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The Complex Representer Theorem

Natanael Alpay, Antonino De Martino, and Kamal Diki

Preprint

Research Summary

We develop a representer theorem for complex reproducing kernel Hilbert spaces. It reduces suitable minimization problems over a function space to finite combinations of kernel functions, connecting the infinite-dimensional theory with computable models.

Applications include constructing input–output data for prescribed regression solutions in Fock and Hardy spaces, with examples involving superoscillations, Gaussian radial basis functions, and finite Blaschke products.

Latest preprint on arXiv

Publication Details

• N. Alpay, A. De Martino, and K. Diki. Representer Theorem in Complex Reproducing Kernel Hilbert Spaces with Applications to Fock and Hardy Spaces and Superoscillations. Preprint, arXiv:2604.22165 (2026).