Interpolation and the Laguerre-Pólya class (Q2743173)

A real sequence \(\{\gamma _k \}^{\infty } _{k=0}\) is called a \textit{multiplier sequence} if the polynomial \(\sum ^n _{k=0} \gamma _k a_k x^k\) has only real zeros for any real polynomial \(\sum ^n _{k=0} a_k x^k\) with only real zeros. These sequences are intrinsically connected with so-called Laguerre-Pólya class of entire functions, i.e., real entire functions admitting the representation NEWLINE\[NEWLINE\phi (x) = c x^n e^{-\alpha x^2 +\beta x} \prod ^{\infty} _{k=1} \left(1+\frac{x}{x_k}\right)e^{-x/x_k}, NEWLINE\]NEWLINE where \(c\neq 0, \alpha \geq 0, \beta, x_k \in R\), \(n\geq 0\) is an integer and \(\sum ^{\infty} _{k=1} x^{-2} _k , \infty\). Regarding the classical history of this topic the reader can read, e.g., B. Levin, \textit{Distribution of zeros of entire functions}, AMS Transl. Vol. 5 (1980), Sect. 8.3, or N. Obreschkoff, \textit{Verteilung und Berechnung der Nullstellen reeller Polynome}, VEB Deutscher Verl. Wissenschaften, Berlin, 1963. Concerning the current developments see the paper under review, where the authors discuss the results and state some open problems. We state their main result, where CZDS stands for the Complex Zero Decreasing Sequences. Theorem 3.6. Let \(f(z), f(0)=1,\) be an entire function of exponential type and \(\{f(k)\}^{\infty} _{k=1}\) be a CZDS. If the Phragmén-Lindelöf indicator function of \(f\) satisfies \(h_f (\pm \pi /2) < \pi\), then \(f\) is in Laguerre-Pólya class and moreover, NEWLINE\[NEWLINEf(z)=e^{az} \prod ^{\infty} _{k=1} \left(1+\frac{z}{x_k}\right),NEWLINE\]NEWLINE where \(a\in R\),\( x_k >0\) and \(\sum^{\infty} _{k=1} 1/x_k < \infty \). Now, let \(H(x) := (1/8)ksi (x/2),\) where NEWLINE\[NEWLINEksi (iz)=(1/2) (z^2 -1/4) \pi^{-z/2 -1/4} \Gamma (1/4 + z/2) \zeta (1/2 +z) NEWLINE\]NEWLINE and \(\zeta\) is the Riemann \(\zeta-\)function. The Riemann hypothesis can be formulated as the statement that \(H\) belongs to the Laguerre-Polya class. Extending NEWLINE\[NEWLINEH(x)=\sum^{\infty} _{m=0} \frac{b_m}{(2m)!} (-x^2)^m, NEWLINE\]NEWLINE the authors establish in particular the equivalence of the Riemann hypothesis to the statement that \(\{F(k)\}^{\infty} _{k=1}\) is CZDS, where \(F(u):= \sum^{\infty} _{m=0} \frac{b_m}{(2m)!} u^m\).