On Zeros of Fourier Transforms

ON ZER OS OF F OURIER TRANSF ORMS RUI MING ZHANG Abstract. In th is work w e v erify the sufficiency of a Jensen’s necessary and sufficien t condition for a class of genu s 0 or 1 entire functions to hav e only real zeros. They are F ourier transforms of ev en, positive, indefinite ly differen- tiable, and ver y f ast decreasing functions. W e also apply our result to sev eral imp ortan t special functions in m athematics, such as mo dified Bessel f unction K iz ( a ) , a > 0 as a function of v ariable z , Riemann Xi function Ξ( z ) , and c haracte r Xi function Ξ( z ; χ ) when χ is a real primitive non-principal c harac - ter sat isfying ϕ ( u ; χ ) ≥ 0 on the real line, w e pro v e these ent ire functions hav e only real zeros. 1. Introduction One of fundamental questio ns in the theo ry of sp ecial functions is whether an ent ire function has o nly real zeros. Some interesting examples are even entire func- tions o f genus 0 or 1 , for example, Jackson’s q -Bes sel function J (2) ( z ; q ) and Bessel function of first kind J ν ( z ) , they hav e only real zeros, and they a re es s ent ially even ent ire functions of genus 0 and 1 res p ectively , [1, 9]. The genu s of an entire function f ( z ) is a n concept mostly rela ted to its Hadamard infinite pro duct expansio n and its or der o f growth. F o r each r > 0 , let [1 0] (1.1) k f k ∞ ,r = sup | z |≤ r | f ( z ) | = max | z | = r | f ( z ) | . Then the order ρ o f f ( z ) can b e defined by (1.2) ρ = lim s up r →∞ log  log k f k ∞ ,r  log( r ) . Given an arbitrary point z 0 ∈ C , the order ρ of f ( z ) can be co mputed from the v alues o f its deriv atives at z 0 , (1.3) ρ = 1 − lim sup n →∞ log   f ( n ) ( z 0 )   n log n ! − 1 . Clearly , formula (1.3) shows that f ′ ( z ) has the same order a s f ( z ) . If the order ρ of f ( z ) is finite, then it ha s the following Hadamard factorization 2000 Mathematics Subje ct Classific ation. 37A45; 26B25; 42A38; 30D10; 33C10; 11M26. Key wor ds and phr ases. F ourier transforms; Bessel functions; Riemann zeta f unction;Riemann h ypothesis; Dirichlet L - s eries; gener alized Riemann h ypothesis. This work is partially supported b y the National Natural Science F oundation of China, gran t No. 1137129 4. The author specially thanks his colleague Dr. Bing He for helping him to c hec k some of the computations. 1 ON ZEROS OF FOURIER TRANSFORMS 2 (1.4) f ( z ) = z m e p ( z ) ∞ Y n =1  1 − z z n  exp z z n + . . . + 1 k  z z n  k ! , where m is an nonnega tive integer, { z n } ∞ n =1 is the set of all nonzero ro ots of f ( z ) , p ( z ) a po lynomial of degree j , and k is the smallest no nneg ative integer such that (1.5) ∞ X n =1 1 | z n | k +1 < ∞ . The g enus of f ( z ) is defined as g = max { j, k } . If the o r der o f f ( z ) is no t a n integer, then g = ⌊ ρ ⌋ is the int eger pa rt of ρ , otherwise, g may b e ρ − 1 o r ρ . In 19 13 Jensen pr oved a s et of necessa ry and sufficient conditions for a cla s s of genus 0 o r 1 entire functions to hav e only r eal zer os, [7, 8, 11]. Gasp er a pplied Jensen’s c onditions to prov e many impor tant sp ecial functions that hav e only real zeros b y directly verifying p ositivities of cer tain sums and in tegrals, [7, 8]. In this work w e verify a J ensen inequality for a cla ss o f functions under muc h stro ng er conditions. More specifica lly , we are going to show the following: Theorem 1. L et ϕ ( u ) b e nonne gative, even, and indefinitely differ entiable function that is not identic al ly zer o. If additional ly for e ach n ∈ Z + ther e exists some d, δ > 0 such t hat (1.6) ϕ ( n ) ( u ) = O  exp  − de δ | u |  as u → ±∞ . Then the F ourier tr ansform Φ( z ) of ϕ ( u ) has only r e al zer os. W e just mention three applications of our results to s pe c ial functions here. F or ℜ ( z ) > 0 , the Bes sel functions K ν ( z ) has in teg ral repr esentation [1, 6, 8, 1 3] K ν ( z ) = 1 2 ˆ ∞ −∞ exp ( − z cosh u − ν u ) du. (1.7) Then for a > 0 (1.8) K iz ( a ) = 1 2 ˆ ∞ −∞ e − a c osh u e izu du defines an entire function o f v ariable z . Clearly , ϕ ( u ) = e − a c osh u satisfies all the conditions o f Theor e m 1. Thus the entire function K iz ( a ) ha s o nly rea l zer o s. The Riemann Xi function is defined by [2, 3 , 5, 7, 8, 1 2] (1.9) Ξ ( s ) = − 1 + 4 s 2 8 π 1+2 is 4 Γ  1 + 2 i s 4  ζ  1 + 2 i s 2  , where Γ( z ) and ζ ( z ) a re analytic contin ua tions of Euler gamma and Riemann zeta functions resp ectively . Then the entire function Ξ ( s ) satisfies [2, 3, 5, 7, 8, 1 1, 1 2] (1.10) Ξ ( s ) = ˆ ∞ −∞ ϕ ( u ) e isu du, s ∈ C , where ϕ ( u ) = 2 π ∞ X n =1 n 2 π n 4 e 9 u/ 2 − 3 n 2 e 5 u/ 2 o exp  − n 2 π e 2 u  , u ∈ R . (1.11) ON ZEROS OF FOURIER TRANSFORMS 3 It is known that ϕ ( u ) is even, po sitive, indefinitely differentiable. F ur thermore, for any ǫ > 0 and in teg ers n ≥ 0 , it satisfies ϕ ( n ) ( u ) = O  exp n − ( π − ǫ ) e 2 | u | o (1.12) as u → ±∞ . Hence, it also satisfies the requirements of our result Theorem 1. Thu s Ξ ( s ) ha s o nly rea l zer o s, which means the Riema nn hypothesis is v alid. Given a positive integer q ≥ 2 , let χ ( n ) b e a primitiv e real c ha racter with resp ect to mo dulus q with parity a , [4, 12] (1.13) a = ( 0 , χ ( − 1) = 1 1 , χ ( − 1) = − 1 . Define the character Xi function Ξ ( s ; χ ) for χ of parity a b y [4] (1.14) Ξ ( s ; χ ) = Γ  2 a +1+2 is 4  L  1 2 + is, χ   π q  (2 a +1+2 is ) / 4 , where Γ( s ) and L ( s, χ ) are the a nalytic contin uations of Euler gamma function and Dirichlet L - series for χ resp ectively . Then the entire function Ξ ( s ; χ ) satisfies (1.15) Ξ ( s ; χ ) = ˆ ∞ −∞ e isu ϕ ( u ; χ ) du, where (1.16) ϕ ( u ; χ ) = 2 e w a u ∞ X n =1 n a χ ( n ) e − n 2 π e 2 u /q , w a = 1 + 2 a 2 , and a is the pa rity of χ . By the transformation for mu las [4] (1.17) ∞ X n = −∞ n a χ ( n ) e − n 2 π u/m = u − w a ∞ X n = −∞ n a χ ( n ) e − n 2 π / ( mu ) , u > 0 we have (1.18) ϕ ( − u ; χ ) = ϕ ( u ; χ ) , u ∈ R . It is clear that for each integer n ≥ 0 we have (1.19) ϕ ( n ) ( u ; χ ) = O  exp  − de 2 | u |  as u → ±∞ for any 0 < d < π q . Hence for those χ such that ϕ ( u ; χ ) > 0 , u ∈ R , ϕ ( u ; χ ) satisfies all the conditions o f Theorem 1. Th us the enti re function Ξ ( s ; χ ) has only r eal zeros, hence the g eneralized Riemann hypo thesis holds for this kind of Dirichlet L - series. 2. Main Resul ts The following theorem only states a s ubs e t of J ensen’s results related to the current work [7, 11] : Theorem 2. L et ϕ ( u ) b e an nonne gative even function such that it is not identic al ly zer o on the r e al line, if it is indefinitely differ entiable and for e ach n ∈ Z + , (2.1) lim u →∞ log   ϕ ( n ) ( u )   u = −∞ . ON ZEROS OF FOURIER TRANSFORMS 4 A dditional ly, the even entir e function (2.2) Φ( z ) = ˆ ∞ −∞ ϕ ( u ) e iuz du = 2 ˆ ∞ 0 ϕ ( u ) co s( z u ) du is of genu s at most 1 . Then the ine qu ality (2.3) ∂ 2 ∂ y 2 | Φ ( x + iy ) | 2 ≥ 0 , x, y ∈ R , is ne c essary and sufficient for Φ( z ) t o have only r e al zer os. Lemma 3. L et f ( z ) , h ( z ) b e two entir e fun ctions of genus at most 1 , then so is f ( z ) h ( z ) . Pr o of. Let ρ f and ρ h be the or ders of f and h r esp ectively . Since they are of g enus at most 1 , 0 ≤ ⌊ ρ f ⌋ , ⌊ ρ h ⌋ ≤ 1 , then we m ust hav e 0 ≤ ρ f , ρ h < 2 . By (1.2) we hav e (2.4) log  log k f k ∞ ,r  log( r ) ≤ a, log  log k h k ∞ ,r  log( r ) ≤ b, as r → ∞ where 0 < a , b < 2 . Thus, (2.5) log k f k ∞ ,r ≤ r a , log k h k ∞ ,r ≤ r b as r → ∞ . F rom (2.6) log  k f h k ∞ ,r  ≤ log  k f k ∞ ,r · k h k ∞ ,r  ≤ log  k f k ∞ ,r  + log  k h k ∞ ,r  we g et (2.7) log  k f h k ∞ ,r  ≤ r a + r b ≤ 2 r max { a,b } for r → ∞ . Then, (2.8) log  log  k f h k ∞ ,r  ≤ log 2 + max { a, b } log r , and (2.9) lim s up r →∞ log  log  k f h k ∞ ,r  log r ≤ max { a, b } < 2 . Hence the order of f h is strictly less 2 , thus the genus of f h is les s than ⌊ max { a, b }⌋ ≤ 1 .  Lemma 4. L et ϕ ( u ) b e an even nonne gative me asur able function that is not iden- tic al ly zer o. If ther e exists some d, δ > 0 such that ϕ ( u ) = O  exp  − de δ | u |  as u → ± ∞ . Then the ent ir e function define d in (2.2) is of genus at most 1 Pr o of. By the a ssumptions we hav e (2.10) b 2 n = ˆ ∞ −∞ ϕ ( u ) u 2 n du > 0 , n ∈ Z + and (2.11) Φ( z ) = 2 ˆ ∞ 0 ϕ ( u ) co s( z u ) du = 2 ˆ ∞ 0 ϕ ( u ) ∞ X n =0  − z 2 u 2  n (2 n )! ! du = ∞ X n =0  − z 2  n (2 n )! b 2 n . ON ZEROS OF FOURIER TRANSFORMS 5 Then, (2.12) sup | z |≤ r | Φ( z ) | = sup | z |≤ r      ∞ X n =0  − z 2  n (2 n )! b 2 n      ≤ ∞ X n =0 r 2 n (2 n )! b 2 n = Φ( ir ) . By (2.1) we hav e (2.13) Φ( ir ) = 2 ˆ ∞ 0 ϕ ( u ) co sh( ru ) du = ˆ ∞ 0 O  exp  − de δu + r u  du = O  ˆ ∞ 0 exp  − de δu + r u  du  = O  1 d r δ ˆ ∞ d exp ( − y ) y r δ − 1 du  = O  1 d r δ ˆ ∞ 0 exp ( − y ) y r δ − 1 du  = O Γ  r δ  d r δ ! . By Stirling’s formula we hav e [1 ] (2.14) log (Φ( ir )) = − r δ log d + log Γ  r δ  + O (1) = O ( r log r ) as r → + ∞ . Hence, (2.15) ρ = lim sup r →∞ log (log Φ( ir )) log( r ) ≤ 1 , which shows that the order ρ o f Φ( z ) is no hig her than 1 , thus Φ( z ) is o f genus 0 or 1 .  Let (2.16) t ( u ) = (  1 − u 2  , − 1 < u < 1 , 0 , | u | ≥ 1 . Clearly , (2.17) d 2 t ( u ) du 2 = ( − 2 , − 1 < u < 1 0 , | u | > 1 and [1, 6, 7, 9, 13] (2.18) √ 8 π z − 3 2 J 3 2 ( z ) = ˆ 1 − 1 e izu  1 − u 2  du. Lemma 5. L et ϕ ( u ) b e nonne gative, even, and indefinitely differ entiable function that is not identic al ly zer o. A dditionally, we assume that for e ach n ∈ Z + ther e exists some d > 0 such that (2.19) ϕ ( n ) ( u ) = O  exp  − de δ | u |  as u → ±∞ . L et us define (2.20) ψ ( u ) = ˆ R ϕ ( u − y ) t ( y ) dy , u ∈ R . Then, ψ ( u ) is nonne gative, even, indefinitely differ entiable function that is not identic al ly zer o and (2.21) ψ (2) ( u ) ≤ 0 , u ∈ R . ON ZEROS OF FOURIER TRANSFORMS 6 F urthermor e, for e ach n ∈ Z + we have (2.22) ψ ( n ) ( u ) = O  exp  − d δ e δ | u |  as u → ±∞ , wher e d δ = d e δ . Pr o of. Since fo r u ∈ R and | y | ≤ 1 we have (2.23) | u − y | ≥ | u | − 1 , e δ | u − y | ≥ e − δ e δ | u | , exp  − de δ | u − y |  ≤ exp  − d e δ e δ | u |  . Then by the asymptotic behavior o f ϕ ( n ) ( u ) , we have (2.24) ˆ R ϕ ( n ) ( u − y ) t ( y ) dy = O  exp  − d e δ e δ | u |  ˆ 1 − 1 t ( y ) dy  = O  exp  − d δ e δ | u |  , where d δ = d e δ . Thus the in tegral ´ R ϕ ( n ) ( u − y ) t ( y ) dy conv erg e s absolutely and uniformly for each n ∈ Z + , hence (2.25) ψ ( n ) ( u ) = ˆ R ϕ ( n ) ( u − y ) t ( y ) dy = O  exp  − d δ e δ | u |  as u → ±∞ . C le a rly , ψ ( u ) is nonnegative and not identically zero if ϕ ( u ) is such, and for each u ∈ R we have (2.26) ψ ( u ) = ˆ R ϕ ( u + y ) t ( − y ) dy = ˆ R ϕ ( − u − y ) t ( y ) dy = ψ ( − u ) . Since (2.27) ψ ( u ) = ˆ R ϕ ( u − y ) t ( y ) dy = ˆ R ϕ ( y ) t ( u − y ) dy , then, (2.28) ψ (2) ( u ) = ˆ R ϕ ( y ) t (2) ( x − y ) dy = − 2 ˆ | x − y |≤ 1 ϕ ( y ) dy ≤ 0 .  Lemma 6. L et (2.29) F ( x ) = ˆ ∞ 0 G ( y ) cos xy dy , wher e G ( y ) ∈ C 2 ([0 , ∞ ) and (2.30) ˆ ∞ 0 {| G ( y ) | + | G ′ ( y ) | + | G ′′ ( y ) |} dy < ∞ . If G ′ (0) = 0 and G ′′ ( y ) ≤ 0 for y ≥ 0 almost everywher e, then F ( x ) ≥ 0 for al l x ≥ 0 . In p articular, if G ( y ) is even and G ′′ ( y ) ≤ 0 for y ≥ 0 almost everywher e, then F ( x ) ≥ 0 for al l x ≥ 0 . Pr o of. Observe that for G ′′ ( y ) ≤ 0 and G ′ (0) = 0 we hav e (2.31) x 2 F ( x ) = x ˆ ∞ 0 G ( y ) d sin xy = − x ˆ ∞ 0 G ′ ( y ) sin xy dy = ˆ ∞ 0 G ′ ( y ) d (1 + co s xy ) = − 2 ˆ ∞ 0 G ′′ ( y ) cos 2  xy 2  dy ≥ 0 . In the case that G ( y ) is even, then G ′ ( y ) is o dd and G ′ (0) = 0 .  ON ZEROS OF FOURIER TRANSFORMS 7 Now we prove our main result Theo rem 1: Pr o of. It is known that the even en tire function √ 8 π z − 3 2 J 3 2 ( z ) is of gen us 1 a nd it has o nly re a l zeros, [1, 7, 8, 9, 13]. Let (2.32) Ψ( z ) = √ 8 π z − 3 2 J 3 2 ( z )Φ( z ) = ˆ R ψ ( u ) e izu du, b y (2 .18) we hav e (2.33) Ψ( z ) = ˆ R ψ ( u ) e izu du, where ψ ( u ) is defined by (2.20). By Lemmas 3, 4 and 5 w e know that in addition to Ψ( z ) and ψ ( u ) satisfy a ll the requirements of Theorem 2 except (2.3). F urthermore, we a lso hav e ψ (2) ( u ) ≤ 0 , u ∈ R . F rom (2.34) Ψ( z ) = ˆ ∞ −∞ ψ ( u ) e − i z u du = ˆ ∞ −∞ ψ ( u ) e izu du = Ψ( z ) , we g et (2.35) 2 | Ψ( z ) | 2 = 2Ψ( z )Ψ( z ) = 2 ˆ R 2 ψ ( u ) ψ ( v ) e − y ( u − v ) e ix ( u + v ) dudv = ˆ R 2 ψ  α + β 2  ψ  β − α 2  e − αy e iβ x dαdβ = ˆ R 2 ψ  α + β 2  ψ  α − β 2  e − αy e iβ x dαdβ = ˆ R 2 ψ  − α + β 2  ψ  − α − β 2  e αy e iβ x dαdβ = ˆ R 2 ψ  α + β 2  ψ  α − β 2  e αy e iβ x dαdβ = ˆ R 2 ψ  α + β 2  ψ  α − β 2  cosh( y α ) e iβ x dαdβ = ˆ R 2 ψ  α + β 2  ψ  α − β 2  cosh( y α ) e − iβ x dαdβ = 2 ˆ ∞ 0  ˆ R ψ  α + β 2  ψ  α − β 2  cosh( y α ) dα  cos( β x ) dαdβ . Then (2.36) ∂ 2 ∂ y 2 | Ψ( z ) | 2 = ˆ ∞ 0  ˆ R ψ  α + β 2  ψ  α − β 2  α 2 cosh( y α ) dα  cos( β x ) dβ . Let (2.37) G ( β ; y ) = ˆ R ψ  α + β 2  ψ  α − β 2  α 2 cosh( y α ) dα. ON ZEROS OF FOURIER TRANSFORMS 8 F rom (2.38) ∂ 2 α  ψ  α + β 2  ψ  α − β 2  + ∂ 2 β  ψ  α + β 2  ψ  α − β 2  = 1 2  ψ (2)  α + β 2  ψ  α − β 2  + ψ  α + β 2  ψ (2)  α − β 2  we g et (2.39) ∂ 2 β G ( β ; y ) = ∂ 2 β ˆ R ψ  α + β 2  ψ  α − β 2  α 2 cosh( y α ) dα = ˆ R ∂ 2 β  ψ  α + β 2  ψ  α − β 2  α 2 cosh( y α ) dα = 1 2 ˆ R  ψ (2)  α + β 2  ψ  α − β 2  + ψ  α + β 2  ψ (2)  α − β 2  α 2 cosh( y α ) dα − ˆ R ∂ 2 α  ψ  α + β 2  ψ  α − β 2  α 2 cosh( y α ) dα = 1 2 ˆ R  ψ (2)  α + β 2  ψ  α − β 2  + ψ  α + β 2  ψ (2)  α − β 2  α 2 cosh( y α ) dα − ˆ R ψ  α + β 2  ψ  α − β 2   (2 + a 2 y 2 ) co sh( ay ) + 4 ( ay ) sinh( ay )  dα ≤ 0 . Then by Lemma 6 we hav e proved that ∂ 2 ∂ y 2 | Ψ( z ) | 2 ≥ 0 . Then by Jensen’s result Theorem 2 w e have shown that Ψ ( z ) = √ 8 π z − 3 2 J 3 2 ( z )Φ( z ) has only real zer os. Since √ 8 π z − 3 2 J 3 2 ( z ) has only real zeros , therefore, Φ( z ) has only r eal ze r os.  References [1] G. E. Andrews, R. Askey and R. Ro y , Sp e cial F unctions, Cambridge Univ ersity Press, Cam- bridge, 1999. [2] Phil i ppe Biane, Jim Pitman and Marc Y or, Probability Laws Related to the Jacobi Theta and Riemann Zeta F unctions, and B rownian Excursions, Bul letin (New Series) of the American Mathematical Society , V olume 38, Number 4, Pages 435-465. [3] G. Csordas, T. S. N orfolk and R. S. V arga, The Riemann Hyp othesis and the T urán Inequa l- ities, T ransactions of the American Mathematical Society , V olume 296, Number 2, August 1986, page 521–541 . [4] H. Dav enport, Multiplicative Number Theory , Springer-V erlag, N ew Y ork, 1980. [5] H. M . Edw ards, Riemann’s Zeta F unction, Do ve r Publications Inc. New Y ork, 1974. [6] A. Erdélyi, Higher T r ansc endental F uncti ons, V ol.I, V ol.I I, V ol.I I I, Rob ert E. 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Póly a, Üb er die algebraisch -funktionen theoret ischen Un tersuc hungen von J. L. W. V. Jensen, Kgl. Danske Videnskab ernes Selskab. Math.-Fys. Med d . 7 (17) (1927), pp. 3–33; reprin ted in his Col lecte d Pap ers , V ol. II, pp. 278–308. [12] E. C. Titchmar sh, The The ory of Riemann Zeta F unction , second edition, Clarendon Press, New Y ork, 1987. [13] G. N. W atson, A T r e atise on the The ory of Bessel F unctions , Cambridge Unive rsity Press, London, 1922. Curr ent addr ess : College of Science, North west A&F Univer sity, Y angling, Shaanxi 712100, P . R. China. E-mail addr ess : ruimin gzhang@ya hoo.com