PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0

FOURIER TRANSFORMS HAVING ONLY REAL ZEROS

DAVID A. CARDON

(Communicated by Joseph A. Ball)

Abstract. Let G(z) be a real entire function of order less than 2 with only real zeros. Then we classify certain distribution functions F such that the Fourier transform H(z) = ∫_{−∞}^{∞} G(it)e^{izt} dF (t) has only real zeros.

1. Introduction

Pólya [13] suggested that determining the class of functions whose Fourier transforms have only real zeros would be a ‘rather artificial question’ if it were not for the Riemann Hypothesis. For ℜ(s) > 1, the Riemann zeta function is defined by ζ(s) = ∑_{n=1}^{∞} n^{−s}. It has an analytic continuation, and the function

ξ(s) = \frac{1}{2}s(s − 1)π^{−s/2}Γ(s/2)ζ(s)

is entire. The Riemann Hypothesis states that all the zeros of ξ(s) satisfy ℜ(s) = 1/2. A proof of the Riemann Hypothesis would be a major advance for analytic number theory. Let Ξ(z) = ξ (\frac{1}{2} + iz). It is well known (see Titchmarsh [18], chapter 10) that

Ξ(z) = ∫_{−∞}^{∞} Φ(x)e^{izx}dx

where

Φ(x) = ∑_{n=1}^{∞} (4n^{4}π^{2}e^{9x/2} − 6n^{2}πe^{5x/2}) exp (−n^{2}πe^{2x}) .

In other words, the Riemann Hypothesis is true if and only if the Fourier transform Ξ(z) has only real zeros.

Pólya wrote several papers (such as [11, 12, 13, 14], which can all be found in [15]) in which he studied the reality of the zeros of various Fourier transforms. A particularly interesting result is the following:

Proposition 1 (Pólya [14]). Let f be an integrable function of a real variable t such that f(t) = \overline{f(−t)} and f(t) = O(e^{−|t|^{b}}) for t → ±∞ and b > 2. Assume that

∫_{−∞}^{∞} f(t)e^{izt}dt
\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors September 23, 2003 and, in revised form, December 23, 2003.

2000 Mathematics Subject Classification. 42A38, 30C15.

Key words and phrases. Fourier transform, zeros of entire functions, Laguerre-Pólya class.

has only real zeros. Let ϕ be a real entire function having only real zeros, and assume that ϕ has a Weierstrass product of the form

ϕ(z) = cz^{m}e^{αz−βz^{2}} ∏_{k} (1 − \frac{z}{α_{k}} ) e^{z/α_{k}}

where c, α, α_{k} are real, β ≥ 0, and m is a nonnegative integer. (In other words, ϕ belongs to the Laguerre-Pólya class.) Then

∫_{−∞}^{∞} ϕ(it)f(t)e^{izt} dt

has only real zeros.

In this paper we are concerned with constructing Fourier transforms with only real zeros in which the measure is not assumed to be the ordinary Lebesgue measure dt. The main result is the following theorem:

Theorem 1. Suppose G is an entire function of order < 2 that is real on the real axis and has only real zeros. Let {a_{k}} be a nonincreasing sequence of positive real numbers, let {X_{k}} be the sequence of independent random variables such that X_{k} takes values ±1 with equal probability, and let F_{n} be the distribution function of the normalized sum Y_{n} = (a_{1}X_{1} + ⋯ + a_{n}X_{n})/s_{n} where s_{n}^{2} = a_{1}^{2} + ⋯ + a_{n}^{2}. The functions F_{n} converge pointwise to a continuous distribution F = lim_{n→∞} F_{n}. Let H be the Fourier transform of G(it) with respect to the measure dF. That is,

H(z) = ∫_{−∞}^{∞} G(it)e^{izt} dF (t).

Then H is an entire function of order ≤ 2 that is real on the real axis. If H is not identically zero, then H has only real zeros.

Theorem 1 includes cases not covered in Proposition 1 because the distribution function F (t), although continuous, need not be differentiable. If F (t) is differentiable, we may write

∫_{−∞}^{∞} G(it)e^{izt} dF (t) = ∫_{−∞}^{∞} G(it)e^{izt}F′(t) dt.

However, since not all functions f(t) in Proposition 1 are of the form f(t) = F′(t) for the types of distributions in Theorem 1, Proposition 1 covers cases not included in Theorem 1. So, while there is some overlap between Proposition 1 and Theorem 1, neither implies the other.

The proof of Theorem 1 is given in §2. Before proceeding with the proof we mention that the proof relies on a result about sums of exponential functions. Let h_{n}(z) be the function of a complex variable z defined by

h_{n}(z) = ∑ G(±ia_{1} ± ⋯ ± ia_{n})e^{iz(±b_{1}±···±b_{n})}

where the summation is over all 2^{n} possible sign combinations, the same sign combination being used in both the argument of G and in the exponent. The numbers a_{1}, a_{2}, a_{3}, … and b_{1}, b_{2}, b_{3}, … are positive, and G is as in Theorem 1. The author shows in [3] that all the zeros of the exponential sum h_{n}(z) are real. Interestingly, the proof of this fact is similar to the proof of the Lee-Yang Circle Theorem from statistical mechanics (cf. [8] Appendix II or [17] Chapter 5). It should be pointed out that this result of the author is related to de Branges’ Hilbert spaces of entire functions [5]. Let a_{k} = b_{k}, s_{n}^{2} = a_{1}^{2} + ⋯ + a_{n}^{2}, and

H_{n}(z) = 2^{−n} ∑ G((±ia_{1} ± ⋯ ± ia_{n})/s_{n})e^{iz(±a_{1}±···±a_{n})/s_{n}}.

All of the zeros of H_{n}(z) are real. Theorem 1 is established by showing that the limit

H(z) = lim_{n→∞} H_{n}(z)

is uniform on compact sets.

2. Proof of Theorem 1

The proof of Theorem 1 is presented in this section as a sequence of lemmas. We begin with some notation.

The Laguerre-Pólya class LP of functions consists of the entire functions having only real zeros with a Weierstrass factorization of the form

az^{q}e^{αz−βz^{2}} ∏(1 − z/α_{n})e^{z/α_{n}}

where a, α, β are real, β ≥ 0, q is a nonnegative integer, and the α_{n} are nonzero real numbers such that ∑ α_{n}^{−2} < ∞. We shall be most interested in the subset LP^{∗} of the Laguerre-Pólya class consisting of all elements of LP of order < 2. Thus, β is necessarily 0 for functions in LP^{∗}.

The distribution function T for a random variable Y is T (x) = Pr(Y ≤ x). We will consider the following types of random variables and their distribution functions: Let {a_{k}} be a nonincreasing sequence of positive real numbers. Let {X_{k}} be a sequence of independent random variables such that X_{k} takes values ±1 with equal probability. Let Y_{n} be the sum

Y_{n} = \frac{a_{1}X_{1} + ⋯ + a_{n}X_{n}}{s_{n}}

where s_{n}^{2} = a_{1}^{2} + ⋯ + a_{n}^{2}. F_{n} will denote the distribution function of Y_{n}, and F will denote the limit F = lim_{n→∞} F_{n}. In Theorem 1 the distribution F has variance 1. However, F could be rescaled to have any other positive value for its variance. The following lemma describes this F.

Lemma 1. The sequence F_{n} converges pointwise to a continuous distribution F. If the sequence s_{n} is unbounded, F is the normal distribution. If the sequence s_{n} is bounded, F is not the normal distribution.

Proof. This is proved in Lemma 1 of [2]. □

Lemma 2 (Pólya [12], Hilfssatz II). Let a be a positive constant, let b be real, and let G(z) be an entire function of genus 0 or 1 that for real z takes real values, has at least one real zero, and has only real zeros. Then the function

e^{ib}G(z + ia) + e^{−ib}G(z − ia)

has only real zeros.

Proof. Pólya’s original statement is Hilfssatz II in [12]. Pólya’s argument is reiterated as Proposition 2 in [4]. □

We need the following important fact about H_{n}(z).

Lemma 3. Suppose G ∈ LP^{∗}. Let a_{1}, …, a_{n}, b_{1}, …, b_{n} be positive real numbers. The exponential sum

h_{n}(z) = ∑ G(±ia_{1} ± ⋯ ± ia_{n})e^{iz(±b_{1}±···±b_{n})}

obtained by summing over all sign combinations, the same combination being used in the argument of G as in the exponent, is in LP^{∗}.

Proof. It is clear that h_{n}(z) is real for real z and has order 1. The fact that h_{n}(z) has real zeros is proved in [3] by a method similar to that of the Lee-Yang Circle Theorem (found in Appendix II in [8]). □

If s_{n}^{2} = a_{1}^{2} + ⋯ + a_{n}^{2} and if we use the notation of Riemann-Steiltjes integration, an immediate corollary to Lemma 3 is the following:

Corollary 4. The function

H_{n}(z) = 2^{−n} ∑ G((±ia_{1} ± ⋯ ± ia_{n})/s_{n})e^{iz(±a_{1}±···±a_{n})/s_{n}} = ∫_{−∞}^{∞} G(it)e^{it} dF_{n}(t)

is in LP^{∗}.

In Lemmas 6 and 7 we will show that the integrals ∫_{−∞}^{∞} G(it)e^{it} dF_{n}(t) converge uniformly to ∫_{−∞}^{∞} G(it)e^{it} dF_{n}(t) for z in compact sets. The proof of Lemma 6 will require the following 1994 result of Pinelis, which is an improvement of a conjecture by Eaton [6]:

Lemma 5 (Pinelis [10], Corollary 2.6). Let X_{k} be independent random variables taking values ±1 with equal probability. Let s_{n}^{2} = a_{1}^{2} + ⋯ + a_{n}^{2}, and let

Y_{n} = \frac{a_{1}X_{1} + ⋯ + a_{n}X_{n}}{s_{n}} .

Then

Pr(|Y_{n}| > u) < 2c(1 − Φ(u))

where c = 2e^{3}/9, Φ(u) = ∫_{−∞}^{u} ϕ(t) dt, and ϕ(t) = (2π)^{−1/2}e^{−t^{2}/2}.

Lemma 6. Let ϵ > 0 be given, suppose G ∈ LP^{∗}, and let K be a compact subset of ℂ. Then there is a positive number A (depending on ϵ and K) such that

∫_{|t|>A} |G(it)e^{izt}| dF_{n}(t) < ϵ

for all n and all z ∈ K.

Proof. Let λ denote the order of G. By hypothesis, λ < 2. Choose δ with max(1, λ) < δ < 2. Then choose A > 0 large enough so that |G(it)e^{izt}| < e^{|t|^{δ}} for all z ∈ K and |t| > A. Such an A exists as follows: Choose δ′ with λ < δ′ < δ. Then for sufficiently large A, |G(it)| < e^{|t|^{δ′}} whenever |t| > A. Since K is compact, there is an R so that |z| < R for all z ∈ K. For sufficiently large A and |t| > A,

|G(it)e^{izt}| ≤ |G(it)|e^{|z||t|} < e^{|t|^{δ′} +R|t|} < e^{|t|^{δ}}.

Thus, A exists as claimed. Then

∫_{A<|t|<B} |G(it)e^{izt}|dF_{n}(t) < 2 ∫_{A}^{B} e^{t^{δ}}dF_{n}(t).

After integration by parts the right hand-side becomes

2e^{B^{δ}}(F_{n}(B) − 1) − 2e^{A^{δ}}(F_{n}(A) − 1) − 2 ∫_{A}^{B}(F_{n}(t) − 1)d(e^{t^{δ}})
< 2e^{A^{δ}}(1 − F_{n}(A)) + 2 ∫_{A}^{B}(1 − F_{n}(t))δt^{δ−1}e^{t^{δ}}dt.

According to Lemma 5 for t ≥ A and if A is sufficiently large, then

1 − F_{n}(t) ≤ β ∫_{t}^{∞} e^{−u^{2}/2}du < \frac{e^{−t^{2}/2}}{2}

where β = 4e^{3}/(9\sqrt{2π}). This gives

∫_{A<|t|<B} |G(it)e^{izt}|dF_{n}(t) < e^{A^{δ}−A^{2}/2} + ∫_{A}^{B} δt^{δ−1}e^{t^{δ}−t^{2}/2}dt

and

∫_{A<|t|} |G(it)e^{izt}|dF_{n}(t) < e^{A^{δ}−A^{2}/2} + ∫_{A}^{∞} δt^{δ−1}e^{t^{δ}−t^{2}/2}dt.

For sufficiently large A the last integral is bounded above by e^{A^{δ}−A^{2}/2}. So,

∫_{A<|t|<B} |G(it)e^{izt}|dF_{n}(t) < 2e^{A^{δ}−A^{2}/2}.

The right-hand side of the last inequality can be made arbitrarily small for sufficiently large A. Therefore, we obtain ∫_{|t|>A} |G(it)e^{izt}|dF_{n}(t) < ϵ as desired. □

Lemma 7. Let K be a compact subset of ℂ. Then

∫_{−A}^{A} G(it)e^{izt} dF_{n}(t) → ∫_{−A}^{A} G(it)e^{izt} dF (t)

uniformly as n → ∞ for z ∈ K.

Proof. By the Helly-Bray Theorem (see [9, p. 182] or [7, p. 298]) it is immediate that convergence occurs pointwise. We must, however, verify uniform convergence for z ∈ K.

Let ϵ > 0 be given, and write g_{z}(t) = G(it)e^{izt}. Choose κ such that κ > |g_{z}(t)| and κ > |g′_{z}(t)| for all z ∈ K and t ∈ [−A, A]. Integration by parts yields

∫_{−A}^{A} G(it)e^{itz} dF (t) = g_{z}(A)F (A) − g_{z}(−A)F (−A) − ∫_{−A}^{A} F (t)g′_{z}(t) dt.

Then

|∫_{−A}^{A} G(it)e^{itz} dF (t) − ∫_{−A}^{A} G(it)e^{itz} dF_{n}(t)|

≤ |g_{z}(A)| |F (A) − F_{n}(A)| + |g_{z}(−A)| |F (−A) − F_{n}(−A)|

+ ∫_{−A}^{A} |F (t) − F_{n}(t)| |g′_{z}(t)| dt

≤ κ|F (A) − F_{n}(A)| + κ|F (−A) − F_{n}(−A)| + κ2A max_{t∈[−A,A]} |F (t) − F_{n}(t)|.

Since the functions F_{n} and F are distributions functions such that F_{n} converges to the continuous distribution F pointwise on [−A, A], F_{n} converges to F uniformly on [−A, A]. Thus for sufficiently large n and all t ∈ [−A, A],

|F_{n}(t) − F (t)| < min (\frac{ϵ}{6Aκ}, \frac{ϵ}{3κ}) .

Therefore, for all sufficiently large n and all z ∈ K,

|∫_{−A}^{A} G(it)e^{itz} dF (t) − ∫_{−A}^{A} G(it)e^{itz} dF_{n}(t)| < \frac{ϵ}{3} + \frac{ϵ}{3} + \frac{ϵ}{3} = ϵ.

This shows that the convergence ∫_{−A}^{A} G(it)e^{itz} dF_{n}(t) → ∫_{−A}^{A} G(it)e^{itz} dF (t) is uniform as claimed. □

Lemma 8. Suppose G ∈ LP^{∗}. Then H(z) = ∫_{−∞}^{∞} G(it)e^{izt} dF (t) is an entire function that is real for real z, and if it does not vanish identically, then it has only real zeros.

Proof. Let H_{n}(z) = ∫_{−∞}^{∞} G(it)e^{izt} dF_{n}(t). By Lemmas 6 and 7, H_{n}(z) converges uniformly to H(z) on compact sets. Since H_{n}(z) is real for real z, its limit H(z) is real for real z. By Hurwitz’s Theorem (see [1, Thm. 2, p. 178]), if H(z) is not identically zero, its zeros are limit points of the zeros of the H_{n}(z). Since, for each n, H_{n}(z) has only real zeros, H(z) also has only real zeros. □

Lemma 9. Suppose G ∈ LP^{∗}. The order of H(z) = ∫_{−∞}^{∞} G(it)e^{izt} dF (t) is ≤ 2.

Proof. Choose δ with λ < δ < 2 where λ is the order of G. Let M be large enough so that |G(z)| < Me^{|z|^{δ}} for all z. By applying Hölder’s Inequality (see [16, p. 63]) we obtain

|∫_{−∞}^{∞} G(it)e^{−izt}dF (t)| ≤ ∫_{−∞}^{∞} Me^{|t|^{δ}+|z||t|}dF (t)

≤ M (∫_{−∞}^{∞} e^{2|t|^{δ}}dF (t))^{1/2} (∫_{−∞}^{∞} e^{2|z||t|}dF (t))^{1/2} .

By Lemma 5 both integrals in the product converge. The first integral is independent of z. We will determine a bound for the second integral. Integration by parts and an application of Lemma 5 give

∫_{−∞}^{∞} e^{2|z||t|}dF (t) = 2 ∫_{0}^{∞} e^{2|z|t}dF (t)
= 2e^{2|z|t}(F (t) − 1)|_{0}^{∞} + 2 ∫_{0}^{∞} 2|z|e^{2|z|t}(1 − F (t)) dt
= 1 + 4|z| ∫_{0}^{∞} |z|e^{2|z|t}(1 − F (t)) dt
≤ 1 + K|z| ∫_{0}^{∞} e^{2|z|t−t^{2}/2}dt where K = \frac{16e^{3}}{9\sqrt{2π}} .

Since

∫_{0}^{∞} e^{2|z|t−t^{2}/2}dt ≤ e^{2|z|^{2}} ∫_{−∞}^{∞} e^{−(t−2|z|)^{2}/2}dt = \sqrt{2π} e^{2|z|^{2}},

we see that |∫_{−∞}^{∞} G(it)e^{−izt}dF (t)| is bounded by a constant times |z|e^{2|z|^{2}}. Thus,

the order of ∫_{−∞}^{∞} G(it)e^{−izt}dF (t) is ≤ 2.

This completes the proof of Theorem 1. □

References

1. Lars V. Ahlfors, Complex analysis: An introduction to the theory of analytic functions of one complex variable, third ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978.

2. David A. Cardon, Convolution operators and zeros of entire functions, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1725–1734.

3. David A. Cardon, Sums of exponential functions having only real zeros, Manuscripta Math. (to appear).

4. David A. Cardon and Pace P. Nielsen, Convolution operators and entire functions with simple zeros, Number theory for the millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 183–196.

5. Louis de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall Inc., Englewood Cliffs, N.J., 1968.

6. M. L. Eaton, A probability inequality for linear combinations of bounded random variables, Ann. Stat. 2 (1974), 609–614.

7. Martin Eisen, Introduction to mathematical probability theory, Prentice-Hall Inc., Englewood Cliffs, N.J., 1969.

8. T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model, Physical Rev. (2) 87 (1952), 410–419.

9. Michel Loève, Probability theory, third ed., D. Van Nostrand Co., Inc., Princeton, N.J.- Toronto, Ont.-London, 1963.

10. Iosif Pinelis, Extremal probabilistic problems and Hotelling’s T^{2} test under a symmetry condition, Ann. Statist. 22 (1994), no. 1, 357–368.

11. George Pólya, On the zeros of an integral function represented by Fourier’s integral, Messenger of Math. 52 (1923), 185–188.

12. George Pólya, Bemerkung über die Integraldarstellung der Riemannschen ξ-Funktion, Acta Math. 48 (1926), 305–317.

13. George Pólya, On the zeros of certain trigonometric integrals, J. London Math. Soc. 1 (1926), 98–99.

14. George Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math. 158 (1927), 6–18.

15. George Pólya, Collected papers, The MIT Press, Cambridge, Mass.-London, 1974, Vol. II: Location of zeros, Edited by R. P. Boas, Mathematicians of Our Time, Vol. 8.

16. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1987.

17. David Ruelle, Statistical mechanics: Rigorous results, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

18. E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., Clarendon Press Oxford University Press, Oxford, 1986, Revised by D. R. Heath-Brown.

Department of Mathematics, Brigham Young University, Provo, Utah 84602

E-mail address: cardon@math.byu.edu