ABSTRACT This paper introduces a weighted ‐deformed Gram matrix framework for quantum‐inspired learning systems, with particular emphasis on applications in biomedical signal analysis. The proposed construction is based on ‐orthogonal basis functions combined with weighted Jacobi‐type kernels, enabling feature representations on nonuniform and memory‐dependent domains. We establish fundamental structural properties of the resulting Gram matrix, including symmetry, positive semi‐definiteness, and, under mild conditions, strict positive definiteness, ensuring stability, invertibility, and robustness in kernel‐based algorithms. A Mercer‐type characterization is further derived, confirming the validity of the associated kernel within a reproducing kernel Hilbert space. The roles of the deformation parameters are explicitly interpreted, where governs structural heterogeneity and discretization effects, while encodes temporal memory and scaling behavior. Numerical experiments, including heatmap visualizations and Mercer spectral decompositions, demonstrate the effectiveness of the proposed framework in capturing long‐range dependencies and improving kernel conditioning. The results indicate that the ‐Gram matrix provides a flexible and physically meaningful tool for advancing quantum‐inspired and data‐driven methodologies in engineering and biomedical applications.
Ibrahim et al. (Mon,) studied this question.