The Paley–Wiener theorem relates growth properties of entire functions on ⺓n to Fourier transforms of dis tributions in Ᏹ'K.. The importance of the Paley– Wiener theorem consists in that
Trang 1Original Russian Text © Ha Huy Bang, Vu Nhat Huy, 2012, published in Doklady Akademii Nauk, 2012, Vol 446, No 5, pp 497–500.
Let Ᏹ(⺢n ) = C∞(⺢n), and let Ᏹ'(⺢n) be the dual
space Then any element in Ᏹ'(⺢n) is a compactly sup
ported distribution, and vice versa Consider a com
pact set K in ⺢n Let us try to construct all elements of
the subspace Ᏹ'(K) of distributions supported on K.
This case differs from the case of (⺢n ) = L q(⺢n) 1 ≤
p < ∞, + = 1 , in which we can specify any ele
ment of the dual space as an element of L q(⺢n); it is
impossible to directly define all elements of Ᏹ'(⺢n),
because, generally, these are generalized functions
The Paley–Wiener theorem relates growth properties
of entire functions on ⺓n to Fourier transforms of dis
tributions in Ᏹ'(K) The importance of the Paley–
Wiener theorem consists in that it makes it possible to
construct all elements of Ᏹ'(K) This theorem and its
versions were studied by many authors (see, e.g., [1–
15]) In this paper, we describe the Fourier image of
the space Ᏹ'(K), where K is any compact set, and apply
this result to construct all elements of Ᏹ'(K) Since
entire functions are usually specified as power series,
and the Fourier transforms of compactly supported
distributions are entire functions, which are uniquely
determined by their Taylor expansion coefficients (at
the origin), we can state the Paley–Wiener theorem in
the language of Taylor coefficients; this is our purpose
First, we prove the Paley–Wiener theorem in the case
of any compact set K Since necessary and sufficient
conditions in this general case are complicated, in
subsequent sections, we introduce certain types of
compact sets K for which the necessary and sufficient
1
p
1
q
⎠⎞
conditions in the corresponding Paley–Wiener theo rem have simpler form Note that the original Paley–
Wiener theorem was proved for L2functions [13] Schwartz was the first to state this theorem for distri
butions (in the case where K is a ball) [14]; then, Hör
mander proved it for convex compact sets K [10] The Paley–Wiener theorem for nonconvex K was studied
in [6, 7]
1 THE CASE OF ARBITRARY
COMPACT SETS
Theorem 1 Let f ∈ Ᏹ'(⺢n ), and let K be any compact set in ⺢n Then supp f ⊂ K if and only if, for any δ > 0, there exists a constant Cδ < ∞ such that
(1)
for any polynomial P(x), where D = (D1, D2, …, D n),
Dα= … , D j = for j = 1, 2, …, n, K(δ) is the δneighborhood in ⺓n of the set K, and = Ff is the Fou rier transform of the function f.
Remark 1 Theorem 1 remains valid if only polyno
mials with real coefficients are considered
Remark 2 Theorem 1 remains valid if only polyno
mials of the form Q p (x)xα, where Q is any polynomial
of degree 2, p ∈ ⺪+, and α ∈ , are considered
2 SETS WITH gPROPERTY
Below, we recall some notions and results from [6, 7] Suppose that 0 ≤ λα ≤ ∞, α ∈ , and
Then G{λα} is referred to as the set generated by the sequence of numbers {λα}
P D ( )fˆ 0 ( ) Cδ P x( )
x∈K( ) δ
sup
≤
D1α1
D nαn i∂
∂x j
f
ˆ
⺪+n
⺪+n
G{ }λα ξ ⺢n
: ξα ≤λα
∈
α ⺪ +
n
∈
∩
=
The Paley–Wiener Theorem in the Language
of Taylor Expansion Coefficients
Ha Huy Banga and Vu Nhat Huyb
Presented by Academician V.S Vladimirov March 15, 2012
Received May 15, 2012
DOI: 10.1134/S1064562412050237
a Institute of Mathematics, 18 Hoang Quoc Viet Street, Cau
Giay, Hanoi, Vietnam
email: hhbang@math.ac.vn
b Hanoi State University, 334 Nguyen Trai Street,
Thanh Xuan, Hanoi, Vietnam
email: nhat_huy85@yahoo.com
MATHEMATICS
Trang 2678 HA HUY BANG, VU NHAT HUY
Let K ⊂ ⺢n We set
We have K ⊂ g(K), and g(K) is called the ghull of the
set K We say that K has gproperty if K = g(K).
The following assertions hold
(i) Any set generated by a sequence of numbers has
gproperty, and vice versa.
(ii) Let I be a set of indices, let and K j = g(K j ) for j ∈ I.
Then has gproperty.
(iii) The set G{λα} may be nonconvex, and any sym
metric convex compact set has gproperty.
It turns out that if K has gproperty, then the formu
lation of the Paley–Wiener theorem becomes very
simple
Theorem 2 Suppose that f ∈ Ᏹ'(⺢n ) and a compact
set K ⊂ ⺢n has gproperty Then supp f ⊂ K if and only
if, for any δ > 0, there exists a constant Cδ < ∞ such that
(2)
where Kδ is the δneighborhood of K.
Remark 3 Theorem 2 remains valid under the
replacement of (2) by the condition
3 SETS GENERATED BY POLYNOMIALS
Let P(x) be a polynomial with real coefficients.
We set
The set Q(P) r is said to be generated by the polynomial
P(x) Note that any tori and balls are sets generated by
polynomials
Theorem 3 Suppose that f ∈ Ᏹ'(⺢n ), r > 0, P(x) is a
polynomial with real coefficients, and Q(P) r is a compact
set Then suppf ⊂ Q(P) r =: K if and only if, for any δ > 0,
there exists a constant Cδ < ∞ such that
If K is a finite intersection of compact sets of the
forms specified above, then it is easy to prove the
Paley–Wiener theorem for K For example, Theorems 2
and 3 imply the following assertion
Theorem 4 Suppose that f ∈ Ᏹ'(⺢n ), r > 0, P1(x),
P2(x), …, P q (x) are polynomials with real coefficients;
Q(P1)r , …, Q(P q)r are compact sets; and H is a compact
set with gproperty Let K := H ∩ Q(P1)r ∩ … Q(P q)r
Then suppf ⊂ K if and only if, for any δ > 0, there exists
g K( ) G ξα
ξ K∈
sup
=
K j
j∈I
∩
Dαˆf ( ) C0 δ ξα, ∀α
ξ K∈ δ
,
∈
≤
Dαˆ 0f ( ) Cδαn
ξα, ∀α
ξ K∈ δ
∈
≤
Pαm ( ) := P x m
x ( )xα, m ⺪+, α ⺪+n
,
Q P( )r := x ⺢n
: P x ( ) r≤
∈
Pαm ( )fˆ 0 D ( ) Cδ(r+δ)m
x ,
x∈K( ) δ
sup
≤
m
∀ ∈⺪+, α ⺪+n
∈
a constant Cδ < ∞ such that, for all (m1 , m2, …, m q) ∈
, and α ∈ ,
Let B(x, ⑀) denote an open ball, and let B[x, ⑀]
denote a closed ball
Remark 4 Suppose that b0, b1, …, b k ∈ ⺢n and
r0, r1, …, r k > 0 are such that B(b0, r0) ∩ B(b j , r j) ≠ for
j = 1, 2, …, k Choose a number R > 0 so that B(b0, r0) ⊂
B(b j , R) for j = 1, 2, …, k Then K1 = B[b0, r0]\ (b j , r j)
is the intersection of B[b0, r0] with the tori B[b j , R]\B(b j , r j),
j = 1, 2, …, k, generated by polynomials Moreover, if
H has gproperty, then K2 = H \ (b j , r j) is the intersection of H with the tori B[b j , R]\B(b j , r j ) for j =
1, 2, …, k, and Theorem 4 can be applied to K1 and K2
4 CONVEX COMPACT SETS Let ⺠1 denote the set of all polynomials of degree
≤1 with real coefficients, and let Φ be the set of all polynomials of the form P m (x)xα, where P(x) ∈ ⺠1,
m∈ ⺪+, and α ∈
Theorem 5 Suppose that f ∈ Ᏹ'(⺢n ) and K is a con vex compact set in ⺢n Then suppf ⊂ K if and only if, for any δ > 0, there exists a constant Cδ < ∞ such that
(3)
for all P(x) ∈ Φ
Remark 5 Theorem 5 remains valid if P has the form Q p (x)xα, where p ∈ ⺪+ and α ∈ , and Q(x) is a
polynomial of degree 1 with complex coefficients Theorems 5 and 2 imply the following assertion
Theorem 6 Suppose that f ∈ Ᏹ'(⺢n ), K1 is a convex compact set, K2 is a compact set with gproperty, and
K := K1 ∩ K2 Then suppf ⊂ K if and only if, for any δ > 0, there exists a constant Cδ < ∞ such that
for all P(x) ∈ Φ
Theorems 5 and 3 imply the following theorem
Theorem 7 Suppose that f ∈ Ᏹ'(⺢n ), r > 0,
P1(x), P2(x), …, P q (x) are polynomials with real coeffi cients; H is a convex compact set; and Q(P1)r , …, Q(P q)r
are compact sets Let K := H ∩ Q(P1)r ∩ … ∩ Q(P q)r
Then suppf ⊂ K if and only if, for any δ > 0, there exists
a constant Cδ < ∞ such that
⺪+q
⺪+n
P1m1
x ( )…P q
m q x ( )xα
Cδ(r+δ)m1+… m+ q
x
xsup∈K( ) δ
≤
䊊
B
j= 1
k
∪
B
j= 1
k
∪
⺪+n
P D ( )fˆ 0 ( ) Cδ P x( )
x∈K( ) δ
sup
≤
⺪+n
P D ( )fˆ 0 ( ) Cδ P x( )
x∈K( ) δ
sup
≤
PP1m1
…P q
m q
( ) D ( )fˆ 0 ( ) Cδ(r+δ)m1+… m+ q
P x( )
x∈K( ) δ
sup
≤
Trang 3THE PALEY–WIENER THEOREM 679
for all P(x) ∈ Φ and (m1 , m2, …, m q) ∈
Remark 6 If K is the intersection of a convex com
pact set with compact sets generated by polynomials or
with compact sets having gproperty, then the Paley–
Wiener theorem can be proved for K For example, let
K = H\ (b j , r j ), where H is a convex compact set;
b1, b2, …., b k ∈ ⺢n ; and r1, r2, …, r k > 0 are such that
H ∩ B(b j , r j) ≠ for j = 1, 2, …, k Then K is the inter
section of H with compact sets generated by polyno
mials, and Theorem 7 applies to such K.
5 AN APPLICATION Recall the Paley–Wiener theorem Suppose that
K is a convex compact set, H is its support function, and
v ∈ Ᏹ'(K) is a distribution of order N.
Then
(*)
Conversely, any entire function satisfying an estimate
of the form (*) is the Fourier–Laplace transform of some
distribution in Ᏹ'(K).
Note that the assumption f ∈ Ᏹ'(⺢n) cannot be dis
pensed with in our theorems Indeed, the function e z
satisfies (2) with K = [–1, 1], but it grows very rapidly
on ⺢+ The Paley–Wiener theorem proved by
Schwartz in [14] and these authors in [6, 7] implies the
following assertion
Corollary 1 For g ∈ C∞(⺢n ), the following assertions
are equivalent:
(i) g belongs to the Fourier image of the space Ᏹ'(⺢n);
(ii) there exist constants C, N, and M such that
(iii) the extension of g(z) is an entire function, and there
exist constants C, N, and R such that, for all z ∈ ⺓n,
(4) Now, let us reconstruct all elements of Ᏹ'(K) in,
e.g., the case where K has gproperty Using Corollary 1
and Theorem 2, we shall see that F(Ᏹ'(K)) is the set of
all entire functions g(z) such that, for some constants
C, N, and M, we have
(5) and, for any δ > 0, there exists a constant Cδ< ∞ such
that
(6)
Now, suppose that g(z) is an entire function satisfy
ing (5) and (6) It follows from (5) that g ∈ S '(⺢n) and
there exists an f ∈ Ᏹ'(K) for which Ff = g Choose δ > 0
and ψ0 ∈ (Kδ) so that ψ0(x) = 1 in some neighbor
hood of K Then, for all ϕ ∈ C∞(⺢n), we have
⺪+q
B
j= 1
k
∪
䊊
vˆ ξ( ) C 1 ξ( + )N
eH Im( ξ), ξ ⺓n
∈
≤
Dα ( ) C 1 ηη ( + )N
Mα, ∀η ⺢n
, α ⺪+n
;
≤
G z ( ) C 1 z( + )N
e R Imz
≤
Dα ( ) C 1 ηη ( + )N
Mα, ∀η ⺢n
, α ⺪+n
,
≤
Dαg 0 ( ) Cδ ξα, ∀α
ξ K∈ δ
∈
≤
C0∞
here, the last integral is finite, because ψ0ϕ ∈ S(⺢n) Therefore,
for all ϕ ∈ C∞(⺢n ) In this way, knowing F(Ᏹ'(K)), we
can construct all elements of Ᏹ'(K).
Consider an entire function f(ξ) = , where
ξ ∈⺓n According to Theorem 2 (provided that K is a
symmetric convex compact set or a compact set with
gproperty), relations (4) and
(7)
imply f(x) ∈ F(Ᏹ'(K)) It is hard to obtain exact esti
mates of the form (*) for at all ξ ∈ ⺓n, because this sum is infinite; in our opinion, the result stated above is valuable in that, instead of (*), it uses
only coarse estimate (4), in which R can be chosen
arbitrarily large, after which the membership of f(x) in F(Ᏹ'(K)) can be established by verifying (7).
6 L pVERSIONS
OF THE PALEY–WIENER THEOREM The original Paley–Wiener theorem was proved for
L2functions Below, we give L pversions of this result (for 1 ≤ p ≤ ∞).
Theorem 8 Suppose that K is any compact set, f ∈
Ᏹ'(⺢n ), and ∈ L p(⺢n ) Then suppf ⊂ K if and only if, for any δ > 0, there exists a constant Cδ < ∞ such that, for all polynomials P(x),
(8)
Theorem 9 Suppose that K is a compact set with g property, f ∈ Ᏹ'(⺢n ), and ∈ L p(⺢n ) Then suppf ⊂ K if and only if, for any δ > 0, there exists a constant Cδ < ∞
such that, for all α ∈ ,
(9)
Theorem 10 Suppose that r > 0, f ∈ Ᏹ'(⺢n), ∈
L p(⺢n ), P(x) is a polynomial, and K := Q(P) r is a com pact set Then suppf ⊂ K if and only if, for any δ > 0, there exists a constant Cδ < ∞ such that, for all m ∈ ⺪+,
(10) Theorems 9 and 10 imply the following assertion
f,ϕ
〈 〉 = 〈f,ψ0ϕ〉 = 〈Ff F, –1(ψ0ϕ)〉
= g F〈 , –1(ψ0ϕ)〉 g x ( )F– 1(ψ0ϕ) x ( ) x, d
⺢∫n
=
f,ϕ
〈 〉 g x ( )F– 1(ψ0ϕ) x ( ) x d
⺢∫n
=
fαξα
∑
α!fα Cδ ξα, ∀α
ξ K∈ δ
,
∈
≤
fαξα
∑
f
ˆ
P D ( )fˆ p Cδ fˆ p P x( )
x∈K( ) δ
sup
≤
f
ˆ
⺪+n
Dαˆf p Cδ ˆf p x
x∈K( ) δ
sup
≤
f
ˆ
P m ( )fˆ D p Cδ ˆf p(r+δ)m
≤
Trang 4680 HA HUY BANG, VU NHAT HUY
Theorem 11 Suppose that r > 0, P1(x), P2(x), …,
P q (x) are polynomials with real coefficients; Q(P1)r, …,
Q(P q)r are compact sets; H is a compact set with gprop
erty; and K := H ∩ Q(P1)r ∩ … ∩ Q(P q)r Suppose also
that ∈ L p(⺢n ) and f ∈ Ᏹ'(⺢n ) Then suppf ⊂ K if and
only if, for any δ > 0, there exists a constant Cδ < ∞ such
that, for any (m1, …, m q) ∈ and α ∈ ,
Theorem 12 Suppose that K is a convex compact set,
f ∈ Ᏹ'(⺢n ), and ∈ L p(⺢n ) Then suppf ⊂ K if and only
if, for any δ > 0, there exists a constant Cδ < ∞ such that
(11)
for all P(x) of degree 1 and all m ∈ ⺪+
Theorems 9 and 12 imply the following assertion
Theorem 13 Suppose that f ∈ Ᏹ'(⺢n), ∈ L p(⺢n),
K1 is a convex compact set, and K2 is a compact set with
gproperty Let K := K1 ∩ K2 Then suppf ⊂ K if and only
if, for any δ > 0, there exists a constant Cδ < ∞ such that,
for all P(x) ∈ Φ,
Theorems 10 and 12 imply the following assertion
Theorem 14 Suppose that f ∈ Ᏹ'(⺢n), ∈ L p(⺢n),
r > 0, P1(x), P2(x), …, P q (x) are polynomials with real
coefficients; H is a convex compact set; and Q(P1)r, …,
Q(P q)r are compact sets Let K := H ∩
Q(P1)r ∩ … ∩ Q(P q)r Then suppf ⊂ K if and only if, for
any δ > 0, there exists a constant Cδ < ∞ such that, for all
P(x) ∈ Φ and (m1 , …, m q)∈ ,
ACKNOWLEDGMENTS This work was supported by the Vietnam State Foundation for the Development of Science and Technology (project no 101.01.50.09)
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f
ˆ
⺪+q
⺪+n
P1m1
x ( )…P q
m q x ( )xα
≤ Cδ ˆf p(r+δ)m1 +… m+ q
x
x∈K( ) δ
sup
f
ˆ
P m ( )fˆ D p Cδ ˆf p P x( )m
x∈K( ) δ
sup
≤
f
ˆ
P D ( )fˆ p Cδ ˆf p P x( )
x∈K( ) δ
≤
f
ˆ
⺪+q
PP1m1
…P q
m q
( ) D ( )fˆ p
≤ Cδ ˆf p(r+δ)m1+… m+ q
P x( )
x∈K( ) δ
sup