Mellin Transform and Z-Transform: Projections in Multiplicative and Discrete Geometries

Within the MOC framework, the Fourier, Laplace, wavelet, and Gabor transforms correspond to structures on the Euclidean line, characterized by additive translation invariance. Two additional fundamental transforms—the Mellin transform and the Z-transform—extend this unification to multiplicative scaling invariance and discrete-time systems, respectively. Both can be rigorously embedded as specialized projections of the MOC kernel under appropriate geometric constraints.

1 Mellin Transform

The Mellin transform is naturally associated with the multiplicative structure of the positive real line, rather than the additive structure of \mathbb{R}.

MOC Configuration

- Manifold: \mathcal{M} = \mathbb{R}_+ (half-line of positive reals, multiplicative group)

- Curvature interpretation: associated with power-law behavior and scale invariance

- Origin: effectively encoded in the scaling structure via coordinate transformation

- Generalized wave number: complex exponent s = \sigma + i\omega

MOC Projection Form

Under the coordinate change x = e^{-t} that converts multiplication to addition, the MOC kernel yields the Mellin transform:

\mathcal{M}\{f\}(s)

=

\int_0^\infty x^{s-1}\,f(x)\,dx

Interpretation

The Mellin transform appears as the MOC projection on a multiplicatively structured manifold. It is the natural counterpart of the Laplace transform for scale-invariant phenomena, frequently appearing in number theory, asymptotic analysis, and fractal geometry.

2 Z-Transform

The Z-transform serves as the cornerstone of discrete-time signal processing and linear systems, representing the discrete analog of the Laplace transform.

MOC Configuration

- Manifold: discrete lattice \mathcal{M} = \mathbb{Z}

- Origin: discrete reference point n = 0

- Geometry: additive translation on integers

- Integral reduces to discrete summation

- Complex variable z replaces the continuous complex frequency

MOC Projection Form

By discretizing the MOC integral into a sum, one recovers the bilateral Z-transform:

\mathcal{Z}\{x\}(z)

=

\sum_{n=-\infty}^\infty x[n]\,z^{-n}

Interpretation

The Z-transform is the discrete restriction of the MOC projection. It corresponds to a sampled, discrete version of the Laplace transform, widely used in digital filtering, control theory, and time-series analysis.

3 Unified Hierarchy

Both transforms fit naturally into the MOC geometric hierarchy:

- Laplace transform: additive continuous geometry

- Mellin transform: multiplicative continuous geometry

- Z-transform: additive discrete geometry

Together with the previously discussed transforms, they demonstrate that the MOC framework unifies the full landscape of classical integral and discrete transforms under a single projective geometric principle.