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This paper introduces a novel framework of creatively equipped convex topological spaces to address challenges in theoretical forward models. We extend the Gelfand–Shilov technique to generalize the Laplace–Stieltjes transform via a simple objective function that combines two transforms in the distributional sense. Appropriate domains support harmonic analysis under distributional conditions. The framework offers tools for modeling and analyzing Cauchy problems and PDEs with distributional data. The primitive operator provides clear explanations in general settings. Extending to higher dimensions yields faster decay rates, even in polynomial cases, relying on strong continuity at the origin implying continuity at every point. It accommodates solutions lacking continuity at zero, common in practical applications.