A Continuous-Order Integral Operator for Maclaurin-type Reconstruction

Abstract

We introduce a continuous-order integral analog of the Maclaurin expansion that reconstructs analytic functions from fractional derivative data. The operator integrates over continuous order, replacing the discrete sum of integer derivatives in the classical Maclaurin series. We identify structural admissibility conditions on the fractional derivative that constrain the order data to form a coherent extension of the classical derivative ladder and to remain finite at the anchor. These conditions restrict admissible definitions to the Riemann-Liouville and Liouville (Fourier-multiplier) derivatives, or to continuations that coincide with them. Under smoothness and decay assumptions on the order data $D^r f(0)$, the continuous-order operator reconstructs $f$ approximately. It differs from the classical Maclaurin series by a systematic sum-integral mismatch. The Euler-Maclaurin summation formula quantifies this mismatch and yields a natural correction hierarchy. In examples drawn from distinct analytic function classes (entire, oscillatory, finite-radius, rapidly decaying, and special functions), the operator with its correction terms yields stable reconstruction, with mean absolute error reduced from $10^{-1}$ to $10^{-3}$ or smaller after three correction terms. Monomials form a degenerate case, as their order spectrum collapses to a single point and cannot be recovered by the continuous-order integral alone. These results establish the continuous-order operator as an integral counterpart to the Maclaurin series, extending a classical discrete construction into a spectral representation that is continuous in derivative order.