Fourier-Probe Analysis of Residue-Class Derivative Sampling

This letter studies a Fourier-probe obstruction test for structured residue-class derivative sampling of analytic functions. At each of two sampling points, only derivatives whose orders lie in prescribed residue classes modulo $q$ are retained. A finite Fourier-probe matrix gives an algebraic test for null functions inside a $q$-dimensional exponential family. Basic Fourier-probe rank regimes are identified for full, undersampled, balanced, and oversampled residue patterns. The case $q=4$ is used to compare the balanced design $R_{0}=\lbrace 0,1\rbrace$, $R_{1}=\lbrace 2,3\rbrace$, with the oversampled design $R_{0}=\lbrace 0,1,2\rbrace$, $R_{1}=\lbrace 1,2,3\rbrace$. The balanced design has a nonzero exponential-family null function, whereas the oversampled design has no nonzero null function inside the same finite exponential test family. Finite Chebyshev reconstruction experiments show that the balanced measurement matrix is severely ill-conditioned and noise-sensitive, whereas the oversampled matrix is substantially better conditioned, yielding much smaller reconstruction errors and improved noise robustness.