Strip-Analytic Abu-Ghuwaleh Transforms: Continuous Dilation-Convolution,Wiener–Mellin Inversion, and Stable Log-Scale Recovery

We develop the strip-analytic sequel to the master-integral-transform program with entire kernels by replacing the discrete Taylor-spectrum model with a continuous spectral model on the dilation side. The central object is a Hardy-strip orbit kernel whose boundary representation induces a continuous dilation-convolution operator acting on the Fourier transform of a weighted signal. In this setting, the Abu-Ghuwaleh transform admits two complementary inversion mechanisms: Mellin contour inversion and contour-free Wiener--Mellin inversion. We prove exact factorization formulas on named weighted function spaces, derive branchwise Mellin diagonalization formulas, obtain inversion theorems under nonvanishing assumptions on the continuous symbol, and show that logarithmic coordinates convert the transform into an additive convolution equation. This yields a practical FFT-based inversion framework together with a stability bound on frequency windows away from zeros of the multiplier. We also prove an explicit injectivity-and-stability proposition for a resolvent-type kernel family with Gamma-type symbol. The paper is designed as the natural continuous-spectrum successor to the entire-kernel and finite-Laurent stages of the program.