
Remark 3.1. If, additionally, the group {U(t)} is bounded (MU (t) ≤ M̃ < ∞, t ∈ R),
then the assumption r ≥ 1 
an be 
hanged to r > 0.
Integral kernels, 
onstru
ted in [10℄, will be used in the proving of the theorem. More-
over, we need additional properties of these kernels, la
king in [10℄. The following lemma
shows how these kernels are 
onstru
ted and 
ontinues the investigation of their proper-
ties.
In what follows we denote as Q the 
lass of fun
tions α : R 7→ R, satisfying the
following 
onditions:
I) α(·) is measurable and bounded on any segment [−T, T ] ⊂ R.
II) α(t) > 0, t ∈ R.
III) α(t1 + t2) ≤ α(t1)α(t2), t1, t2 ∈ R.
|ln(α(t))|
dt < ∞.
4 YA. GRUSHKA AND S. TORBA
Lemma 3.1. Let α ∈ Q. Then there exists su
h entire fun
tion Kα : C 7→ C that
1) Kα(t) ≥ 0, t ∈ R;
−∞ Kα(t) dt = 1;
3) ∀r > 0 ∃cr = cr(α) > 0 ∀z ∈ C |Kα(rz)| ≤ cr e
r|Im z|
α(|z|)
Proof. Without lost of generality we may assume that the fun
tion α(t) satis�es addi-
tional 
onditions:
V) α(t) ≥ 1, t ∈ R; 1
VI) α(t) is even on R and is monotoni
ally in
reasing on R+;
∥∥α−1
L1(R)
−∞ |α
−1(t)|dt < ∞.
It is easy to verify that assumptions V),VII) and 
ondition that the fun
tion α(t) is
even in VI) don't 
on�ne the general 
ase if one examined the fun
tion α1(t) = α̃(t)α̃(−t),
where α̃(t) = (1 + α(t))(1 + t2). In [11, theorems 1 and 2℄ it has been proved that the
monotony 
ondition on α(t) in VI) doesn't 
on�ne the general 
ase too.
It follows from VII) that
(3.2) α(t) → ∞, t → ∞.
Let β(t) = lnα(t), t ∈ R. Conditions III)-VII) and (3.2) lead to 
on
lusion that
β(t) > 0, β(−t) = β(t), β(t) → ∞, t → ∞;
β(t1 + t2) ≤ β(t1) + β(t2), t1, t2 ∈ R(3.3)
dt < ∞(3.4)
Be
ause of (3.3) there exists limit limt→∞
. And, by virtue of (3.4):
(3.5) lim
Also, using (3.4) it is easy to 
he
k that
(3.6)
moreover, all terms of the series (3.6) are positive. From the 
onvergen
e of series (3.6)
follows the existen
e of su
h sequen
e {Qn}∞n=1 ⊂ R that Qn > 1, Qn → ∞, n → ∞
(3.7)
Qk = S < ∞.
We set
ak :=
β(k)Qk
, k ∈ N.
The de�nition of ak and (3.7) result in equality
(3.8)
ak = 1.
We 
onstru
t the sequen
e of fun
tions, whi
h, obviously, are entire for every n ∈ N:
fn(z) :=
Pk(z), where Pk(z) =
sin akz
, z ∈ C, n ∈ N.
As shown in [8℄, for non-quasianalyti
 groups the 
ondition ‖U(t)‖ ≥ 1 always holds, therefore in
this paper the 
ondition V) automati
ally takes pla
e.
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 5
Similarly to the proof of the Denjoy-Carleman theorem [12, p.378℄ it 
an be 
on
luded
that the sequen
e of (entire) fun
tions fn(z) 
onverges uniformly to the fun
tion
f(z) =
sin akz
, z ∈ C
in every disk {z ∈ C | |z| ≤ R}. Thus, by Weierstrass theorem, the fun
tion f(z) is entire.
Using the inequality | sin z| ≤ min(1, |z|)e|Imz|, z ∈ C and taking (3.8) into a

ount,
when z ∈ C and r > 0, we re
eive
|f(rz)| =
sin akrz
akr|z|
akr|z|
akr|Im z|
= er|Im z|
akr|z|
≤ er|Im z|
akr|z|
for every N ∈ N. Using the inequality min(1, a) ·min(1, b) ≤ min(1, ab), we get:
(3.9)
|f(rz)| ≤ er|Im z|min2
akr|z|
= er|Im z|min2
2N(∏N
β(k)Qk
(r|z|)N
= er|Im z|min2
2NN !
· · · β(N)
)N |z|NQ1 · · ·QN
Be
ause of the 
ondition Qn → ∞, n → ∞ there exists su
h number n(r) ∈ N that:
(3.10) ∀n > n(r) Qn ≥
It follows from (3.5) that there is T0 ∈ (0,∞) su
h that:
(3.11) ∀ t > T0
In [10℄ the following statement was proved:
(3.12) ∀ t1, t2 ∈ R+ t1 ≤ t2 ⇒
β(t1)
β(t2)
Let z ∈ C and |z| ≥ max
β[−1](n(r)), T0
, where β[−1] is the inverse fun
tion of
the fun
tion β on [0,∞) (the inverse fun
tion β[−1] exists due to monotony of β on
[0,∞)). We substitute as N in (3.9) N := [β(|z|)], where [·] denotes the integer part
of a number. Then for k ∈ {1, . . . , N}, in a

ordan
e with (3.11) and (3.12), we obtain
k ≤ N ≤ β(|z|) ≤ |z| and
(3.13)
β(|z|)
Using (3.9),(3.10),(3.13), we �nd
|f(rz)| ≤ er|Im z|
2NN !
β(|z|)
)N ( r
)N |z|NQ1 · · ·QN
≤ er|Imz|
2NN !
)N ( r
)N |z|NQ1 · · ·QN
= er|Im z|
22NN !
Q1 · · ·QN
6 YA. GRUSHKA AND S. TORBA
≤ er|Im z|
Q1 · · ·QN
= er|Im z|
Q1 · · ·QN
Sin
e Qn ≥ 1, the last inequality leads to
(3.14) |f(rz)| ≤ er|Im z|
)N−n(r)
= er|Im z|
)2n(r)
e−[β(|z|)] ≤
≤ er|Im z|
)2n(r)
e−(β(|z|)−1) = C(1)r
er|Im z|
α(|z|)
where C
r = e
)2n(r)
. When z ∈ C and |z| < max
β[−1](n(r)), T0
, using (3.9),
we get
(3.15) |f(rz)| ≤ er|Im z| = er|Im z|α(|z|)
α(|z|)
≤ er|Im z| C
α(|z|)
where C
r = α(max(β
[−1](n(r)), T0)). It follows from (3.14), (3.15) that
(3.16) |f(rz)| ≤ er|Imz|
α(|z|)
, z ∈ C, where C(0)r = max(C(1)r , C(2)r ).
Inequality (3.16) and Condition VII) imply that ‖f‖L1(R) < ∞. Thus it is enough to
set Kα(z) := 1‖f‖
L1(R)
f(z), z ∈ C and use (3.16) to �nish the proof. �
Let α ∈ Q, and Kα : C 7→ C is the fun
tion 
onstru
ted by the fun
tion α in lemma
3.1. We set
Kα,r(z) := rKα(rz), z ∈ C, r ∈ (0,∞).
The lemma 3.1 ensures us that the fun
tion Kα,r has the following properties:
1) Kα,r(t) ≥ 0, t ∈ R;
−∞ Kα,r(t) dt = 1;
3) ∀z ∈ C |Kα,r(z)| ≤ rcr e
r|Im z|
α(|z|) ; r > 0.
Lemma 3.2. ∀r ∈ (0,∞) there exists 
onstant c̃r = c̃r(α) > 0, su
h that ∀n ∈ N the
following inequality holds:
|K(n)α,r(t)| ≤ c̃r
α(|t|)
rn, t ∈ R
Proof. In what follows in this proof we assume t ∈ R, r ∈ (0,∞), n ∈ N. Let
γn,r(t) :=
ζ ∈ C : |ζ − t| = n
Using Cau
hy's integral theorem and Stirling's approximation for n!, we get
|K(n)α,r(t)| ≤
γn,r(t)
|Kα,r(ξ)|
|ξ − t|n+1
|dξ| = n!
γn,r(t)
|Kα,r(ξ)||dξ| ≤
(!)rn+1√
γn,r(t)
|Kα,r(ξ)||dξ|, where c(!) = sup
< e1/12.
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 7
Using property 3) of the fun
tion Kα,r, the 
ondition t ∈ R and 
onditions III), VI) of
the fun
tion α, one 
an �nd from the last inequality
|K(n)α,r(t)| ≤
c(!)rn+1√
e−nrcr
γn,r(t)
er|Im ξ|
α(|ξ|) |dξ| =
c(!)rn+1√
α(|t|)
γn,r(t)
er|Im (ξ−t)|α(|(t− ξ) + ξ|)
α(|ξ|)
|dξ| ≤
(!)rn+1√
α(|t|)
γn,r(t)
er|Im (ξ−t)|α(|t− ξ|)|dξ| ≤
(!)rn+1√
α(|t|)
γn,r(t)
|dξ| = c̃r
α(|t|)
where c̃r = c
(!)rcr . �
Remark 3.2. If the fun
tion α(t) satis�es the 
onditions of lemma 3.1, but, moreover,
has the polynomial order of growth at in�nity, i.e. ∃m ∈ N0, ∃M > 0:
(3.17) α(t) ≤ M(1 + |t|)2m, t ∈ R,
another integral kernel may be used:
K̃α(z) =
sin z
, Km =
sin x
In mu
h the same way to the proving of the lemmas 3.1 and 3.2 one 
an show that
∣∣K̃α(rz)
∣∣ ≤ C̃r
er|Im z|
α(|z|)
, where C̃r =
∣∣K̃(n)α,r(t)
∣∣ ≤ c̃r
2πnα(n
α(|t|)
rn, where c̃r = c
(!)rC̃r ,
that is to say, de�ned in su
h a way integral kernel satis�es lemmas 3.1 and 3.2.
Proof of theorem 3.1. Let the group {U(t) : t ∈ R} satis�es (1.1). Then it follows from
[11, theorems 1 and 2℄ that
(3.18)
ln (MU (|t|))
1 + t2
dt < ∞.
We �x arbitrary k ∈ N and set
α(t) :=
MU (|t|)
(1 + |t|)k+2, t ∈ R.
The fun
tion α is, obviously, even on R. Condition (3.18) and the properties of the
fun
tion MU (·) imply α ∈ Q, and, moreover,
(3.19)
(1 + |t|)MU (|t|)
(1 + |t|)2
Using lemma 3.1 (or remark 3.2 if α(t) ≤ M(1+ |t|)m) for the fun
tion α(t), we 
onstru
t
the family of kernels Kα,r.
In what follows, we assume x ∈ X, r ∈ (0,∞) and n ∈ {1, . . . , k}. We de�ne
xr,n :=
Kα,r(t)U(nt)x dt.
8 YA. GRUSHKA AND S. TORBA
Let ν ∈ N0. Let's prove that xr,n ∈ C∞(A) =
ν∈N0 D(A
ν ) and
(3.20) Aνxr,n =
(−1)ν
K(ν)α,r(t)U(nt)x dt.
It follows from the property 3) of the fun
tion Kα,r and from lemma 3.2 that there exists
su
h 
onstant C̃(ν, r) > 0 that K(ν)α,r(t) ≤
eC(ν,r)
, t ∈ R. Thus, using (3.19), we get
(3.21)
∥∥∥K(ν)α,r(t)U(nt)x
∥∥∥ dt ≤
C̃(ν, r)
‖U(t)‖n ‖x‖ dt ≤
≤ C̃(ν, r) ‖x‖
MU (|t|)k
dt ≤ 2C̃(ν, r) ‖x‖ < ∞.
Therefore the integral
α,r(t)U(nt)x dt 
onverges. We de�ne
x(ν)r,n =
(−1)ν
K(ν)α,r(t)U(nt)x dt.
Then, using 
losedness of the operator A and integration by parts, one 
an �nd for
x ∈ D(A) that x(ν)r,n ∈ D(A) and
(3.22) Ax(ν)r,n =
(−1)ν
K(ν)α,r(t)U(nt)Axdt =
(−1)ν
K(ν)α,r(t)(U(nt)x)′dt =
= − (−1)
K(ν+1)α,r (t)U(nt)x dt = x(ν+1)r,n .
Let x is an arbitrary element of the spa
e X. Then there exists the sequen
e {xm}∞m=1 ⊂
D(A) su
h that ‖xm − x‖ → 0, m → ∞. Consequently, using inequality (3.21) and
relation (3.22), one 
an get
∥∥∥(xm)(ν)r,n − x(ν)r,n
∥∥∥ ≤
∥∥∥K(ν)α,r(t)U(nt)(xm − x)
∥∥∥ dt ≤
2C̃(ν, r)
‖xm − x‖ → 0;
∥∥∥A(xm)(ν)r,n − x(ν+1)r,n
∥∥∥ =
∥∥∥(xm)(ν+1)r,n − x(ν+1)r,n
∥∥∥→ 0, m → ∞.
Hen
e, taking into a

ount 
losedness of the operator A, we have:
(3.23) x(ν)r,n ∈ D(A), Ax(ν)r,n = x(ν+1)r,n .
One 
an get (3.20) from (3.23) by indu
tion.
Using relation (3.20) and lemma 3.2, one 
an �nd:
(3.24) ‖Aνxr,n‖ ≤
∣∣∣K(ν)α,r(t)
∣∣∣ ‖U(nt)‖ dt ≤
≤ ‖x‖
2πν α
α(|t|)
rν ‖U(t)‖n dt ≤
≤ c̃r ‖x‖
2πν α
)(∫ ∞
‖U(t)‖n
where, a

ordingly to (3.19) and due to n ≤ k,
‖U(t)‖n
‖U(t)‖k
dt ≤ 2 < ∞.
Sin
e β(t) = ln(α(t)), t ∈ R, as was mentioned in the proof of lemma 3.1, limτ→∞ β(τ)τ =
0 (
f. (3.5)). Thus
))1/ν
= lim
β( νr )) = e
·0 = 1.
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 9
Therefore from relation (3.24) one 
an get:
lim sup
‖Aνxr,n‖
)1/ν ≤ r
The last inequality brings us to the 
on
lusion that
(3.25) xr,n ∈ E(A) and σ(xr,n, A) ≤
For arbitrary x ∈ X we de�ne
(3.26) x̃r,k :=
Kα,r(t)(x + (−1)k−1(U(t)− I)kx)dt =
Kα,r(t)
(−1)n+1
U(nt)x dt
(the absolute 
onvergen
e by the norm of X of the integral in the right part of (3.26)
follows from inequality (3.21), so the de�nition of the ve
tor x̃r,k is 
orre
t). Using
de�nition (3.26) one 
an get:
x̃r,k =
(−1)n+1
Kα,r(t)U(nt)xdt =
(−1)n+1
xr,n.
Therefore, a

ordingly to (3.25),
x̃r,k ∈ E(A) and σ(x̃r,k, A) ≤ r.
Hen
e for an arbitrary x ∈ X we have:
Er(x,A) = inf
y∈E(A) :σ(y,A)≤r
‖x− y‖ ≤ ‖x− x̃r,k‖
Using (3.26), the property 2) of the kernel Kα,r and (2.3), the last inequality implies:
Er(x,A) ≤
Kα,r(t)xdt−
Kα,r(t)
x+ (−1)k−1(U(t)− I)kx
∥∥∥∥ ≤
Kα,r(t)
∥∥(U(t)− I)kx
∥∥ dt ≤
Kα,r(t)ω̃k(|t|, x, A) dt.
So, in a

ordan
e with the property 4) of the fun
tion ω̃k(|t|, x, A),
(3.27) Er(x,A) ≤
Kα,r(t)ω̃k
|rt|1
, x, A
≤ ω̃k
, x, A
1 + |rt|MU (|t|)
)kKα,r(t)dt.
Taking into a

ount properties of the fun
tion MU (·), the de�nition of Kα,r, lemma 3.1
and equality (3.19), one 
an �nd for r ≥ 1:
1 + |rt|MU (|t|)
)kKα,r(t)dt ≤
1 + |rt|MU (rt)
rKα(rt)dt ≤
(1 + τ)MU (τ)
)kKα(τ)dτ ≤ c1
(1 + |τ |)MU (|τ |)
dτ = 2c1 < ∞.
In a

ordan
e with (3.27), inequality (3.1) holds for all r ∈ [1,∞) with a 
onstant
mk = 2c1. It should be noted that 
onstant mk, indeed, depends on k, be
ause due to
3.1, the 
onstant c1 = c1(α) depends on the fun
tion α(t) = (MU (|t|))k(1 + |t|)k+2.
10 YA. GRUSHKA AND S. TORBA
Moreover, let the group {U(t)} is bounded (MU (t) ≤ M̃, t ∈ R, M̃ ≥ 1). Taking into
a

ount properties of the fun
tion MU (·), the de�nition of Kα,r, lemma 3.1 and equality
(3.19), one 
an �nd for r ∈ (0,∞)
(1 + |rt|MU (|t|))kKα,r(t)dt ≤
(1 + |rt|M̃MU (rt))krKα(rt)dt ≤
≤ M̃k
((1 + τ)MU (τ))
kKα(τ)dτ ≤ 2M̃kc1 < ∞,
whi
h proves remark 3.1 with the 
onstant mk = 2M̃
kc1. �
Theorem 3.1 allows us to prove the analogue of the 
lassi
 Ja
kson's inequality for m
times di�erentiable fun
tions:
Corollary 3.1. Let x ∈ D(Am), m ∈ N0. Then ∀k ∈ N0
(3.28) Er(x,A) ≤ mk+m
, Amx,A
, r ≥ 1,
where the 
onstants mn (n ∈ N) are the same as in theorem 3.1.
Proof. Let x ∈ D(Am) and r ≥ 1. By theorem 3.1,
Er(x,A) ≤ mk+m · ω̃k+m
, x, A
Let t ∈ R, 0 ≤ |t| ≤ 1
. Then, using properties of the groups of the C0 
lass and properties
of the fun
tion MU (t), one 
an get:
∥∥(U(t)− I)k+mx
∥∥(U(t)− I)m(U(t)− I)kx
· · ·
‖U(ξ1 + · · ·+ ξm)‖
∥∥(U(t)− I)kAmx
∥∥ dξ1 . . . dξm ≤
≤ MU (m|t|)
∥∥(U(t)− I)kAmx
∥∥ tm ≤
, Amx,A
This implies ω̃k+m
, x, A
= sup|t|≤ 1
∥∥(U(t)− I)k+mx
∥∥ ≤ MU (
, Amx,A
, whi
h
proves inequality (3.28). �
By setting in 
orollary 3.1 k = 0 and taking into a

ount that ω̃0 (·, Amx,A) ≡ ‖Amx‖,
one 
an 
on
lude the following inequality:
Corollary 3.2. Let x ∈ D(Am), m ∈ N0. Then
(3.29) Er(x,A) ≤
MU (1/r)
)m‖Amx‖ r ≥ 1,
where the 
onstants mn (n ∈ N) are the same as in theorem 3.1.
4. The examples of appli
ation of the abstra
t Ja
kson's inequality in
parti
ular spa
es
Lets 
onsider several examples of appli
ation of theorem 3.1 in parti
ular spa
es.
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 11
4.1. Ja
kson's inequalities in Lp(2π) and C(2π).
Example 4.1. Let the spa
e X and the operator A are the same as in the example 2.1.
Then for x ∈ X the quantity Er(x,A) is the value of the best approximation of fun
tion
x by trigonometri
 polynomials whose degree does not ex
eed r with respe
t to the norm
in X. It is generally known that di�erential operator A is a generator of (isometri
) group
of shifts in the spa
e X:
(U(t)x)(ξ) = x(t+ ξ), x ∈ X; t, ξ ∈ R
‖U(t)‖ ≡ 1, t ∈ R,(4.1)
where ‖U(·)‖ = ‖U(·)‖L(X) is the norm of the operator U(t) in the spa
e L(X) of linear

ontinuous operators over X. It follows from (4.1) that
ω̃k(t, x, A) = ωk(t, x, A) = sup
0≤h≤t
(−1)k−j
x(·+ jh)
, t ∈ R+, x ∈ X.
I.e., in that 
ase, ω̃k(t, x, A) 
oin
ides with 
lassi
 modulus of 
ontinuity of k-th degree
in the spa
e X.
Thus, from theorem 3.1 and 
orollary 3.1 one 
an 
on
lude all 
lassi
 Ja
kson-type
inequalities in the spa
es C(2π) and Lp(2π), 1 ≤ p < ∞.
4.2. Ja
kson's inequalities of the approximation by exponential type entire
fun
tions in the spa
e Lp(R, µ
p). We 
onsider the real-valued fun
tion µ(t) satisfying
the following 
onditions:
1) µ(t) ≥ 1, t ∈ R;
2) µ(t) is even, monotoni
ally non-de
reasing when t > 0;
3) µ(t) satis�es naturally o

urring in many appli
ations 
ondition µ(t+ s) ≤ µ(t) ·
µ(s), s, t ∈ R.
lnµ(t)
dt < ∞,
or alternatively, instead of 4), the equivalent 
ondition holds:
lnµ(k)
Lets 
onsider several important 
lasses of fun
tions satisfying 
onditions 1)�4).
1. Constant fun
tion µ(t) ≡ 1, t ∈ R.
2. Fun
tions with polynomial order of growth at in�nity. It is easy to 
he
k that for
su
h fun
tions following estimate holds: ∃k ∈ N, ∃M ≥ 1
µ(t) ≤ M(1 + |t|)k, t ∈ R.
3. Fun
tions of the form
µ(t) = e|t|
, 0 < β < 1, t ∈ R.
4. µ(t) represented as a power series for t > 0. I.e.,
µ(t) =
where {mn}n∈N is the sequen
e of positive real numbers satisfying two 
onditions:
• m0 = 1, m2n ≤ mn−1 ·mn+1, n ∈ N;
• ∀k, l ∈ N (k+l)!
The fun
tion µ(t), de�ned above, obviously satis�es 
onditions 1) and 2). The 
ondition
∀k, l ∈ N (k+l)!
implies
(4.2)
tksn−kn!
k!(n− k)!mn
tksn−k
mkmn−k
12 YA. GRUSHKA AND S. TORBA
and it is easy to see that 
ondition 3) follows from inequality (4.2). The Denjoy -
Carleman theorem [12, p.376℄ asserts that the following 
onditions are equivalent:
a) µ(t) satis�es 
ondition 4);
5. µ(t) as a module of an entire fun
tion with zeroes on the imaginary axis. We

onsider
ω(t) = C
, t ∈ R,
where C ≥ 1, 0 < t1 ≤ t2 ≤ . . . ,
< ∞. We set µ(t) := |ω(t)|. Then µ(t)
satis�es 
onditions 1) � 3), and, as shown in [8℄, µ(t) satis�es 
ondition 4) also.
Lets pro
eed to the des
ription of the spa
es Lp(R, µ
p). Let the fun
tion µ(t) satis�es

onditions 1) � 4). One 
an 
onsider the spa
e Lp(R, µ
p) of the fun
tions x(s), s ∈ R,
integrable in p-th degree with the weight µp:
Lp(R,µp)
|x(s)|pµp(s) ds.
Lp(R, µ
p) is the Bana
h spa
e. We 
onsider the di�erential operator A (D(A) = {x ∈
Lp(R, µ
p) ∩ AC(R) : x′ ∈ Lp(R, µp)}, (Ax)(t) = dxdt ). As in example 4.1, the operator
A generates the group of shifts {U(t)}t∈R in the spa
e Lp(R, µp). But in 
ontrast to
example 4.1, this group isn't bounded. Indeed, lets 
onsider
x(s) =
1, s ∈ [0, 1],
0, s 6∈ [0, 1].
Obviously, x(s) ∈ Lp(R, µp), but for t > 1
‖U(t)x‖p =
|x(t+ s)|pµp(s) ds =
µp(s) ds ≥ µp(t− 1) → ∞, t → ∞.
On the other hand, be
ause of the property 3),
‖U(t)x‖p =
|x(t+s)|pµp(s) ds ≤ µp(−t)
|x(t+s)|pµp(t+s) ds =
µ(−t)
)p‖x‖p,
so ‖U(t)‖Lp(R,µp) ≤ µ(−t) = µ(|t|), t ∈ R. 2
By the same way as in the example 4.1, modules of 
ontinuity ωk and ω̃k 
oin
ides
with 
lassi
 ones, but in 
ontrast to the example 4.1, they don't equal mutually. The
spa
e E(A) 
onsists of fast de
res
ent at the in�nity entire fun
tions. The examples of
su
h fun
tions have been given in [8℄. By applying theorem 3.1 one 
an get
Corollary 4.1. ∀k ∈ N there exists 
onstant mk(p, µ) > 0 su
h that ∀f ∈ Lp(R, µp)
Er(f) ≤ mk · ω̃k
, x, A
, r ≥ 1.
If µ(t) is 
ontinuous and µ(0) = 1, it is possible to show in a similar manner that ‖U(t)‖Lp(R,µp) =
µ(|t|).
DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 13
Referen
es
1. N. P. Kup
ov, Dire
t and inverse theorems of approximation theory and semigroups of operators
(Russian), Uspekhi Mat.Nauk. 23 (1968), No. 4, 118-178.
2. A. P. Terehin, A bounded group of operators, and best approximation (Russian), Di�eren
ial'nye
Uravneniya i Vy�
isl.Mat., Vyp.2 (1975), 3-28.
3. G. V. Radzievskii, On best approximations and the rate of 
onvergen
e of expansions in root ve
tors
of an operator, Ukrainian Math.J. 49 (1997), No. 6, 844-864 (1998).
4. G. V. Radzievskii, Dire
t and inverse theorems in problems of approximation by ve
tors of �nite
degree, Sb.Math. 189 (1998), No.3�4, 561-601.
5. M. L. Gorba
huk and V. I. Gorba
huk, On the approximation of smooth ve
tors of a 
losed operator
by entire ve
tors of exponential type, Ukrainian Math.J. 47 (1995), No. 5, 713-726 (1996).
6. M. L. Gorba
huk and V. I. Gorba
huk Operator approa
h to approximation problems, St.Petersburg
Math.J. 9 (1998), No. 6, 1097-1110.
7. M. L. Gorba
huk, Ya. I. Grushka and S. M. Torba, Dire
t and inverse theorems in the theory of
approximations by the Ritz method, Ukrainian Math.J. 57 (2005), No. 5, 751-764 (arXiv:0709.4243
[math.FA℄).
8. Ju. I. Ljubi�
 and V. I. Ma
aev, Operators with separable spe
trum (Russian), Mat.Sb.(N.S.) 56 (98)
(1962), No. 4, 433-468.
9. M. L. Gorba
huk, On analyti
 solutions of operator-di�erential equations, Ukrainian Math.J. 52
(2000), No. 5, 680-693 (2001).
10. V. A. Mar�
enko, On some questions of the approximation of 
ontinuous fun
tions on the whole real
axis (Russian), Zap. Mat. Otd. Fiz-Mat. Fak. KHGU i KHMO 22 (1951), No. 4, 115-125.
11. O. I. Inozem
ev and V. A. Mar�
enko, On majorants of genus zero (Russian), Uspekhi Mat.Nauk
(N.S.) 11 (1956), 173-178.
12. Walter Rudin, Real and Complex Analysis, M
Grow-Hill, New York, 1970.
E-mail address: sergiy.torba�gmail.
om, grushka�imath.kiev.ua
Institute of Mathemati
s of the National A
ademy of S
ien
es of Ukraine, Teresh
henkovskaya
3, 01601 Kiev (Ukraine)
http://arxiv.org/abs/0709.4243
	1. Introduction
	2. Preliminaries
	3. Abstract Jackson's inequality in a Banach space
	4. The examples of application of the abstract Jackson's inequality in particular spaces
	4.1. Jackson's inequalities in Lp(2) and C(2)
	4.2. Jackson's inequalities of the approximation by exponential type entire functions in the space Lp(R, p)
	References
