We define a degenerate q-Laplace transform on the bilateral Jackson grid using Jackson’s improper q-integral and the kernel \((1+\lambda t)^{-s/\lambda}\). Estimating the two grid tails separately yields explicit convergence conditions and weighted Jackson data spaces on which the transform is bounded and holomorphic. On a vertical line, the change of variable \(x = \lambda^{-1}\log(1+\lambda t)\) turns the transform into an absolutely convergent pure-point Fourier–Stieltjes series; mean coefficient extraction then recovers every Jackson-grid value and identifies the exact off-grid null space. This is recovery on the sampled Jackson grid, not ordinary inversion at off-grid points. The kernel also determines the semigroup law \(t \oplus_\lambda u = t + u + \lambda t u\). With this law, finite measures form a convolution Banach algebra and the transform is multiplicative. A small-total-variation hypothesis gives a Neumann resolvent, together with norm and perturbation bounds, for a linear hereditary measure equation. For the quadratic hereditary Volterra equation, local well-posedness follows from a contraction argument. Numerical experiments examine multi-parameter Neumann convergence, the \(q \to 1^-\) Jackson-to-Riemann limit, the \(\lambda \to 0^+\) deformed-to-additive convolution limit, symmetric and asymmetric bilateral truncations, and transform-domain Picard convergence for the nonlinear model.
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A Degenerate q-Laplace Transform on the Jackson Grid: Fourier–Mean Recovery, Deformed Convolution, and Computational Hereditary Models
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Received: 29 August 2026 | Revised: 15 September 2026 | Accepted: 22 October 2026 | Published: 26 October 2026
Abstract
Keywords
degenerate q-Laplace transform | Jackson q-integral | Fourier–mean recovery | deformed convolution | hereditary Volterra equation | numerical experiments
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