Vector versions of Prony’s algorithm and vector-valued rational approximations.

Given the scalar sequence { f m } m = 0 ∞ that satisfies f m = ∑ i = 1 k a i ζ i m , m = 0 , 1 , … , <graphic href="10444_2020_9751_Article_Equa.gif"></graphic> where a i , ζ i ∈ ℂ and ζi are distinct, the algorithm of Prony concerns the determination of the ai and the ζi from a finite number of the fm. This algorithm is also related to Padé approximants from the infinite power series ∑ j = 0 ∞ f j z j . In this work, we discuss ways of extending Prony’s algorithm to sequences of vectors { f m } m = 0 ∞ in ℂ N that satisfy f m = ∑ i = 1 k a i ζ i m , m = 0 , 1 , … , <graphic href="10444_2020_9751_Article_Equb.gif"></graphic> where a i ∈ ℂ N and ζ i ∈ ℂ . Two distinct problems arise depending on whether the vectors <bold>a</bold>i are linearly independent or not. We consider different approaches that enable us to determine the <bold>a</bold>i and ζi for these two problems, and develop suitable methods. We concentrate especially on extensions that take into account the possibility of the components of the <bold>a</bold>i being coupled. One of the applications we consider concerns the case in which f m = ∑ i = 1 r a i ζ i m , m = 0 , 1 , … , r large , <graphic href="10444_2020_9751_Article_Equc.gif"></graphic> and we would like to approximate/determine of a number of the pairs (ζi, <bold>a</bold>i) for which |ζi| are largest. We present the related theory and provide numerical examples that confirm this theory. This application can be extended to the more general case in which f m = ∑ i = 1 r p i (m) ζ i m , m = 0 , 1 , … , <graphic href="10444_2020_9751_Article_Equd.gif"></graphic> where p i (m) ∈ ℂ N are some (vector-valued) polynomials in m, and ζ i ∈ ℂ are distinct. Finally, the methods suggested here can be extended to vector sequences in infinite dimensional spaces in a straightforward manner. [ABSTRACT FROM AUTHOR]