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 ζ_i are distinct, the algorithm of Prony concerns the determination of the a_i and the ζ_i from a finite number of the f_m. This algorithm is also related to Padé 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 a_i are linearly independent or not. We consider different approaches that enable us to determine the a_i and ζ_i for these two problems, and develop suitable methods. We concentrate especially on extensions that take into account the possibility of the components of the a_i 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 (ζ_i, a_i) for which |ζ_i| 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.