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Sum-of-Squares Decomposition via Generalized KYP Lemma

The Kalman-Yakubovich-Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) of a proper rational function and a linear matrix inequality (LMI). A recent result generalized the KYP lemma to characterize an FDI of a possibly nonproper rational function on a portion of a curve on the complex plane. This note examines implications of the generalized KYP result to sum-of-squares (SOS) decompositions of matrix-valued nonnegative polynomials of a single complex variable on a curve in the complex plane. Our result generalizes and unifies some existing SOS results, and also establishes equivalences among FDI, LMI, and SOS.