Lectures for DAAD course, Novi Sad, September 2013.
Gelfand-Shilov type spaces
Filip Tomić, Faculty of tehnical sciences, University of Novi Sad
• Lecture 1,2 : Basic notions and inequalities. Properties of the
Fourier transform on Schwarz space S. Definition of Gelfand-Shilov
type spaces S µ , Sν , Sνµ .
• Lecture 3,4 : Some equivalent characterizations of spaces Sνµ . Topology and triviality of these spaces.
• Lecture 5,6 : Simplest bounded operations in spaces Sνµ .
1
Lecture 1
Basic notions and
inequalities. Properties of
the Fourier transform on
Schwarz space S.
Let bf N = {0, 1, 2, . . .}. Given α, β ∈ Nd set |α| = α1 + . . . + αd and
α! = α1 ! . . . αd !. We say that α ≤ β if αj ≤ βj for¡ every
j = ¡1, .¢. . , d. α < β
¢ Q
α!
,
if α 6= β and α ≤ β. With this notation we have αβ = dj=1 αβjj = β!(α−β)!
for β ≤ α. For x ∈ Rd , partial derivatives are denoted by ∂j = ∂xj or
Dj = −i∂j . More generally we set ∂ α = ∂1α1 . . . ∂dαd or Dα = D1α1 . . . Ddαd .
For x ∈ Rd and α ∈ Nd we write xα = xα1 1 xα2 2 . . . xαd d .
Example: Let x ∈ Rd and α, β ∈ Nd . Then
(
β
α
∂ (x ) =
¡α¢ α−β
, β≤α
β x
β!
0, β > α
.
(1.1)
Functions are always understood as complex-valued. With C n (Rd ) we
denote space of continuous functions with n continuous derivatives on Rd .
T
C ∞ (Rd ) = N C n (Rd ) is space of infinitely differentiable functions. Using
multiindeks notation we can express Leibniz’ formula for multidimensional
case:
∂ α (f g) =
X
à !
α β α−β
∂ f∂
g
β
β≤α
2
(1.2)
for f, g ∈ C n (Rd ). The support of a continuous function f is defined to be
the closure in Rd of the set {x ∈ Rd : f (x) 6= 0} and is denoted by supp(f ).
With C0∞ (Rd ) we denote the space of C ∞ functions on Rd with compact
support.
If f, g : Rd → [0, ∞), we set
f (x) . g(x),
x ∈ Rd ,
if there exists C > 0 such that f (x)C ≤ g(x), for all x ∈ Rd . Moreover,
if f and g depends on a futher variable z ∈ Rd , the statement that, for all
z ∈ Rd ,
f (x, z) . g(x, z), x ∈ Rd ,
means that for every z ∈ Rd there exists Cz > 0 such that f (x, z) ≤
Cz g(x, z).
1.1
Basic inequalities
Let us start with generalized Newton’s formula:
(x1 + . . . + xd )N =
X N!
xα .
|α|=N
(1.3)
α!
If we put x1 = . . . = xd = 1 in (1.3) we get dN =
we get
P
N!
|α|=N α! .
|α|! ≤ dN α!.
In particular,
(1.4)
If d = 2, then by (1.4) we have (k + j)! ≤ 2k+j k!j!, or for α, β ∈ Nd we have
(α + β)! ≤ 2|α|+|β| α!β!.
(1.5)
On the other hand, the inequalities α!β! ≤ (α ¡+ ¢β)! and α! ≤ |α|! are trivial.
P
Moreover, using formula (1.3) we have β≤α αβ = 2|α| and so
à !
α
β
≤ 2|α| ,
β ≤ α.
(1.6)
If we denote by #A cardinality of set A, following simple formulas are true:
Ã
m+d
#{(α1 , . . . , αd ) ∈ N : |α| ≤ m} =
m
d
3
!
(1.7)
and
Ã
d
#{(α1 , . . . , αd ) ∈ N : |α| = m} =
!
m+d−1
.
d−1
(1.8)
We will end this section with Stirling’s formula:
√
θN
N ! = N N e−N 2πN e 12N ,
(1.9)
for some 0 < θN < 1, N=1,2,. . . . Formula (1.10) implies that
N N ≤ eN N !,
(1.10)
whereas obviously we have N ! ≤ N N .
1.2
Schwartz space S(Rd ) and Fourier Transform
For x, ξ ∈ Rd we set x · ξ = xξ =
Pd
j=1 xj ξj and |x| =
³P
that a measurable function f belongs to Banach space
³R
´1
´1
d
2 2.
j=1 xj
Lp (Rd ), 1 ≤
We say
p < ∞,
p
p
< ∞ (dx is standard Lesbegue measure on Rd ).
Rd |f (x)| dx
f belongs to L∞ (Rd ) if ||f ||∞ = supRd |f (x)|
< ∞. For p = 2 we get Hilbert
R
2
d
space L (R ) with scalar product (f, g) = Rd f (x)g(x)dx.
Let f be a measurable function defined on Rd . Fundamental operators
if ||f ||p =
of time-frequency analysis are translation, modulation and dilatation defined
by
Ty f (x) = f (x − y), y ∈ Rd ,
Mσ f = eiσx f (x),
Ds f (x) = f (sx),
σ ∈ Rd ,
s > 0,
respectively.
Definition 1.2.1 The Schwartz space of rapidly decreasing functions S(Rd )
consists of C ∞ (Rd ) functions such that for every α, β ∈ Nd there exists
Cαβ > 0 such that
sup |xα ∂ β f (x)| ≤ Cαβ .
x∈Rd
4
(1.11)
Topology of space S is given by family of seminorms at the left hand
side of (1.11). By using simple inequalities
|xα | ≤ |x||α|
and |x|n ≤ Cdn
X
|xα | α ∈ Nd , x ∈ Rd ,
(1.12)
|α|=n
one can show that equivalent definition of S is that for every n ∈ N and
α ∈ Nd the exists Cn,α > 0 such that
sup hxin |∂ α f (x)| ≤ Cn,α ,
(1.13)
x∈Rd
1
where h·i = (1 + | · |2 ) 2 are Japanese brackets.
Example: By elementary computation it can be shown that Gaussian
|x|2
function f (x) = e− 2 is in S(Rd ). Function g(x) = e−|x| is not in S because
it is not C ∞ on Rd .
We would like to point out some basic properties of S:
• It is an algebra with respect to pointwise multiplication (direct use of
Leibniz’ formula).
• It is closed under multiplication by polynomials.
• It is closed under differentiation, translation and modulation operators.
• It is continuously embedded in all Lp , 1 ≤ p ≤ ∞, i.e., there exists
C > 0 and m ∈ N such that
||f ||p ≤ C
X
sup |xα ∂ β f (x)|,
f ∈ S.
d
|α|,|β|≤m x∈R
• Moreover, it is dense in Lp spaces for 1 ≤ p < ∞, but not in L∞ .
Definition 1.2.2 For f ∈ S(Rd ) we define Fourier transform by
d
fˆ(ξ) = Ff (ξ) = (2π)− 2
Z
Rd
f (x)e−ixξ dx,
ξ ∈ Rd .
Next theorem captures basic properties of Fourier transform on S.
Theorem 1.2.1 Let f, g ∈ S(Rd ). Then
5
d
i) ||fˆ||∞ ≤ (2π)− 2 ||f ||1 .
α f (ξ) = ξ α fˆ(ξ), for every α ∈ Nd .
ii) D\
\
β f (ξ), for every β ∈ Nd .
iii) Dβ fˆ(ξ) = (−x)
−d
b
ˆ
ˆ \
ˆ
iv) T\
y f (ξ) = M−y f , Mσ f (ξ) = Tσ f (ξ) and Ds f (ξ) = s D 1 f (ξ), for
s
every y, σ ∈ Rd and s > 0.
v) fˆ ∈ S.
vi) (Adjoint formula)
R
Rd
R
fˆ(x)g(x) dx = Rd f (x)ĝ(x) dx.
vii) (Plancherel identity) ||f ||2 = ||fˆ||2 .
Using the fact iv) in theorem 1.2.1 we can define inverse Fourier transform
for elements in S.
Definition 1.2.3 For f ∈ S(Rd ) we define inverse Fourier transform by
∨
f (ξ) = F
−1
− d2
f (ξ) = (2π)
Z
Rd
fˆ(ξ)eixξ dξ,
x ∈ Rd .
It holds that F −1 Ff = FF −1 f = f forf ∈ S. Moreover, Fourier transform
is topological isomorphism from S onto S (i.e. FS = S).
x2
Example: Consider ϕ(x) = e− 2 on R. Then ϕ satisfies differential
equation
ϕ0 + xϕ = 0.
(1.14)
Applying Fourier transform to equation (1.14) and using i) and ii) from
theorem 1.2.1) we obtain equation ξ ϕ̂ + ϕ̂0 (ξ) = 0 (modulo a multiplicative
ξ
constant). Thus, we conclude that ϕ̂(ξ) = e− 2 , ξ ∈ Rd . For multidimensional case, observe that
− d2
(2π)
Z
Rd
e
−
|x|2
2
−ixξ
e
− d2
dx = (2π)
6
d Z
Y
j=1 R
e
−
x2
j
2
e−ixj ξj dxj .
1.3
Tempered distributions
In the sequel we will give a short review of a space of tempered distributions.
Definition 1.3.1 A sequence of functions {ϕj }j∈N tends to zero in S if for
every α, β ∈ Nd
sup |xα ∂ β f (x)| → 0, j → ∞.
x∈Rd
Definition 1.3.2 A linear functional T on S is called tempered distribution
if for any sequence {ϕj }j∈N that tends to zero in S, T (ϕj ) tends to zero as
sequence of numbers. Space of all tempered distributions i denoted by S 0 (Rd ).
Definition 1.3.3 Let f be a measurable function defined on Rd such that
Z
|f (x)|
hxiN
Rd
dx < ∞,
for some positive integer N . Then we call f a tempered function.
It is clear from definition that tempered functions have at most polynomial
growth at infinity. If f is a tempered function, then it can be shown that
Z
Tf (ϕ) =
Rd
f (x)ϕ(x)dx := hf, gi,
(1.15)
is a tempered distribution.
It is customary to identify the tempered distribution Tf with a function f
and to say that such tempered distributions are functions. Duality between
spaces S and S 0 is given by (1.15).
Example: Define δ(ϕ) = ϕ(0). To show that δ ∈ S 0 choose {ϕj }j∈N in
S that tends to zero. Since convergence in S imply uniform convergence we
have that δ(ϕj ) = ϕj (0) tends to zero.
Delta distribution is an example of tempered distribution that is not a
tempered function. Explicitly it is given by
(
δ(x) =
+∞, x = 0
.
0, x =
6 0
We can use iv) and v) from theorem 1.2.1 to define Fourier transform
on S 0 .
Definition 1.3.4
T̂ (ϕ) = T (ϕ̂),
7
ϕ ∈ S.
(1.16)
It can be shown that Fourier transform defined by (1.16) is topological
isomorphism on S 0 . S 0 is endowed with the weak-star topology defined by
the (uncountable) family of seminorms
|T |ϕ = |hT, ϕi|, ϕ ∈ S.
We say that two tempered distributions T and S are equal if hT, ϕi = hS, ϕi
for every ϕ ∈ S.
Example: Let us compute Fourier transform of δ distribution. Using
(1.16) we have
− d2
hδ̂, ϕi = hδ, ϕ̂i = (2π)
Z
Rd
d
ϕ(x)dx = h(2π)− 2 , ϕi .
d
Thus, δ̂ = (2π)− 2 .
At the end of this section we introduce Sobolev spaces.
Definition 1.3.5 Let s ∈ R. Sobolev spaces H s (Rd ) consists of f ∈ S 0 such
that
||f ||H s := ||h·ifˆ||2
(1.17)
is finite, where h·i are Japanese brackets.
Remark: When s = k ∈ N equivalent norm to (1.17) is given by
||f ||H k ≡
X
||∂ α f ||2 .
(1.18)
|α|≤k
Theorem 1.3.1 (Sobolev embedding theorem) For every s >
Cs > 0 such that
||f ||∞ ≤ Cs ||f ||H s .
8
d
2
there exists
Lecture 2
Definition of Gelfand-Shilov
type spaces S µ, Sν , Sνµ. Some
equivalent characterizations
and triviality of spaces Sνµ.
2.1
Definition of Gelfand-Shilov type spaces
Let us recall: S(Rd ) consists of C ∞ (Rd ) functions such that for every α, β ∈
Nd there exists Cαβ > 0 such that
sup |xα ∂ β f (x)| ≤ Cαβ .
(2.1)
x∈Rd
In the sequel S will be our universal set.
For special choice of constants Cα,β in (2.1) we get appropriate subspaces
of S. In particular we define
Definition 2.1.1 Let f ∈ S and µ, ν > 0.
i) f ∈ S µ (Rd ) if for suitable constant C > 0 and all α ∈ Nd
sup |xα ∂ β f (x)| . C |β| (β!)µ
, β ∈ Nd .
(2.2)
x∈Rd
ii) f ∈ Sν (Rd ) if for suitable constant C > 0 and all β ∈ Nd
sup |xα ∂ β f (x)| . C |α| (α!)ν ,
x∈Rd
9
α ∈ Nd .
(2.3)
iii) f ∈ Sνµ (Rd ) if for suitable constant C > 0
sup |xα ∂ β f (x)| . C |α|+|β| (α!)ν (β!)µ ,
α, β ∈ Nd .
(2.4)
x∈Rd
Theorem 2.1.1 For f ∈ S the following conditions are equivalent:
i) There exists constant C > 0 such that
sup |xα f (x)| . C |α| (α!)ν .
(2.5)
x∈Rd
ii) There exists constant ε > 0 such that
1
|f (x)| . e−ε|x| ν ,
x ∈ Rd .
(2.6)
Remark: If we apply theorem (2.1.1) to function f (x) = (∂ β g)(x), for
β ∈ Nd and g ∈ S we will get the bounds that does not depend on β. If
function satisfy inequality (2.5) we say that it decreases exponentially with
an order ≥ ν1 and a type ≥ ε. Connection between constants C in (2.5) and
ε in (2.6) is given by
1
νC − ν
.
(2.7)
ε=
2
Next theorem gives us equivalent characterizations of spaces Sνµ in terms
of Fourier transform.
Theorem 2.1.2 Assume that µ, ν > 0 and that µ + ν ≥ 1. For f ∈ S(Rd )
the following conditions are equivalent:
i) f ∈ Sνµ .
ii) There exists C > 0 such that
sup |xα f (x)| . C |α| (α!)ν ,
α ∈ Nd ,
x∈Rd
sup |ξ β fˆ(ξ)| . C |β| (β!)µ ,
ξ∈Rd
10
β ∈ Nd .
iii) There exists ε > 0 such that
1
|f (x)| . e−ε|x| ν ,
1
µ
|fˆ(ξ)| . e−ε|ξ| ,
x ∈ Rd ,
(2.8)
ξ ∈ Rd .
(2.9)
v) There exists C > 0 such that
||xα f ||2 . C |α| (α!)ν ,
α ∈ Nd ,
||ξ β fˆ||2 . C |β| (β!)µ ,
β ∈ Nd .
vi) There exists C > 0 such that
||xα f ||2 . C |α| (α!)ν ,
α ∈ Nd ,
||∂ β f ||2 . C |β| (β!)µ ,
β ∈ Nd .
vi) There exists C > 0 such that
||xα ∂ β f ||2 . C |α|+|β| (α!)ν (β!)µ ,
α, β ∈ Nd ,
Assuming that µ + ν ≥ 1, by theorem 2.1.2 we immediatly obtain following properties of spaces Sνµ :
0
• Sνµ ⊆ Sνµ0 for µ ≤ µ0 , ν ≤ ν 0 .
• f ∈ Sνµ if and only if fˆ ∈ Sµν . In particular, Sµµ are Fourier transorm
invariant.
• f ∈ Sνµ if and only if
1
sup eε|x| ν |∂ β f (x)| . C |β| (β!)µ
(2.10)
x∈Rd
for some ε, C > 0.
The case ν = µ plays an important role in applications, because of the
invariance under Fourier transform. We give another useful characterization
of these spaces.
11
Theorem 2.1.3 A function f ∈ S(Rd ) belongs to Sµµ , µ ≥ 21 , if and only if
there exists a constant C > 0 such that
sup |xα ∂ β f (x)| . C N N N µ ,
|α| + |β| ≤ N , N = 0, 1, 2 . . . ..
(2.11)
x∈Rd
Proof: From (2.11) we have
sup |xα ∂ β f (x)| . C |α|+|β| (|α| + |β|)(|α|+|β|)µ ,
α, β ∈ Nd .
x∈Rd
Applying (1.10), (1.4), (1.5) we get
(|α| + |β|)(|α|+|β|)µ ≤ (2ed)|α|+|β| α!β! .
On the other hand, for |α| + |β| ≤ N , by (2.4) (for ν = µ) we have
sup |xα ∂ β f (x)| . C |α|+β (α!β!)µ . N N µ
N = 0, 1, 2 . . . ,
x∈Rd
where we assume C ≥ 1. This proves the theorem.
2
2k
Example: Consider functions g(x) = e−t , x ∈ R, k ∈ N, which
1
satisfies (2.8) for ν = 2k
. Note that g satisfies equation
g 0 + 2kx2k−1 g = 0.
(2.12)
Let h(ξ) = ĝ(ξ). Applying Fourier transform to equation (2.12) we obtain
h2k−1 +
(−1)k+1
ξh = 0,
2k
(2.13)
modulo a multiplicative constant.
It can be shown that bounded solutions of (2.13) must satisfy
|h(ξ)| . e−ε|ξ|
2k
2k−1
,
1
1− 2k
for some ε > 0. Thus we conclude that g ∈ S 1
2k
12
2
1
. In particular, e−|x| ∈ S 12 .
2
2.2
Triviality of spaces Sνµ
In the sequel we will show that condition µ + ν ≥ 1 in the theorem 2.1.2 is
not restrictive. In fact, for µ + ν < 1 spaces Sνµ are trivial (i.e. they contain
only zero function). We will use the following theorem.
Theorem 2.2.1 Assume that f ∈ S(Rd ) satisfies (2.10) for 0 < µ < 1
and ν > 0. Then f extends to an entire analytic function f (x + iy) in Cd
with
1
|f (x + iy)| . e−ε|x| ν +δ|y|
1
1−µ
,
x, y ∈ Rd ,
(2.14)
where δ > 0 is a suitable constant. In the case µ = 1, ν > 0 f extends to
an analytic function f (x + iy) in the strip {x + iy ∈ Cd : |y| < T } for a
suitable T > 0 with
1
|f (x + iy)| . e−ε|x| ν ,
x ∈ Rd , |y| < T .
Next theorem is of Liouville-type. Proof is a consequnce of Liouville theorem from complex analysis which states that every bounded entire function
is constant.
Theorem 2.2.2 Assume that 0 < λ < θ. Let f (x + iy) be an entire finction
in Cd satisfying
|f (x + iy)| . e−ε|x|
θ +δ|y|λ
,
x, y ∈ Rd ,
(2.15)
for suitable constants ε, δ > 0. Then f ≡ 0.
Proof: Consider the entire function f (iz) = f (ix − y). Then we have
θ +δ|y|λ
|f (z)f (iz)| . e−ε|x|
θ +δ|x|λ
e−ε|y|
x, y ∈ Rd .
Since λ < θ, right-hand side tends to zero for x + iy → ∞. By Lioville
theorem, the entire function f (z)f (iz) is identically zero. This implies that
f ≡ 0.
2
Now we may prove theorem concerning triviality of spaces Sνµ . The result
can be read as a version of Heinserberg uncertainty principle, generically
expressed by the statement that a function f and its Fourier transform fˆ
cannot both be small at infinity.
13
Theorem 2.2.3 Let f ∈ S(Rd ). For µ + ν < 1 classes Sνµ are trivial.
Moreover, if µ + ν < 1 each of conditions in theorem 2.1.2, as well as 2.10,
implies f ≡ 0.
We finally observe that definition that the definition of spaces Sνµ extends
in a natural way to the cases when ν and µ takes values 0 and ∞. First
µ = S µ and S ∞ = S , where spaces S µ and S ν are defined by
observe that S∞
ν
ν
(2.2) and (2.3), respectively. The class S0µ , µ > 1, consists of C0∞ functions
which satisfies
sup |∂ β f (x)| . C β (β!)µ ,
β ∈ Nd .
x∈Rd
It coincides with Gµ0 , the space of compactly supported Gevrey functions.
Similarly we define Sν0 , which consists of bandlimited functions (i.e. functions
with compactly supported Fourier transform.)
Remark: It can be shown that Sν ∩ S µ = Sνµ . Inclusion ” ⊇ ” is trivial
while ” ⊆ ” follows from theorem 2.1.2.
14
Lecture 3
Simplest bounded operations
in spaces Sνµ
3.1
Topology of spaces Sνµ
µ,B
For simplicity we will assume that d = 1. Fix A, B > 0. Let Sν,A
(R) denote
the set of all functions ϕ ∈ Sνµ such that
sup |xk ϕ(q) (x)| . Ak1 B1q k kν q qµ
k, q = 0, 1, 2 . . .
x∈R
is valid for any A1 > A and B1 > B.
µ,B
In other words, Sν,A
(R) consists of those ϕ ∈ Sνµ which for any δ, ρ > 0
satisfies
sup |xk ϕ(q) (x)| . (A + δ)k (B + ρ)q k kν q qµ
k, q = 0, 1, 2 . . . .
(3.1)
x∈R
Constants in (3.1) are only depending on δ, ρ and ϕ. We say that
µ,B
µ,B
(3.1) holds and constants are
F ⊆ Sν,A
is bounded if for every ϕ ∈ Sν,A
independent of ϕ. By using (2.10) and the fact (2.7) we get that
1
sup e(a−δ)|x| ν |xk ϕ(q) (x)| . (B + ρ)q q qµ
k, q = 0, 1, 2 . . . .
(3.2)
x∈R
µ,B
is equivalent definition of spaces Sν,A
(R). For convenience, we will denote
µ,B
by Kν,a space of all functions which satisfies (3.2).
15
If we introduce family of norms
||ϕ||δ,ρ =
|xk ϕ(q) (x)|
k
q kν qµ
k,q∈N∪{0} x∈R (A + δ) (B + ρ) k q
sup
sup
δ, ρ > 0 ,
µ,B
it can be shown that Sν,A
(R) are complete countably normed spaces. We
may also construct the union of these spaces for A, B = 1, 2, . . . and obtain
Sνµ =
[
µ,B
Sν,A
.
A,B
Remark: We say that sequence ϕj , j ∈ N, converges to zero in Sνµ if all
µ,B
the functions ϕj belongs to some Sν,A
and converges to zero in topology of
µ,B
Sν,A
.
Remark: For ν = µ we get Fourier transform invariant space Sµµ . Dual
space of Sµµ is denoted by Sµµ 0 and known as space of tempered ultradistributions.
3.2
Bounded operations in spaces Sνµ
.
Many linear operators of importance to anlysis are defined and bouded
(therefore also continuos) in spaces Sνµ . We will prove some of the following
statements:
• Operation of multiplication by x is defined and bounded in any space
Sνµ for µ, ν ≥ 0.
• Differentiation operator is defined and bounded in any space Sνµ for
µ, ν ≥ 0.
• Translation operator Th ϕ(x) = ϕ(x−h) , h ∈ R, is defined by bounded
in any space Sνµ for ν > 0 and µ ≥ 0. For ν = 0 translation operator
µ,B
µ,B
transform S0,A
into S0,A+|h|
, so the space S0µ,B remains unchanged.
• Dilatation operator Ds ϕ(x) = ϕ(sx) , s > 0, is defined and bounded in
µ,B
µ,λB
any space Sνµ for µ, ν ≥ 0. It transorms space Sν,A
into Sν,
A .
λ
At the end, we will consider two imprtortant operators:
Multiplication by an infinetely differentiable function: Let us
consider function f satisfying
16
1
|f (q) (x)| . εq q qµ eε|x| ν ,
x ∈ R,
(3.3)
for every ε > 0 and for ν > 0. We will prove that multilplication by function
µ,B
f transform Sν,A
into itself, and hence it is a bounded operator on Sνµ .
If f satisfies inequality
1
|f (q) (x)| . B0q q qµ ea0 |x| ν ,
x ∈ R,
(3.4)
µ,B for
for some B0 , a0 > 0, then multiplication by f is defined in spaces Kν,a
µ,B+B0
which a0 < a, and transorms them into Kν,a−a
.
0
Exmaple: Consider the function f (x) = eiσx . It is easy to see that
function f satisfies (3.3) for any µ > 0. For µ = 0 function f satisfies
condition (3.4) for B0 = |σ|.
Infinite order differential operators: Let
f (s) =
∞
X
cn sn ,
n=0
be some entire function. We will consider infinite oreder differential operators of the form
∞
X
dn
d
cn n .
f( ) =
dx
dx
n=0
Definition 3.2.1 We say that the entire function f has an order of growth
≤ λ and a type < b i it satisfy the inequality
λ
|f (s)| . eb1 |x| ,
s ∈ R,
where b1 < b is some constant.
The following theorem holds:
P∞
Theorem 3.2.1 Let µ > 0. If f (s) =
function of order of growth ≤ µ1 and type <
n
n=0 cn s
µ
1
B µ e2
is an entire analytic
d
the the operator f ( dx
) is
1
µ
µ,B
µ,Be
.
defined and bounded in the space Sν,A
, and transorm this space into Sν,A
17