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 p < ∞, + = ⎞ , 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: hhbang@math.ac.vn b Hanoi State University, 334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam e mail: nhat_huy85@yahoo.com 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] THE CASE OF ARBITRARY COMPACT SETS Theorem 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( ) ≤ 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 … 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 Theorem remains valid if only polyno mials with real coefficients are considered Remark Theorem 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 SETS WITH g PROPERTY Below, we recall some notions and results from n [6, 7] Suppose that ≤ 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 ( x )…P q q ( x )x ) ( D )fˆ( ) ≤ 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 Suppose that b0, b1, …, bk ∈ ‫ޒ‬n and r0, r1, …, rk > are such that B(b0, r0) ∩ B(bj, rj) ≠ ᭺ for j = 1, 2, …, k Choose a number R > 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 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 ( ) ≤ 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 Theorem remains valid under the replacement of (2) by the condition n α n α D ˆf ( ) ≤ C α sup ξ , ∀α ∈ ‫ ޚ‬ 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, SETS GENERATED BY POLYNOMIALS Let P(x) be a polynomial with real coefficients We set m ∈ ‫ޚ‬+, and α ∈ ‫ ޚ‬+ Theorem Suppose that f ∈ Ᏹ'(‫ޒ‬n) and K is a 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( ) ≤ 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 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 > 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 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( ) ≤ C ( r + δ ) sup x , for all P(x) ∈ Φ Remark Theorem 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 with complex coefficients Theorems and imply the following assertion Theorem 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( ) ≤ 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 and imply the following assertion Theorem 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 and imply the following theorem Theorem 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 …P q ) ( D )fˆ( ) ≤ C ( r + δ ) sup P ( x ) n α δ x ∈ K( δ ) n ‫ޚ‬+ δ q x ∈ K( δ ) δ DOKLADY MATHEMATICS x ∈ K( δ ) Vol 86 No 2012 THE PALEY–WIENER THEOREM –1 q for all P(x) ∈ Φ and (m1, m2, …, mq) ∈ ‫ ޚ‬+ Remark 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 ϕ〉 = 〈 Ff, F ( ψ ϕ )〉 ∫ g ( x )F –1 = 〈 g, F ( ψ ϕ )〉 = ‫ޒ‬ –1 ( ψ 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 > 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 applies to such K 679 ∫ g ( x )F 〈 f, ϕ〉 = ‫ޒ‬ –1 ( ψ 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) = 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 ( + 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 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 ( + 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 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 ( + η ) M , ∀η ∈ ‫ ޒ‬, α ∈ ‫ ޚ‬+ , (5) and, for any δ > 0, there exists a constant Cδ < ∞ such that α α D g ( ) ≤ 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 > ∞ and ψ0 ∈ C (Kδ) so that ψ0(x) = in some neighbor hood of K Then, for all ϕ ∈ C ∞(‫ޒ‬n), we have DOKLADY MATHEMATICS Vol 86 No 2012 ∑f αξ α , where ξ ∈ ‫ ރ‬According to Theorem (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) ∑ 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 ≤ p ≤ ∞) Theorem 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 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 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 …P q q ) ( D )fˆ p m + … + mq ≤ C δ ˆf p ( r + δ ) 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 ( x )…P q q ( x )x ) ( D )fˆ p m + … + mq α ≤ C δ ˆf p ( r + δ ) 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 and all m ∈ ‫ޚ‬+ Theorems 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) N B Andersen and M Jeu, Trans Am Math Soc 362 (7), 3613–3640 (2010) J Arthur, Acta Math 150 (1/2), 1–89 (1983) J Arthur, Am J Math 104 (6), 1243–1288 (1982) L D Abreu and F Bouzeffour, Proc Am Math Soc 138 (8), 2853–2862 (2010) E P Ban and H Schlichtkrull, Ann Math 164 (3), 879–909 (2006) H H Bang, Tr Mat Inst im V.A Steklova, Ross Akad Nauk 214, 298–319 (1996) H H Bang, Dokl Math 55 (3), 353–355 (1997) H H Bang, Trans Am Math Soc 347 (3), 1067– 1080 (1995) 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) DOKLADY MATHEMATICS Vol 86 No 2012 ... 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 and imply the following assertion Theorem. .. 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. .. 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