Caputo–FRFT Coupling Induced by Compatible Fractional Convolution: Representation and Discrete Realization

Caputo fractional derivatives describe causal power-law memory, whereas the fractional Fourier transform represents continuous rotations in the time–frequency plane. This study derives an Abel–Gaussian-regularized coupling for endpoint-compatible causal signals. Unlike a direct composition of the two operators, the coupling is induced by embedding the causal Caputo convolution into a chirp-conjugated fractional convolution compatible with the adopted transform phase convention. Evaluation of the two transform branches shows that their quadratic phases cancel the additional convolution phase. On a compactly supported smooth core, removal of the regularization yields a rotation–memory–rotation factorization in which the fractional-memory multiplier acts at the ordinary Fourier node and the outer rotation maps the weighted spectrum to the prescribed fractional domain. At the discrete level, modulo-four reduction of the transform order is kept separate from decomposition of the derivative order, while a one-sided Grünwald–Letnikov approximation preserves causal history accumulation. Numerical experiments confirm fixed-spectral rotation identities and forward–inverse reconstruction at double-precision roundoff, approximately first-order convergence to an analytic Caputo reference, and decreasing low-frequency discrepancies under grid refinement. Further tests distinguish the effects of the sampling interval, observation length, fractional order, and nonzero endpoint traces. Exact boundary-term retention preserves the finite record, whereas compatible tapering produces a modified surrogate. For the tested dense implementations, the spectral implementation is faster than the direct full-history Grünwald–Letnikov implementation, while the shared dense operators dominate the storage requirements.