Telescoping Approximations for Partial Differential Equations via Semigroup Methods

We extend the telescoping approximation framework developed for constants, functions, ordinarydifferential equations, and integral operators to partial differential equations. Focusingon linear evolution equations generated by differential operators, we construct telescoping approximantsfor PDE solutions via semigroup theory.For the heat equation, wave equation, and Schr¨odinger equation, we show that replacing theexact semigroup etA by optimally telescoping exponential approximants yields solution sequenceswhose successive differences decay as O(n−(k+1)) uniformly on compact time intervals. Wefurther establish telescoping approximations for inhomogeneous PDEs via Duhamel’s principle.This completes the linear telescoping theory within the present framework, from constantsto PDEs, and provides a foundation for future work on nonlinear equations and perturbativemethods.