Carleson Frame Methods for Pantograph‐Type Delay Differential Equations: Theory and Applications

ABSTRACT We propose a unified analytic‐numerical method for pantograph‐type delay differential equations based on Carleson frames. The construction exploits the hyperbolic separation of Carleson sequences in the unit disk together with the diagonal action of contraction‐induced composition operators on Hardy spaces. Within this setting, solutions are represented through truncated orbits of a bounded operator, leading to redundant but stable frame expansions that naturally reflect the intrinsic self‐similarity of proportional‐delay dynamics. We establish well‐posedness of the discrete scheme, stability of the expansion coefficients, and quasi‐optimal convergence, and we further obtain spectral (geometric) convergence for analytic solutions. For a canonical generator, we derive explicit closed‐form expressions for the Gram, delay, and derivative matrices, enabling an efficient and numerically robust implementation. Numerical experiments for both linear and nonlinear pantograph equations confirm the theoretical predictions, demonstrating high accuracy, favorable conditioning, and stable behavior under refinement. To the best of our knowledge, this work provides one of the first systematic numerical frameworks for proportional‐delay systems based on Carleson frames, and highlights their potential as a mathematically transparent and effective tool for the analysis and computation of such problems.