Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$):
$$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$
The functional equation for $\zeta(s)$ is equivalent to $\Xi(z)=\Xi(-z)$.
Riemann $\Xi(z)$ function can be expressed as a Fourier transformation:
$$\Xi(z)=2\int_0^{\infty}\Phi(u)\cos(z u){\rm d}u$$
where $$\Phi(u)=\sum_{n=1}^{\infty}\left(4\pi^2n^4\exp(9u/2)-6\pi n^2\exp(5u/2)\right)\exp\left(-\pi n^2 \exp(2u)\right)=\Phi(-u)$$
Polya approximated $\Phi(u)$ with $\Phi_{*}(u)$ and $\Phi_{**}(u)$:
$$\Phi_{*}(u)=8\pi^2\cosh(9u/2)\exp\left(-2\pi \cosh(2u)\right)$$
$$\Phi_{**}(u)=\left(8\pi^2\cosh(9u/2)-6\pi\cosh(5u/2)\right)\exp\left(-2\pi \cosh(2u)\right)$$
This is because he noticed that when $u\to\infty$, $\Phi(u)\to\Phi_{*}(u)$ and $\Phi(u)\to\Phi_{**}(u)$.
Polya proved that the resulting $\Xi_*(z)$ and $\Xi_{**}(z)$ have real zeros only.
de Bruijn approximated $\Phi(u)$ with $\Phi_d(u)$:
$$\Phi_d(u)=2\cosh(5u/2)\left(2\pi^3-3\pi+4\pi^2\cosh(u)\right)\exp\left(-2\pi \cosh(2u)\right)$$
de Bruijn proved that the resulting $\Xi_d(z)$ has real zeros only.
Hejhal approximated $\Phi(u)$ with $\Phi_N(u)$:
$$\Phi_N(u)=\sum_{n=1}^{N}\left(8\pi^2n^4\cosh(9u/2)-12\pi n^2\cosh(5u/2)\right)\exp\left(-2\pi n^2 \cosh(2u)\right)$$
Hejhal proved that almost all the zeros of the resulting $\Xi_N(z)$ are real. However when $N\to\infty$ $\Phi_N(u) \not\to \Phi(u)$.
For more details please refer to two excellent review papers by Dimitrov and Rusev [1] and Ki[2].
[1]: Dimitrov and Rusev, The zeros of entire Fourier transforms, EAST JOURNAL ON APPROXIMATIONS Volume 17, Number 1 (2011), 1-108 [2]: Ki, The Zeros of Fourier Transforms
Question 1: Are there any new research results on approximating Riemann $\Xi(z)$ by Fourier transformations?
Question 2: Why approximating Riemann $\Xi(z)$ by Fourier transformations does not seem to be an active research field towards a possible proof of Riemann hypothesis$?
Best regards- mike