Spectral Reconstruction in Fractional Derivative Order

Abstract

We introduce the spectral reconstruction operator $\mathcal{S}$, which treats derivative order as a continuous variable and integrates fractional derivative data across all nonnegative real orders to represent analytic functions. Because these derivatives draw on function values across an infinite stretch of the real line, the spectral integrand encodes information about $f$ that no individual integer derivative at the origin can capture. We define an admissible class of fractional derivative families that recover the classical derivative ladder, satisfy the semigroup property, and keep $D^r f(0)$ finite (a condition that excludes monomials), and prove that the operator asymptotically reconstructs the exponential function as $x\to\infty$. The spectral operator and the classical Maclaurin series evaluate the same underlying function of derivative order, one by integration and the other by summation; classical summation formulas quantify this discrepancy under their respective hypotheses. Euler--Maclaurin gives a finite correction series with remainder; Abel--Plana gives an exact correction integral. Examples from entire, oscillatory, rapidly decaying, special-function, and finite-radius analytic classes, including a real pole and a complex-conjugate pole pair, demonstrate stable reconstruction; the operator's Abel--Plana correction matches $f$ to full computational precision in every case. The spectral integral and Maclaurin series share the same convergence boundary in the finite-radius examples, though through different mathematics. The discrepancy remains bounded, so the uncorrected operator is already useful where $f(x)$ is large; the leading term $\mathcal{E}_0=\frac{1}{2}f(0)$ and higher terms account for the rest near the origin. These results establish derivative order as a meaningful, non-local continuous domain for representing and reconstructing analytic functions.