<p>This paper introduces the extended Hermite wavelet and presents a novel computational framework for solving systems of linear Fredholm integral equations using the extended Hermite wavelet technique. The method transforms integral equations into a system of algebraic equations with unknown wavelet coefficients, thereby reducing analytical complexity to linear algebraic computations. The proposed approach ensures rapid convergence, high numerical stability, and remarkable accuracy, with error estimates rigorously established for functions in Hölder’s class and its generalized forms. A key theoretical contribution of this work lies in establishing convergence properties and error estimates for functions belonging to Hölder’s class and its generalized variants. By deriving rigorous approximation theorems, the study demonstrates that extended Hermite wavelets yield highly accurate approximations with errors decaying rapidly as resolution parameters increase.</p>