5.2 Convergence of Padé approximants

Padé approximants are rational functions used to approximate complex functions. They consist of polynomial ratios that match Taylor series expansions, often capturing function behavior better than polynomials alone, especially for functions with poles or singularities.

Convergence of Padé approximants refers to how well they approximate the original function as polynomial degrees increase. Different types include convergence in capacity, measure, pointwise, and uniform convergence. Understanding these properties is crucial for determining the effectiveness of Padé approximants in various applications.

Definition of Padé approximants

• Rational functions used to approximate more complex functions
• Consists of a ratio of two polynomials $P_m(x)$ and $Q_n(x)$ of degrees $m$ and $n$ respectively
• Constructed to match the Taylor series expansion of the function being approximated up to a certain order
• Provides a way to extend the accuracy of approximations beyond what is possible with Taylor series alone
• Can often capture the behavior of functions with poles or other singularities better than polynomial approximations

Convergence properties

• Convergence of Padé approximants refers to how well they approximate the original function as the degrees of the polynomials increase
• Different types of convergence, such as convergence in capacity, measure, pointwise, and uniform convergence
• Understanding these convergence properties is crucial for determining the effectiveness and reliability of Padé approximants in various applications

Convergence in capacity

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• Measures how well the Padé approximants converge to the original function on average over a given set
• Defined in terms of the Lebesgue measure of the set where the approximation error exceeds a certain threshold
• Provides a global sense of convergence, taking into account the behavior of the approximants over the entire domain
• Useful for understanding the overall effectiveness of Padé approximants in approximating a function

Convergence in measure

• Assesses the convergence of Padé approximants by considering the measure of the set where the approximation error is large
• Convergence in measure implies that the measure of the set where the approximation error exceeds a given threshold tends to zero as the degrees of the polynomials increase
• Stronger than convergence in capacity but weaker than pointwise convergence
• Helps determine the extent to which Padé approximants accurately represent the original function over most of the domain

Pointwise convergence

• Evaluates the convergence of Padé approximants at individual points in the domain
• Pointwise convergence means that for each fixed point $x$, the sequence of Padé approximants converges to the value of the original function at that point
• Does not necessarily imply uniform convergence, as the rate of convergence may vary from point to point
• Important for understanding the local behavior of Padé approximants and their ability to approximate the function at specific locations

Uniform convergence

• Strongest form of convergence for Padé approximants
• Uniform convergence means that the maximum approximation error over the entire domain tends to zero as the degrees of the polynomials increase
• Implies pointwise convergence and convergence in measure
• Guarantees that the Padé approximants provide a uniformly accurate approximation of the original function over the whole domain
• Crucial for applications where a globally reliable approximation is required

Montessus de Ballore theorem

• Fundamental result in the theory of Padé approximants, named after Robert de Montessus de Ballore
• Establishes conditions under which a sequence of Padé approximants converges uniformly to a given function

Statement of theorem

• Let $f(z)$ be a function analytic in a disk $D_R = \{z : |z| < R\}$ except for a finite number of poles in the disk
• Denote the poles by $z_1, z_2, \ldots, z_k$ with multiplicities $m_1, m_2, \ldots, m_k$ respectively
• If $n \geq \sum_{i=1}^k m_i$, then the sequence of Padé approximants $[n/n]_f(z)$ converges uniformly to $f(z)$ on compact subsets of $D_R \setminus \{z_1, z_2, \ldots, z_k\}$

Proof of theorem

• Involves complex analysis techniques such as Cauchy's integral formula and the residue theorem
• Key steps include representing the error between the function and its Padé approximant using contour integrals and bounding the error using estimates on the growth of the function and its Padé approximant
• Relies on the fact that the denominator polynomial of the Padé approximant increasingly captures the pole structure of the function as the degree increases

Applications of theorem

• Provides a theoretical foundation for the use of Padé approximants in approximating functions with poles
• Justifies the effectiveness of Padé approximants in applications such as analytic continuation, where the goal is to extend a function beyond its initial domain of definition
• Helps guide the choice of the degrees of the numerator and denominator polynomials in constructing Padé approximants for specific functions

Nuttall-Pommerenke theorem

• Generalizes the Montessus de Ballore theorem to functions with branch points instead of poles
• Deals with the convergence of Padé approximants for functions with algebraic or logarithmic singularities

Statement of theorem

• Let $f(z)$ be a function analytic in a domain $\Omega$ except for a finite number of branch points $z_1, z_2, \ldots, z_k$
• Assume that $f(z)$ has a Puiseux expansion (a generalized power series) around each branch point
• If certain conditions on the degrees of the numerator and denominator polynomials are satisfied, then the sequence of Padé approximants converges uniformly to $f(z)$ on compact subsets of $\Omega \setminus \{z_1, z_2, \ldots, z_k\}$

Proof of theorem

• Builds upon the ideas used in the proof of the Montessus de Ballore theorem
• Involves a careful analysis of the Puiseux expansions of the function around the branch points
• Uses estimates on the growth of the function and its Padé approximants to bound the approximation error

Applications of theorem

• Extends the applicability of Padé approximants to functions with more general types of singularities
• Relevant for approximating functions arising in physical problems, such as the solution of certain differential equations with singular coefficients
• Provides a theoretical basis for the use of Padé approximants in the study of multivalued complex functions and their Riemann surfaces

Padé approximants vs power series

• Padé approximants offer several advantages over traditional power series approximations, but also have some limitations

Advantages of Padé approximants

• Can provide better approximations than truncated power series, especially near poles or other singularities
• Often exhibit faster convergence than power series, requiring fewer terms to achieve a given level of accuracy
• Can be used to approximate functions that are not analytic or have a finite radius of convergence
• Useful for extrapolation, as they can capture the behavior of functions outside the domain where the power series is valid

Limitations of Padé approximants

• Construction of Padé approximants requires solving a linear system of equations, which can be computationally expensive for high degrees
• The choice of the degrees of the numerator and denominator polynomials is not always straightforward and may require trial and error
• Padé approximants are not guaranteed to converge for all functions, and their convergence properties can be sensitive to the location and nature of singularities
• In some cases, Padé approximants may exhibit spurious poles or other artifacts that are not present in the original function

Computation of Padé approximants

• Several methods exist for computing Padé approximants, each with its own advantages and trade-offs

Recursive algorithms

• Based on the recursive relations satisfied by the coefficients of the numerator and denominator polynomials
• Efficient for computing Padé approximants of low to moderate degrees
• Examples include the Viskovatov method and the Thacher-Tukey algorithm
• Can be implemented using simple arithmetic operations and avoid the need for solving large linear systems

Linear algebra methods

• Involve setting up and solving a linear system of equations to determine the coefficients of the Padé approximant
• Suitable for high-degree Padé approximants or when the recursive algorithms become unstable
• Can be solved using standard linear algebra techniques such as Gaussian elimination or QR factorization
• May require the use of high-precision arithmetic to avoid numerical instabilities, especially for ill-conditioned systems

Applications of convergence properties

• The convergence properties of Padé approximants make them valuable tools in various areas of applied mathematics and scientific computing

Analytic continuation

• Padé approximants can be used to extend the domain of a function beyond its initial region of definition
• By approximating a function in a region where it is analytic and then evaluating the approximant outside that region, one can estimate the values of the function in a larger domain
• Particularly useful for functions with singularities or branch cuts that limit the applicability of power series methods

Solving differential equations

• Padé approximants can be employed to find approximate solutions to ordinary and partial differential equations
• By representing the solution as a rational function and matching the derivatives of the approximant with the differential equation, one can obtain a system of equations for the coefficients of the Padé approximant
• The resulting approximation can provide insights into the qualitative behavior of the solution and serve as a starting point for more refined numerical methods

Approximating special functions

• Many special functions, such as the gamma function, Bessel functions, and hypergeometric functions, have complicated series expansions or integral representations that can be difficult to evaluate numerically
• Padé approximants offer a way to construct accurate and efficient approximations to these functions over a wide range of arguments
• By using the known series expansion or asymptotic behavior of the function to generate the Padé approximant, one can obtain approximations that are valid in regions where the original series may converge slowly or diverge
• These approximations can be used to speed up computations and to develop robust numerical algorithms for evaluating special functions in scientific and engineering applications