= Fourier-cutoff oscillator functional determinant
{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.
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