Asian Research Journal of Mathematics

Hyperfunctions provide a complex analytic framework for representing singular objects that cannot always be treated adequately by ordinary functions or distributions. The Mellin transform is a central tool for analysing scaling behaviour and asymptotic properties, while Tauberian theorems give conditions under which information about a transform determines corresponding properties of the original object. This paper establishes a Tauberian theorem for the Mellin transform of measurable hyperfunctions having bounded exponential growth and support contained in [1,∞). After recalling the necessary notions concerning hyperfunctions, support, singular support, bounded exponential growth, Mellin transforms, and Dirichlet integrals, the study relates the Mellin transform of a hyperfunction on the positive real axis to the Laplace transform under the substitution y = e^−s. The main result assumes a measurable hyperfunction g(y) with |g(y)| ≤ \(\frac{N}{y}\) for y > 0 and a Mellin transform that extends holomorphically to an open set containing the half-plane Re t < 1. Under these hypotheses, the integral \(\int ^∞_0\) g(y)dy is shown to converge, and its value is identified with the holomorphic continuation at t = 1, namely ˆg(1). The result adapts a Tauberian theorem for Dirichlet integrals to the setting of hyperfunctions and clarifies the link between Mellin-transform behaviour and convergence of the original hyperfunction integral.