DSpace at VNU: The Paley–Wiener Theorem in the Language of Taylor Expansion Coefficients

ISSN 1064 5624, Doklady Mathematics, 2012, Vol. 86, No. 2, pp. 677–680. © Pleiades Publishing, Ltd., 2012.
Original Russian Text © Ha Huy Bang, Vu Nhat Huy, 2012, published in Doklady Akademii Nauk, 2012, Vol. 446, No. 5, pp. 497–500.

MATHEMATICS

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

Let Ᏹ(‫ޒ‬n) = C∞(‫ޒ‬n), and let Ᏹ'(‫ޒ‬n) be the dual
space. Then any element in Ᏹ'(‫ޒ‬n) is a compactly sup
ported distribution, and vice versa. Consider a com
pact set K in ‫ޒ‬n. Let us try to construct all elements of
the subspace Ᏹ'(K) of distributions supported on K.
This case differs from the case of L 'p (‫ޒ‬n) = Lq(‫ޒ‬n) ⎛ 1 ≤
⎝
1
1
p < ∞, + = 1 ⎞ , in which we can specify any ele
⎠
p
q
ment of the dual space as an element of Lq(‫ޒ‬n); it is
impossible to directly define all elements of Ᏹ'(‫ޒ‬n),
because, generally, these are generalized functions.
The Paley–Wiener theorem relates growth properties
of entire functions on ‫ރ‬n 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

a

Institute of Mathematics, 18 Hoang Quoc Viet Street, Cau
Giay, Hanoi, Vietnam
e mail:
b Hanoi State University, 334 Nguyen Trai Street,
Thanh Xuan, Hanoi, Vietnam
e mail:

conditions in the corresponding Paley–Wiener theo
rem have simpler form. Note that the original Paley–

Wiener theorem was proved for L2 functions [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'(‫ޒ‬n), and let K be any compact
set in ‫ޒ‬n. Then supp f ⊂ K if and only if, for any δ > 0,
there exists a constant Cδ < ∞ such that
P ( D )fˆ( 0 ) ≤ C sup P ( x )
(1)
δ

x ∈ K( δ )

for any polynomial P(x), where D = (D1, D2, …, Dn),
α
α
i∂
Dα = D 1 1 … D n n , Dj =
for j = 1, 2, …, n, K(δ) is the
∂x j
δ neighborhood in ‫ރ‬n of the set K, and ˆf = 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

n
of degree 2, p ∈ ‫ޚ‬+, and α ∈ ‫ ޚ‬+ , are considered.
2. SETS WITH g PROPERTY
Below, we recall some notions and results from
n
[6, 7]. Suppose that 0 ≤ λα ≤ ∞, α ∈ ‫ ޚ‬+ , and
G { λα } =

∩ {ξ ∈ ‫ ޒ‬:
n

α

ξ ≤ λ α }.

n

α ∈ ‫ޚ‬+

Then G{λα} is referred to as the set generated by the
sequence of numbers {λα}.
677


678

HA HUY BANG, VU NHAT HUY

Let K ⊂ ‫ޒ‬n. We set

a constant Cδ < ∞ such that, for all (m1, m2, …, mq) ∈

⎧
α ⎫
g ( K ) = G ⎨ sup ξ ⎬.
⎩ξ ∈ K
⎭
We have K ⊂ g(K), and g(K) is called the g hull of the
set K. We say that K has g property if K = g(K).
The following assertions hold.
(i) Any set generated by a sequence of numbers has
g property, and vice versa.
(ii) Let I be a set of indices, let and Kj = g(Kj) for j ∈ I.
Then
K j has g property.

∩
j∈I

q

n

‫ ޚ‬+ , and α ∈ ‫ ޚ‬+ ,
m
m
α
( P 1 1 ( x )…P q q ( x )x ) ( D )fˆ( 0 )

≤ Cδ ( r + δ )

m1 + … + mq

α

sup x .

x ∈ K( δ )

Let B(x, ⑀) denote an open ball, and let B[x, ⑀]
denote a closed ball.
Remark 4. Suppose that b0, b1, …, bk ∈ ‫ޒ‬n and
r0, r1, …, rk > 0 are such that B(b0, r0) ∩ B(bj, rj) ≠ ᭺ for
j = 1, 2, …, k. Choose a number R > 0 so that B(b0, r0) ⊂
k

∪ B (b , r )

(iii) The set G{λα} may be nonconvex, and any sym
metric convex compact set has g property.
It turns out that if K has g property, then the formu
lation of the Paley–Wiener theorem becomes very
simple.
Theorem 2. Suppose that f ∈ Ᏹ'(‫ޒ‬n) and a compact
set K ⊂ ‫ޒ‬n has g property. Then supp f ⊂ K if and only
if, for any δ > 0, there exists a constant Cδ < ∞ such that
n
α
α
D ˆf ( 0 ) ≤ C sup ξ , ∀α ∈ ‫ ޚ‬,

(2)

B(bj, R) for j = 1, 2, …, k. Then K1 = B[b0, r0]\

where Kδ is the δ neighborhood of K.
Remark 3. Theorem 2 remains valid under the
replacement of (2) by the condition
n
α
n
α
D ˆf ( 0 ) ≤ C α sup ξ , ∀α ∈ ‫ ޚ‬.

4. CONVEX COMPACT SETS
Let ‫ސ‬1 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) ∈ ‫ސ‬1,

3. SETS GENERATED BY POLYNOMIALS
Let P(x) be a polynomial with real coefficients.
We set

m ∈ ‫ޚ‬+, and α ∈ ‫ ޚ‬+ .
Theorem 5. Suppose that f ∈ Ᏹ'(‫ޒ‬n) and K is a con
vex compact set in ‫ޒ‬n. Then suppf ⊂ K if and only if, for
any δ > 0, there exists a constant Cδ < ∞ such that
P ( D )fˆ( 0 ) ≤ C sup P ( x )
(3)

δ

m

m

+

ξ ∈ Kδ

α

P α ( x ) := P ( x )x ,

j

is the intersection of B[b0, r0] with the tori B[bj, R]\B(bj, rj),
j = 1, 2, …, k, generated by polynomials. Moreover, if
k

H has g property, then K2 = H \

∪ B (b , r ) is the
j

j

j=1

intersection of H with the tori B[bj, R]\B(bj, rj) for j =
1, 2, …, k, and Theorem 4 can be applied to K1 and K2.

+

ξ ∈ Kδ

δ

j

j=1

n

m ∈ ‫ޚ‬+ ,

δ

n

α ∈ ‫ޚ‬+ ,

x ∈ K( δ )

Q ( P ) r := { x ∈ ‫ ޒ‬: P ( x ) ≤ r }, r > 0.
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'(‫ޒ‬n), 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
m
m
α
P ( D )fˆ( 0 ) ≤ C ( r + δ ) sup x ,

for all P(x) ∈ Φ.
Remark 5. Theorem 5 remains valid if P has the
n
form Qp(x)xα, where p ∈ ‫ޚ‬+ and α ∈ ‫ ޚ‬+ , 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'(‫ޒ‬n), K1 is a convex
compact set, K2 is a compact set with g property, and
K := K1 ∩ K2. Then suppf ⊂ K if and only if, for any δ > 0,
there exists a constant Cδ < ∞ such that
P ( D )fˆ( 0 ) ≤ C sup P ( x )

∀m ∈ ‫ ޚ‬+ , α ∈
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'(‫ޒ‬n), r > 0, P1(x),
P2(x), …, Pq(x) are polynomials with real coefficients;
Q(P1)r, …, Q(Pq)r are compact sets; and H is a compact
set with g property. Let K := H ∩ Q(P1)r ∩ … Q(Pq)r.
Then suppf ⊂ K if and only if, for any δ > 0, there exists

for all P(x) ∈ Φ.

Theorems 5 and 3 imply the following theorem.
Theorem 7. Suppose that f ∈ Ᏹ'(‫ޒ‬n), r > 0,
P1(x), P2(x), …, Pq(x) are polynomials with real coeffi
cients; H is a convex compact set; and Q(P1)r, …, Q(Pq)r
are compact sets. Let K := H ∩ Q(P1)r ∩ … ∩ Q(Pq)r.
Then suppf ⊂ K if and only if, for any δ > 0, there exists
a constant Cδ < ∞ such that
m + … + mq
m
m
( PP 1 …P q ) ( D )fˆ( 0 ) ≤ C ( r + δ ) 1
sup P ( x )

n

α

δ

x ∈ K( δ )
n
‫ޚ‬+ .

δ

1

q

x ∈ K( δ )

δ

DOKLADY MATHEMATICS

x ∈ K( δ )

Vol. 86

No. 2

2012


THE PALEY–WIENER THEOREM
–1

q

for all P(x) ∈ Φ and (m1, m2, …, mq) ∈ ‫ ޚ‬+ .
Remark 6. If K is the intersection of a convex com
pact set with compact sets generated by polynomials or
with compact sets having g property, then the Paley–
Wiener theorem can be proved for K. For example, let
k

K = H\

∪

B (bj, rj), where H is a convex compact set;

〈 f, ϕ〉 = 〈 f, ψ 0 ϕ〉 = 〈 Ff, F ( ψ 0 ϕ )〉

∫ g ( x )F

–1

= 〈 g, F ( ψ 0 ϕ )〉 =

‫ޒ‬

–1

( ψ 0 ϕ ) ( x ) dx,

n

here, the last integral is finite, because ψ0ϕ ∈ S(‫ޒ‬n).
Therefore,

j=1

b1, b2, …., bk ∈ ‫ޒ‬n; and r1, r2, …, rk > 0 are such that
H ∩ B(bj, rj) ≠ ᭺ 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.

679

∫ g ( x )F

〈 f, ϕ〉 =

‫ޒ‬

–1

( ψ 0 ϕ ) ( x ) dx

n

for all ϕ ∈ C (‫) ޒ‬. In this way, knowing F(Ᏹ'(K)), we
can construct all elements of Ᏹ'(K).
∞

n

Consider an entire function f(ξ) =
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
ˆ ( ξ ) ≤ C ( 1 + ξ ) N e H ( Imξ ) , ξ ∈ ‫ ރ‬n .
v
(*)
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'(‫ޒ‬n) cannot be dis
pensed with in our theorems. Indeed, the function ez
satisfies (2) with K = [–1, 1], but it grows very rapidly
on ‫ޒ‬+. The Paley–Wiener theorem proved by
Schwartz in [14] and these authors in [6, 7] implies the
following assertion.
Corollary 1. For g ∈ C∞(‫ޒ‬n), the following assertions
are equivalent:
(i) g belongs to the Fourier image of the space Ᏹ'(‫ޒ‬n);
(ii) there exist constants C, N, and M such that
α

α

N

n

n

D g ( η ) ≤ C ( 1 + η ) M , ∀η ∈ ‫ ޒ‬, α ∈ ‫ ޚ‬+ ;
(iii) the extension of g(z) is an entire function, and there
exist constants C, N, and R such that, for all z ∈ ‫ރ‬n,
N R Imz

G(z) ≤ C(1 + z ) e
.
(4)
Now, let us reconstruct all elements of Ᏹ'(K) in,
e.g., the case where K has g property. 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
α

N

α

n

n

D g ( η ) ≤ C ( 1 + η ) M , ∀η ∈ ‫ ޒ‬, α ∈ ‫ ޚ‬+ , (5)
and, for any δ > 0, there exists a constant Cδ < ∞ such
that
α

α

D g ( 0 ) ≤ C δ sup ξ ,
ξ ∈ Kδ

n

∀α ∈ ‫ ޚ‬+ .

(6)

Now, suppose that g(z) is an entire function satisfy

ing (5) and (6). It follows from (5) that g ∈ S '(‫ޒ‬n) and
there exists an f ∈ Ᏹ'(K) for which Ff = g. Choose δ > 0
∞
and ψ0 ∈ C 0 (Kδ) so that ψ0(x) = 1 in some neighbor
hood of K. Then, for all ϕ ∈ C ∞(‫ޒ‬n), we have
DOKLADY MATHEMATICS

Vol. 86

No. 2

2012

∑f

αξ

α

, where

ξ ∈ ‫ ރ‬According to Theorem 2 (provided that K is a
symmetric convex compact set or a compact set with
g property), relations (4) and
n.

α

n

α!f α ≤ C δ sup ξ ,

∀α ∈ ‫ ޚ‬+ ,

ξ ∈ Kδ

(7)

imply f(x) ∈ F(Ᏹ'(K)). It is hard to obtain exact esti
α
mates of the form (*) for
f α ξ at all ξ ∈ ‫ރ‬n,
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. Lp VERSIONS
OF THE PALEY–WIENER THEOREM
The original Paley–Wiener theorem was proved for
L2 functions. Below, we give Lp versions of this result
(for 1 ≤ p ≤ ∞).
Theorem 8. Suppose that K is any compact set, f ∈
Ᏹ'(‫ޒ‬n), and ˆf ∈ Lp(‫ޒ‬n). Then suppf ⊂ K if and only if,
for any δ > 0, there exists a constant Cδ < ∞ such that, for
all polynomials P(x),
P ( D )fˆ ≤ C fˆ sup P ( x ) .

(8)
δ

p

p

x ∈ K( δ )

Theorem 9. Suppose that K is a compact set with g
property, f ∈ Ᏹ'(‫ޒ‬n), and ˆf ∈ Lp(‫ޒ‬n). Then suppf ⊂ K if
and only if, for any δ > 0, there exists a constant Cδ < ∞
n

such that, for all α ∈ ‫ ޚ‬+ ,
α
D ˆf p ≤ C δ ˆf

p

α

sup x .

(9)

x ∈ K( δ )

Theorem 10. Suppose that r > 0, f ∈ Ᏹ'(‫ޒ‬n), ˆf ∈
Lp(‫ޒ‬n), 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 ∈ ‫ޚ‬+,
m
m
P ( D )fˆ ≤ C ˆf ( r + δ ) .
(10)
p

δ

p

Theorems 9 and 10 imply the following assertion.


680

HA HUY BANG, VU NHAT HUY

Theorem 11. Suppose that r > 0, P1(x), P2(x), …,
Pq(x) are polynomials with real coefficients; Q(P1)r, …,
Q(Pq)r are compact sets; H is a compact set with g prop
erty; and K := H ∩ Q(P1)r ∩ … ∩ Q(Pq)r. Suppose also
that ˆf ∈ L (‫ޒ‬n) and f ∈ Ᏹ'(‫ޒ‬n). Then suppf ⊂ K if and

m
m
( PP 1 1 …P q q ) ( D )fˆ

p

m + … + mq
≤ C δ ˆf p ( r + δ ) 1
sup P ( x ) .
x ∈ K( δ )

p

only if, for any δ > 0, there exists a constant Cδ < ∞ such
that, for any (m1, …, mq) ∈

q
‫ޚ‬+

and α ∈

n
‫ޚ‬+ ,

m
m
α
( P 1 1 ( x )…P q q ( x )x ) ( D )fˆ p
m + … + mq
α
≤ C δ ˆf p ( r + δ ) 1
sup x .

REFERENCES

x ∈ K( δ )

Theorem 12. Suppose that K is a convex compact set,
f ∈ Ᏹ'(‫ޒ‬n), and ˆf ∈ Lp(‫ޒ‬n). Then suppf ⊂ K if and only
if, for any δ > 0, there exists a constant Cδ < ∞ such that
m
m
P ( D )fˆ ≤ C ˆf sup P ( x )
(11)
δ

p

p

x ∈ K( δ )

for all P(x) of degree 1 and all m ∈ ‫ޚ‬+.
Theorems 9 and 12 imply the following assertion.
Theorem 13. Suppose that f ∈ Ᏹ'(‫ޒ‬n), ˆf ∈ L (‫ޒ‬n),
p

K1 is a convex compact set, and K2 is a compact set with
g property. 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,
P ( D )fˆ ≤ C ˆf sup P ( x ) .
p

δ

p

x ∈ K(δ )

Theorems 10 and 12 imply the following assertion.
Theorem 14. Suppose that f ∈ Ᏹ'(‫ޒ‬n), ˆf ∈ L (‫ޒ‬n),
p

r > 0, P1(x), P2(x), …, Pq(x) are polynomials with real
coefficients; H is a convex compact set; and Q(P1)r, …,
Q(Pq)r are compact sets. Let K := H ∩
Q(P1)r ∩ … ∩ Q(Pq)r. Then suppf ⊂ K if and only if, for
any δ > 0, there exists a constant Cδ < ∞ such that, for all
q

P(x) ∈ Φ and (m1, …, mq) ∈ ‫ ޚ‬+ ,

ACKNOWLEDGMENTS
This work was supported by the Vietnam State
Foundation for the Development of Science and
Technology (project no. 101.01.50.09).

1. N. B. Andersen and M. Jeu, Trans. Am. Math. Soc. 362
(7), 3613–3640 (2010).
2. J. Arthur, Acta Math. 150 (1/2), 1–89 (1983).
3. J. Arthur, Am. J. Math. 104 (6), 1243–1288 (1982).
4. L. D. Abreu and F. Bouzeffour, Proc. Am. Math. Soc.
138 (8), 2853–2862 (2010).
5. E. P. Ban and H. Schlichtkrull, Ann. Math. 164 (3),

879–909 (2006).
6. H. H. Bang, Tr. Mat. Inst. im. V.A. Steklova, Ross.
Akad. Nauk 214, 298–319 (1996).
7. H. H. Bang, Dokl. Math. 55 (3), 353–355 (1997).
8. H. H. Bang, Trans. Am. Math. Soc. 347 (3), 1067–
1080 (1995).
9. S. Helgason, Math. Ann. 165 (4), 297–308 (1966).
10. L. Hörmander, The Analysis of Linear Partial Differen
tial Operators (Springer Verlag, Berlin, 1983).
11. M. Leu, Trans. Am. Math. Soc. 358 (10), 4225–4250
(2006).
12. G. Olafsson and H. Schlichtkrull, Adv. Math. 218 (1),
202–215 (2008).
13. R. Paley and N. Wiener, Fourier Transform in the Com
plex Domain (Am. Math. Soc. Coll. Publ., New York,
1934).
14. L. Schwartz, Commun. Sém. Math. Univ. Lund., 196–
206 (1952).
15. E. M. Stein, Ann. Math. 65 (2), 582–592 (1957).

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Vol. 86

No. 2

2012