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Spectral measures give rise to a natural harmonic analysis on the unit discvia a boundary representation of a positive matrix arising from a spectrum ofthe measure. We consider in this paper the reverse: for a positive matrix inthe Hardy space of the unit disc we consider which measures, if any, yield aboundary representation of the positive matrix. We prove a characterization ofthose representing measures via a matrix identity by introducing a new operatorproduct called the Abel Product.

A Characterization of Boundary Representations of Positive Matrices in the Hardy Space via the Abel Product