Commun. Korean Math. Soc. 17 (2002), No. 1, pp. 1 14, en-. כ Some recent results on the distribution of zeros of real tire functions are surveyed. 1 1930 Fourier Pólya, 1822 Fourier כ [19]., Fourier de Gua. Pólya Fourier Poisson Pólya 1920. כ, Fourier כ, Fourier., Pólya, כ., Pólya Fourier-Pólya Pólya-Wiman. כ. 1937, Ålander, Pólya, Wiman, 1987. ( [7].) Pólya 1937 Received December 10, 2001. 2000 Mathematics Subject Classification: 30D15, 30D20, 33C10. Key words and phrases: linear operator, invariant subspace, transitive algebra.
2,. ([4], [5], [19], [20], [21], [24].) 1987 1996, Craven, Csordas, Smith Pólya-Wiman Fourier-Pólya [7], [8], [11], [12], [13], [14]. (2000 ) Fourier-Pólya Fourier [15]. כ : 2 de Gua Fourier, 3 Pólya, 4 5 Pólya-Wiman Fourier-Pólya, 6. 2 De Gua Fourier כ f I. f I.,,., f. f I., Fourier f f כ f כ (critical zero). f f כ 0. f f K, f 2K +1., f Rolle, f., f. f m, f m, m 1 ; f כ 0. f a m. f(a) 0,f (a) = = f (m) (a) =0 f (m+1) (a) 0. m, f, a m/2. m, f(a)f (m+1) (a) > 0 a f
3, f(a)f (m+1) (a) < 0 a f. a f (m +1)/2, (m 1)/2, (m 1)/2, (m +1)/2. f K f f K ;., f Rolle כ. n =2,3,... f (n) f (n 1) f (n). I, n =1, 2,... f n כ f, I f כ f I., coshx 0 ; cos x. cosh x cos x. P. N N P P K P, x P (x), N = N 1+2K. J J P P,, N +2J = N +2J 1. J = J K.., de Gua. 2.1 (de Gua).. Pólya [19], Fourier J 0 (2 x), J 0 Bessel,. Cauchy (de Gua ), Poisson de Gua כ. DeGua,. e x 1, cosh x de Gau.
4, Fourier כ : Fourier. f, f. כ (entire function). כ. J 0 (2 x) (order) 1/2, 1893 Hadamard,. ( 3.) Fourier J 0 (2 x). 3 Pólya כ. f ρ כ. log log M(r; f) ρ = lim r ; log r M(r; f) z = r f כ. 0<ρ<, f τ. log M(r; f) τ = lim r r ρ. (ρ, τ) כ ρ, ρ τ. 0- כ z (1 az) כ ; 1- כ כ z, e az, (1 az)e az כ. Hadamard, 1 0-, 0- (1, 0) ; 2 1-, 1- (2, 0). 1-1 - כ 0 α 0 1- g(z) e αz2 g(z) כ ; (2,α). ( (order), (type), (genus), Hadamard [2] [17].) Pólya [19].
5 A. 0-. B. 1 -. C. f 1 -, m f (m),f (m+1),...., A C Fourier-Pólya Pólya- Wiman. Pólya B C ([19] I). 3.1. 1 -,. Fourier-Pólya Fourier כ. J 0 (2 x) 0-, Fourier-Pólya Fourier. Fourier-Pólya Pólya-Wiman, כ כ. Pólya-Wiman [23]. Levin [17] 8. 3.2 (Laguerre-Pólya-Schur). N N f, f 1 - כ N {P n } f כ. 1 - כ LP, כ LP. f LP כ {P n } f כ, f LP כ P LP g f = Pg כ. 3.1.
6, 4 Pólya-Wiman Pólya-Wiman כ Jensen. a a ā a Jensen. f, f Jensen J (f). f 1 -. f : z, (1 az)e az (a R), exp ( αz 2 + βz ) (α 0, β R), (1 cz)(1 cz)exp(cz + cz). f f /f., Im z>0 0 Im z<0 0. Im z 0, c Jensen Im z. z/ J(f) Im z 0 Im z>0 0 Im z<0 0.. 4.1 (Jensen). f 1 -, f J (f). Pólya-Wiman. [15] 2 [12]. f f(z 0 )=f (z 1 )= = f (n 1) (z n 1 )=0, : z ζ1 ζn 1 (4.1) f(z) = f (n) (ζ n )dζ n dζ 2 dζ 1 (z C). z 0 z 1 z n 1 f 1 -. f. 3.1 f, f,...,f (n),... 1 -. n, f (n), Jensen n +1 z 0,z 1,...,z n : z j f (j), j = 1,...,n z j z j 1 Jensen. f, {z n } n=0 : z n f (n), n =1, 2,... z n z n 1 Jensen. z n f (n), m, n f (m) (4.1) f (m) (z) f (m+n) (z) z,z m,...,z m+n 1. f α 0 (2,α) n =1, 2,...
7 z n z n 1 Jensen m z 1 f (m) (z) =0, כ f. Pólya-Wiman 3.1 Jensen LP :.. LP [11]. (,.) 4.2. LP f, A N n N f (n) ( A n, A n). f ρ( 2), ( A n, A n) ( An 1/rho,An 1/rho ). LP LP Pólya-Wiman. LP +, β 0 0 0- g e βz g(z). LP + LP +,, [11]. 4.3. f LP + n f (n) LP +. Δ 0 Im z Δ 1 - LP Δ, Laguerre-Pólya-Schur ([17] 8 ): 4.4. f LP Δ כ Im z Δ {P n } f כ. Gauss-Lucas LP Δ. Pólya-Wiman Pólya-Wiman.
8, 4.5 ( Pólya-Wiman ). Δ 0. f Im z Δ 1 -, m f (m),f (m+1),... Im z Δ. Pólya-Wiman 3.1 [13], [14]. 4.6 ( Pólya-Wiman ). ρ 2. (ρ, 0) 1 - f Δ Im z Δ, A N n N f (n) Re z An 1/rho. 4.7. f 1 -, E J (f). 2J 2J f E, K f E, J J = K. 4.6 1 - כ. LP Δ כ,. n f f (n) D n f,pólya-wiman D. [6] f f(d) כ. exp ( λd 2) [3], [6], [16], [18].. [16] 4.8. f(z) (2,α) 1 -. Δ 0, λ>0 λα < 1/4. f(z) Im z Δ, e λd2 f(z) 1 - Im z max{δ 2 2λ, 0}. Δ 2 < 2λ, e λd2 f(z). 4.9. f(z) (2,α) 1 -. Δ 0, λ > 0 λα < 1/4. ɛ Im z Δ+ɛ f(z), e λd2 f(z) 1 -, ɛ Im z max{δ 2 2λ, 0} + ɛ.
9 Δ 2 < 2λ,. 4.9 1950 de Bruijn ([3] p. 199, 205). 5 Fourier-Pólya f 0-. f, Pólya-Wiman 3.1 f כ. Fourier-P olya 0-. Fourier-P olya ([15] 3, 4 ). כ 0-. f 0-. [15] 3 f. 5.1. {f n } K f, n f n K. 5.2. P f K, Pf K. f, f x f(x) 0. f. f, x f(x) 0,. f. f f, (f (1, 0) ) Phragmén-Lindelöf ([2] 1.4.3). f., n =1, 2,... f (n). x n =0, 1, 2,... f(x) > 0. (absolutely monotone), a, b > 0 ae bx
10,. Bernstein-Widder ([26] 3.9.2), : 5.2 (Bernstein-Widder). (, 0] f (, 0), [0, ) F : (5.1) f(x) = 0 e tx df (t). f (5.1), (1, 0). ( [15] pp. 59 60.) כ f (1, 0). f. Fourier-Pólya. cos x, sinx, coshx, sinhx, Ξx, 0-. 1, 0- f zf(z 2 ) f(z 2 ). de Gua. [15] 4.3. 5.3. f(z) 0-, zf(z 2 ) f(z 2 ). ([15] pp. 63 65).. f(z) 0-, g(z) =f(z 2 ). g(z). 5.2 g(z) zg(z). g(z). f(z) Maclaurin. f(z) = a n z n a k 0. n=k g(z) Maclaurin. g(z) = a n z 2n. n=k a k 0, n k a n a n+1 0 g(z) כ. n k
11 a n a n+1 0. 0 f(z) m =0, 1, 2, x<0 f (m) (z) 0. f(z) 0, f(z). c g(z) c 2 f(z) f(z) כ f(z). f(z) 0-, Fourier-Pólya f(z). כ a, a>0. m =0, 1,... g m (z) =f (m) (z 2 ). g 0 (z) כ. 5.2 g m (z) = 2zf (m+1) (z 2 )=2zg m+1 (z), g m+1 (z) g m (z). m g m (z). a f(z), m f (m) (x) x = a 0 כ כ. a>0. x 2 x>0, g m (x) x = a 0 כ. g m (x). [13] [14] 0- de Gua. f 1 -. f., f f de Gua. f, f. n f (n). ([9], [10] [25] ) f (n) de Gua., Fourier-Pólya 1 - כ. 6 Fourier-Pólya :, כ. כ כ כ.. 1887 Pólya 1912 70 200 [1]. 4 Location of zeros 37 [22]. כ
12,. : 1968 1913 1943., Ξ- כ. ([22] p. 166, 237, 241, 243, 265 ). 20,. כ, כ כ. Fourier-Pólya, (general theorem). Fourier-Pólya.. de Gua כ ([27] pp. 133 134)., Fourier-Pólya : ( ),. Pólya, Fourier-Pólya Ξ- כ : Ξ- Ξ-, 5.3 Ξ-. [15] 5 Fourier-Pólya. כ. Fourier-Pólya. [1] G. K. Alexanderson, The Random Walk of George Pólya, MAA, Washington DC, 2000. [2] R. P. Boas, Entire Functions, Academic Press, New York, 1954.
13 [3] N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197 226. [4] R. P. Boas and G. Pólya, Generalizations of completely convex functions, Proc. Nat. Acad. Sci. 27 (1941), 323 325. [5], Influence of signs of the derivatives of a function on its analytic character, Duke Math. J. 9 (1942), 406 424. [6] T. Craven and G. Csordas, Differential operators of infinite order and the distribution of zeros of entire functions, J. Math. Anal. Appl. 186 (1994), 799 820. [7] T. Craven, G. Csordas, and W. Smith, The zeros of derivatives of entire functions and the Pólya Wiman conjecture, Ann. of Math. 125 (1987), no. 2, 405 431. [8], Zeros of derivatives of entire functions, Proc. Amer. Math. Soc. 101 (1987), 323 326. [9] S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227 249. [10], Derivatives of entire functions and a question of Pólya. II, Trans. Amer. Math. Soc. 234 (1977), 497 503. [11] Y. O. Kim, A proof of the Pólya Wiman conjecture, Proc. Amer. Math. Soc. 109 (1990), 1045 1052. [12], On a theorem of Craven, Csordas and Smith, Complex Variables 22 (1993), 207 209. [13], Critical points of real entire functions and a conjecture of Pólya, Proc. Amer. Math. Soc. 124 (1996), 819 830. [14], Critical points of real entire functions whose zeros are distributed in an infinite strip, J. Math. Anal. Appl. 204 (1996), 472 481. [15] H. Ki and Y.-O. Kim, On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture, Duke Math. J. 104 (2000), 45 73. [16], De Bruijn s question on the zeros of Fourier transforms, preprint. [17] B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Mono., Amer. Math. Soc., Providence, R.I. 5 (1964). [18] C. M. Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61 (1976), 245 251. [19], Some problems connected with Fourier s work on transcendental equations, Quart. J. Math. Oxford Ser. 1 (1930), 21 34. [20], On functions whose derivatives do not vanish in a given interval, Proc. Nat. Acad. Sci. 27 (1941), 216 217. [21], On the zeros of derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178 191. [22], Collected Papers. Vol. II. Location of Zeros (ed. R. P. Boas), MIT Press, Cambridge, 1974. [23] G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, Crelle s Journal für die reine u. agew. Mathematik 144 (1914), 89 113. [24] G. Pólya and N. Wiener, On the oscillation of the derivatives of a periodic function, Trans. Amer. Math. Soc. 52 (1942), 245 256. [25] T. Sheil Small, On the zeros of the derivatives of real entire functions and Wiman s conjecture, Ann. of Math. 129 (1989), 179 193.
14, [26] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Dover Publ., New York, 1994. [27] Sz. Nagy, Über die Lage der nichtreellen Nullstellen von reellen Polynomen und von gewissen reellen ganzen Funktionen, Crelle s Journal für die reine u. agew. Mathematik 170 (1934), 133 147. 134 E-mail: haseo@bubble.yonsei.ac.kr 98 E-mail: kimyo@kunja.sejong.ac.kr