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

Abstract

Given the scalar sequence $\{f_{m}\}^{\infty }_{m=0}$ that satisfies $$ f_{m} = \sum\limits_{i=1}^{k} {a_{i}}{\zeta}_{i}^{m},\quad m=0,1,\ldots, $$ where $a_{i}, \zeta _{i}\in \mathbb {C}$ 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 Pade approximants from the infinite power series ${\sum }^{\infty }_{j=0}f_{j}z^{j}$. In this work, we discuss ways of extending Prony’s algorithm to sequences of vectors ${\{\boldsymbol {f}_{m}\}}^{\infty }_{m=0}$ in $\mathbb {C}^{N}$ that satisfy $$ \boldsymbol{f}_{m} = \sum\limits_{i=1}^{k} \boldsymbol{a}_{i} {\zeta}_{i}^{m}, \quad m=0,1,\ldots, $$ where $\boldsymbol {a}_{i}\in \mathbb {C}^{N}$ and $\zeta _{i}\in \mathbb {C}$. Two distinct problems arise depending on whether the vectors ai are linearly independent or not. We consider different approaches that enable us to determine the ai 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 ai being coupled. One of the applications we consider concerns the case in which $$ \boldsymbol{f}_{m} = \sum\limits_{i=1}^{r} \boldsymbol{a}_{i} {\zeta}_{i}^{m}, \quad m=0,1,\ldots,\quad r \text{ large}, $$ and we would like to approximate/determine of a number of the pairs (ζi, ai) 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 $$ \boldsymbol{f}_{m} = \sum\limits_{i=1}^{r} \boldsymbol{p}_{i} (m){\zeta}_{i}^{m}, \quad m=0,1,\ldots, $$ where $\boldsymbol {p}_{i}(m)\in \mathbb {C}^{N}$ are some (vector-valued) polynomials in m, and $\zeta _{i}\in \mathbb {C}$ are distinct. Finally, the methods suggested here can be extended to vector sequences in infinite dimensional spaces in a straightforward manner.