The Teju-Nyquist Brownian Interpolation Theorem: Optimal Low-Frequency Reconstruction via Polyphase Spectral Extrapolation under a Brownian Prior

This paper presents the Teju-Nyquist Brownian Interpolation Theorem, which establishes the minimum mean-square error (MMSE) framework for reconstructing missing low-frequency content in band-limited signals under a Brownian (1/f^2) spectral prior. By combining polyphase decomposition with stochastic estimation theory, we derive a closed-form optimal interpolation filter that exploits the statistical structure of Brownian noise to extrapolate spectral content below the decimated Nyquist frequency. The method leverages the Brownian bridge formulation and Wiener estimation to achieve bandwidth extension without introducing spectral artifacts. We demonstrate that polyphase spectral extrapolation under a Brownian prior yields a principled, signal-theoretically optimal approach to low-frequency reconstruction, with applications in audio interpolation, sub-band coding, and stochastic signal processing.