On Reciprocal Sequences of Matricial Carathéodory Sequences and Associated Matrix Functions
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This paper provides a detailed discussion of reciprocal sequences of finite and infinite matricial Carathodory sequences, including an examination of the relationship a reciprocal sequence has to its original sequence. The properties of such reciprocal sequences are described. Of particular interest is the fact that the reciprocal sequence of a Carathodory sequence is again a Carathodory sequence. Later, these results are applied to matrix-valued functions. The natural focus is, in particular, on the matricial Carathodory functions associated with a matricial Carathodory sequence and its reciprocal sequence. These matrix functions are then shown to be Moore-Penrose inverses of each other. The implications of these results for non-negative Hermitian matrix measures on the unit circle are also discussed.
KeywordsReciprocal sequences Moore-Penrose inverse matrix-valued Carathodory functions
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