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In this paper functions f belonging to L2 (R) are considered which spectrum is contained in a 'multi-band' set E, i.e. in a subset of the real axis which is the union of finite many intervals. For such functions a generalization of the Whittaker-Kotelnikov-Shannon sampling formula is given. The considered problem is also related to Riesz bases of exponentials in L2 (E). In the first part of this work we consider sets E consisting of regularly positioned intervals of the same length.

Abstract. A theory of matrix-valued functions from the matricial Smirnov class 1J1;t (]]JJ) is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman's theorem are presented for matrix-valued functions from the Smirnov class 1J1;t(]]JJ). We also consider a family F>.. = F -AI of functions belonging to the matricial Smirnov class which is indexed by a complex parameter A. We show that with the exception of a "very small" set of such A the corresponding inner factor in the inner-outer factorization of the function F>.. is a Blaschke-Potapov product.The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class. Notationse -the complex plane. '1:= {t E e : It I = I} -the unit circle. IIJ):= {z E e : Izl < I} -the unit disc.~1l' -the a-algebra of Borel subsets of'l. m -normalized Lebesgue measure on the measurable space ('I, ~1l').en -the n-dimensional complex space equipped with the usual Euclidean In -the n X n unit matrix.

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