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2010 Mathematics Subject Classification: Primary: 41A21 [MSN][ZBL]

The field concerned with best rational approximation to power series. Let

$$\tag{1 } f( z) = \sum _ { k= } 0 ^ \infty f _ {k} z ^ {k}$$

be an arbitrary power series (formal or convergent), and let $R _ {n,m}$( $n, m \geq 0$) be the class of all rational functions of type $p/q$ where $p$ and $q$ are polynomials in $z$, $\mathop{\rm deg} q \leq m$, $\mathop{\rm deg} p \leq n$ and $q \not\equiv 0$. A Padé approximant of type $( n, m)$ to the power series (1) (the function $f$) is a rational function $\pi _ {n,m} \in R _ {n,m}$ having the maximum possible order of contact in the class $R _ {n,m}$ with the power series (1) at the point $z= 0$. More precisely, the function $\pi _ {n,m}$ is determined by the condition

$$\sigma ( f - \pi _ {n,m} ) = \max \{ {\sigma ( f- r) } : {r \in R _ {n,m} } \} ,$$

where $\sigma ( \phi )$ is the index of the first non-zero coefficient of the series

$$\phi = \sum _ { k= } 0 ^ \infty \phi _ {k} z ^ {k} .$$

It is also possible to determine the function $\pi _ {n,m}$ as the quotient $p/q$ of arbitrary polynomials $p$ and $q$ $( q \not\equiv 0)$ satisfying the conditions

$$\mathop{\rm deg} p \leq n ,\ \mathop{\rm deg} q \leq m ,$$

$$\tag{2 } ( qf - p)( z) = A _ {n,m} z ^ {n+} m+ 1 + \dots .$$

For fixed $n, m$ there exists a unique Padé approximant $\pi _ {n,m}$ to the power series (1). The table $\{ \pi _ {n,m} \} _ {n,m=} 0 ^ \infty$ is called the Padé table of the series (1). The sequences of type $\{ \pi _ {n,m} \} _ {n=} 0 ^ \infty$ are called the rows of the Padé table (the zero row coincides with the sequence of Taylor polynomials of $f$); $\{ \pi _ {n,m} \} _ {m=} 0 ^ \infty$ are called the columns of the Padé table; and $\{ \pi _ {n+} j,n \} _ {j=} 0 ^ \infty$ are called the diagonals of the Padé table. The most important special case $j= 0$ is the principal diagonal.

The calculation of the functions $\pi _ {n,m}$ reduces to the solution of a system of linear equations whose coefficients are expressed in terms of the coefficients $f _ {k}$, $k = 0 \dots n+ m$, of the given power series. If the Hankel matrix $\Delta _ {n,m}$,

$$\Delta _ {n,m} = \left | \begin{array}{cccc} f _ {n-} m+ 1 &f _ {n-} m+ 2 &\dots &f _ {n} \\ \dots &\dots &\dots &\dots \\ f _ {n} &f _ {n+} 1 &\dots &f _ {n+} m- 1 \\ \end{array} \right | ,$$

has determinant non-zero, then the denominator $q _ {n,m}$ of the function $\pi _ {n,m}$ is given by the formula

$$q _ {n,m} ( z) = \frac{1}{ \mathop{\rm det} ( \Delta _ {n,m} ) } \left | \begin{array}{cc} \Delta _ {n,m} &f _ {n+} 1 \\ {} &\cdot \\ {} &\cdot \\ {} &\cdot \\ {} &f _ {n+} m \\ z ^ {m} \dots z & 1 \\ \end{array} \right |$$

(the normalization is $q _ {n,m} ( 0) = 1$; an explicit formula can also be written down for the numerator of the function $\pi _ {n,m}$). Moreover,

$$( f - \pi _ {n,m} )( z) = A _ {n,m} z ^ {n+} m+ 1 + \dots .$$

The latter relation is sometimes taken as the definition of a Padé approximant; in this case a Padé approximant need not exist for certain $( n, m)$. The Padé approximant of type $( n, m)$ of the given power series $f$ is often denoted by the symbol

$$[ n/m ] = [ n/m ] _ {f} .$$

For an effective calculation of Padé approximants it is more convenient to make use not of explicit formulas but of recurrence relations existing in the Padé table. A large number of algorithms have been constructed for the automatic calculation of a Padé approximant; these problems are of specific importance in connection with applications (see [17], [18]).

The first general problem concerning the interpolation of given values of a function at $n+ m+ 1$ different points by means of rational functions of class $R _ {n,m}$ was considered by A.L. Cauchy [1]; C.G.J. Jacobi [2] extended Cauchy's results to the case of multiple-point interpolation. The case of one $( n+ m+ 1)$- multiple point corresponds to Padé approximation. The idea of Padé approximation was formulated at the end of the 19th century within the classical theory of continued fractions (G. Frobenius [3], H. Padé [4]). Fundamental results on diagonal Padé approximants were obtained by P.L. Chebyshev, A.A. Markov and T.J. Stieltjes in terms of continued fractions. They discovered and studied the relations of diagonal Padé approximants with orthogonal polynomials, quadrature formulas, moment problems, and other problems of classical analysis (see [7]–). The origin of the study of the rows of the Padé table lays in the work on the radius of meromorphy of a function defined by a power series and on the convergence of the row of the Padé table in the discs of meromorphy (see [5], [6]).

From the beginning of the 20th century onwards Padé approximation has become an independent object of analysis and constitutes an important chapter in the theory of rational approximation of analytic functions. Using local data (coefficients of a power series) for their construction, they allow one to study global properties of the corresponding analytic function (analytic continuation, the character and distribution of singularities, etc.) and to compute the value of a function outside the disc of convergence of the power series.

Along with classical Padé approximation, various generalizations have been considered: general interpolation processes by means of rational functions with free poles (multiple-point Padé approximation); rational approximation of series with respect to given systems of polynomials (e.g. with respect to orthogonal polynomials); joint Padé approximation (Padé–Hermite approximation); rational (Padé type) approximation of power series of several variables, and other topics.

#### References

Over the last ten years there has been an enormous increase in the number of new results concerning Padé and Hermite–Padé approximation. One of the main influences has been the interplay between this field and the fields of continued fractions, orthogonal polynomials, moment problems, potential theory, and functional analysis.

Also, the generalizations in the directions of two-point Padé approximation, simultaneous approximation (already indicated in the work by Ch. Hermite and Padé) and multivariate approximation have led to new results.

Most of the relevant publications can be found in the proceedings of the many conferences on the subject and in recent volumes of the "Journal of Approximation Theory" or the "Journal of Constructive Approximation"

Reference [a2] is a valuable and interesting historical survey of the subject. The article by C. Brezinski in [a3] contains a rather extensive bibliography; a much more complete one has been compiled by Brezinski and is available on request.