In this study, we established a general theorem regarding the equivalence of convolution operators restricted to a finite spectral band. We demonstrated that two kernels with identical Fourier transforms over the resolved band act identically on all band-limited functions, even if their kernels differ outside the band. This property is significant in applied mathematics and computational physics, particularly in scenarios where measurements or simulations are spectrally truncated. As an application, we examine the proportionality relation $S(\boldsymbol {r}) \approx \zeta\,\omega(\boldsymbol{r})$ in filtered vorticity dynamics and clarify why real-space diagnostics can underestimate the spectral proportionality due to unobservable degrees of freedom. Our theoretical findings were supported by numerical illustrations using synthetic data.

翻译：本研究建立了关于限制在有限谱带内的卷积算子等价性的一般定理。我们证明了两个在解析带内具有相同傅里叶变换的核函数，对所有带限函数的作用完全一致，即使它们的核函数在带外存在差异。该性质在应用数学和计算物理学中具有重要意义，特别是在测量或模拟存在谱截断的场景中。作为应用，我们研究了滤波涡量动力学中的比例关系$S(\boldsymbol {r}) \approx \zeta\,\omega(\boldsymbol{r})$，并阐明了为何实空间诊断会因不可观测的自由度而低估谱比例关系。我们的理论发现通过使用合成数据的数值算例得到了验证。