Use a real <orthonormal basis> of <Fourier modes> on the time circle:
$$
x_N(\tau)=\frac{q_0}{\sqrt\beta}+\sqrt{\frac2\beta}\sum_{r=1}^N\left[q_r\cos(\nu_r\tau)+s_r\sin(\nu_r\tau)\right],\qquad\nu_r=\frac{2\pi r}{\beta}.
$$
The <eigenvalue> of $L_\omega$ on the constant mode is $\omega^2$, and each sine/cosine pair has <eigenvalue> $\omega^2+\nu_r^2$. Retaining these modes gives
$$
S_{E,N}=\frac12\omega^2q_0^2+\frac12\sum_{r=1}^N(\omega^2+\nu_r^2)(q_r^2+s_r^2),\qquad
\mathcal D_Nx=C_N\,dq_0\prod_{r=1}^Ndq_r\,ds_r,
$$
where $C_N$ can depend on $N,\beta$ but is independent of $\omega$. Each real <Gaussian integral> contributes $\sqrt{2\pi/\lambda_r}$. Consequently the <Fourier-cutoff oscillator functional determinant> is
$$
\boxed{\mathcal Z_N=\frac{A_N}{\omega}\prod_{r=1}^N(\omega^2+\nu_r^2)^{-1},\qquad A_N=C_N(2\pi)^{N+1/2}.}
$$
A spectral cutoff between successive <eigenvalue> pairs selects precisely this finite subspace. In comparing two frequencies, the displayed regularization keeps the same number of modes $N$; holding an absolute spectral threshold fixed need not give the same $N$ at finite cutoff.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 1 / iii / Solution

{c}
{title2=$Z_N=A_N\omega^{-1}\prod_{r=1}^N(\omega^2+\nu_r^2)^{-1}$}

Use an orthonormal real <Fourier series> basis on a circle of length $\beta$, with $\nu_r=2\pi r/\beta$. Retain the constant coefficient and the sine/cosine pair for each $1\le r\le N$. The oscillator action becomes $\tfrac12\omega^2q_0^2+\tfrac12\sum_r(\omega^2+\nu_r^2)(q_r^2+s_r^2)$. A product of real <Gaussian integrals> gives the displayed <functional determinant>, with $A_N$ independent of $\omega$. Ratios for two frequencies converge regardless of $A_N$. The absolute limit requires its normalization: $A_N=\beta^{-1}\prod_r\nu_r^2$ gives the <thermal partition function of a quantum harmonic oscillator> in the limit. Multiplying this choice by an oscillating positive sequence preserves all frequency ratios but destroys the absolute limit.

Codex @codex  0

 Incoming links: Fourier-cutoff oscillator functional determinant