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Exact barycentric rational interpolation of analytic functions with an application to the Bessel functions

Brett R. Green

arXiv:2609.39805v1math.NAcs.NAmath.CA

TL;DR

The paper develops exact analytic-function expansions from barycentric rational interpolation, with an analytic weight determining the nodes and finite truncations providing approximations. Demonstrations on cosine and Bessel functions show that the approximations can have relatively consistent accuracy across the interpolation region, while their errors and convergence can be quantified.

  • Problem

    Barycentric rational interpolation has mainly been studied for arbitrary data rather than as a method of exact function expansion with general node placement and weighting.

    Source: §1 · p. 2

  • Method

    The method uses an analytic weight to set interpolation nodes and derives exact and finite-node barycentric expansions whose analytic remainders yield error bounds.

    Source: §1 · p. 2

  • Results

    The approximate barycentric expansion has relatively consistent accuracy across the interpolation region, unlike Taylor and asymptotic expansions, which prioritize small and large |z|, respectively.

    Source: §4.3 · pp. 23–24

  • Takeaways & Limitations

    The method was demonstrated for cosine and Bessel functions, and the error and convergence of its approximations are readily quantifiable.

    Source: §5 · p. 24

    Higher powers of the weight tend to improve accuracy but make the interpolation formulas more complex, and this accuracy gain is not guaranteed.

    Source: §2.2 · p. 8

Abstract

from arXiv · show

We derive a method of analytic function expansion using barycentric rational interpolation, in which an arbitrary analytic weight function defines the interpolation and nodes. Truncating these expansions yields successive approximations, and the derivation of the expansions gives the approximation error as a byproduct. We demonstrate the method with cosine and the Bessel functions of the first kind for several choices of weight functions, and we examine the accuracy and convergence of the approximate interpolations as a function of the weight and the number of nodes used in the interpolation.

1 Introduction

The paper develops exact function expansions from barycentric rational interpolation, extending the approach beyond arbitrary-data interpolation and equally spaced nodes. Choosing an analytic weight sets the nodes; truncation yields approximations whose errors can be bounded.

1 Introduction

  • The approach contrasts single-point expansions, such as Taylor and Padé, with expansions using limited information at many points.The passage presents Weierstrass products as one example of a nonlocal function expansion.Source: §1 · p. 1
  • The paper’s contribution is an exact barycentric expansion for general node placement and weighting, with error bounds for finite-node approximations.The authors contrast this with Berrut’s exact expansion for equally spaced nodes.Source: §1 · p. 2
  • An analytic weight function G(z) determines the interpolation nodes through its zeros.Residue theory is then used to express F(z)/G(z) and 1/G(z) in barycentric rational form.Source: §1 · p. 2
  • The method separates each meromorphic function into Laurent principal parts and an analytic remainder, with the remainders calculated by contour integration.Those contour integrals also provide bounds on finite-node approximation error.Source: §1 · p. 2
  • The method is demonstrated on cosine and the Bessel functions of the first kind.For the Bessel functions, the paper derives closed-form barycentric rational interpolation formulas.Source: §1 · p. 2

2 Constructing barycentric rational expansions

The construction expresses meromorphic functions as sums of pole-centered Laurent principal parts plus an analytic remainder, then applies this representation to numerator and denominator functions to form exact and finite-node barycentric expansions. The framework also characterizes interpolation error, convergence, node derivatives, and the effects of weight choice.

2 Constructing barycentric rational expansions

  • The construction proceeds by first separating meromorphic functions into principal parts and a remainder, then using that result to derive the barycentric expansion.This is the paper’s stated two-step construction.Source: §2 · p. 3

2.1 Expanding meromorphic functions as a sum over principal parts and an analytic remainder

  • For a finite region, the meromorphic function equals its pole-centered Laurent principal parts plus an analytic remainder, and enlarging the region adds pole terms.In the applications described, the remainder integrals can be evaluated in closed form only as the region tends to the whole complex plane.Source: §2.1 · p. 4
  • The exact expansion is obtained as the limit of expanding regions that eventually cover the complex plane, provided the remainder-integral limits converge and yield a convergent series.The proof invokes dominated convergence.Source: §2.1 · p. 4

2.2 Exact and approximate barycentric rational expansions in practice

  • The exact expansion is formed by applying the principal-part-plus-remainder representation to F(z)/G(z) and 1/G(z), while the finite-node version retains only poles within the chosen region.The poles of these functions provide the interpolation nodes.Source: §2.2 · p. 6
  • The approximate interpolation reproduces the function value and a number of derivatives at each node determined by the order of the weight’s zero there.The proposition states that a pole of order o_k guarantees the first o_k − 1 derivatives.Source: §2.2 · p. 6
  • Convergence is governed by whichever analytic remainder, numerator or denominator, converges more slowly.The remainder integrals can therefore be used to assess the convergence rate as nodes are added.Source: §2.2 · pp. 7–8
  • Increasing the power of the weight tends to reduce interpolation error, but higher powers produce more complex formulas with longer Laurent principal parts.The authors note that improved accuracy is not guaranteed because changing the weight can also increase the expansion errors.Source: §2.2 · p. 8
  • For square contours and functions with even or odd symmetry, the contour integral can be reduced to the right and top sides, with opposite sides canceling for opposite parities.The cancellation applies when the relevant indices have opposite parity.Source: §2.2 · pp. 8–9

3 Barycentric rational expansions for cosine

The paper derives exact barycentric expansions for cosine and examines how weight powers and node placement affect finite approximations. For the sine-squared weight, the analytic remainders vanish; changing the weight or nodes changes approximation accuracy and convergence.

3.1 Expansion weighted by sin2 z

  • With the sin² z weight, cosine’s analytic remainders vanish in the limit, leaving an exact expansion in principal parts at the nodes.Finite approximations converge at the rate of the slower numerator or denominator remainder; both scale as R_m^-1 in this case.Source: §3.1 · p. 11

3.2 Additional barycentric expansions of cosine

  • The cosine expansions vary the weight power or node set, and finite approximations are compared as nodes are added.The paper also considers nodes at zeros, extrema and zeros together, and midpoints between extrema and zeros.Source: §3.2 · p. 11, §3.2.2 · p. 14

3.2.1 Varying powers of the weight function

  • Increasing p decreases overall error while lengthening the Laurent principal part; on the real axis, with more nodes, p = 1 and p = 2 converge as R^-1.Source: §3.2.1 · p. 12

3.2.2 Weighting at different nodes

  • Among the tested node choices, p = 1 weighting at extrema and zeros has better convergence scaling and is most accurate except when relatively few nodes are used.Its error scales as R_m^-2 because the j = 0 term vanishes; symmetry and node spacing affect efficiency.Source: Figure 3 · p. 14, §3.2.2 · p. 15

4 Barycentric rational expansions for the Bessel functions

The Bessel expansions use zeros of a chosen weight function as interpolation nodes, with extrema, zeros, or both considered for different weights. Their accuracy and convergence depend on the weight and node count, and the extrema-weighted p = 2 expansion performs best in the reported comparisons.

4.1 Calculating the Laurent series principal parts

  • For Bessel-function expansions, the zero of Jq(z) at z = 0 depends on q, so the pole order there does too; elsewhere, coefficient systems have dimension 1 or 2.Principal-part coefficients are obtained in closed form at nodes zk except z0, where they are best found computationally.Source: §4 · p. 16, §4.1 · p. 18

4.2 Calculating the analytic remainders

  • The analytic remainders are evaluated using contour integrals, Bessel asymptotics, inequalities, and parity; for the p = 2 denominator example, the j = 0 integral gives a nonvanishing remainder.The corresponding integral vanishes as m grows for j ≥ 1, whereas the j = 0 case does not.Source: §4.2 · pp. 19–20

4.3 Comparison of Bessel function expansions

  • Among the tested Bessel interpolations, weighting at extrema with p = 2 has the greatest accuracy and fastest convergence.The comparison includes weighting at extrema, zeros, or both, with p = 1 or 2, except that the method is invalid for both extrema and zeros with p = 2.Source: §4 · p. 15, §4.3 · p. 20
  • Across Bessel orders, the p = 2 extrema-weighted error profile shifts with the nodes but retains essentially unchanged magnitude and shape.The comparison holds all even q and plots nodes z0 through z4; the broader accuracy comparison examines q = 0 and q = 1 near the origin as node counts increase.Source: Figure 5 · p. 21, §4.3 · p. 22, Figure 6 · p. 22
  • Analytic-remainder estimates are generally close to fitted convergence rates, though actual convergence exceeds the prediction in a few cases.Table 2 compares fitted slopes with exponents from the analytic-remainder integrals; the authors note that the bounds are sometimes loose.Source: §4.3 · p. 23, Table 2 · p. 23
  • The extrema-weighted p = 2 barycentric expansion maintains relatively consistent accuracy across the interpolation region, unlike the small- or large-|z| specialization of Taylor and asymptotic expansions.Its denominator's analytic remainder also gives the correct large-|z| limit of 0; for higher Bessel orders, asymptotic expansions are inaccurate over larger regions around z = 0.Source: §4.3 · pp. 23–24, Figure 8 · p. 24

5 Conclusion

The paper concludes that complex-analysis techniques yield barycentric rational expansions with quantifiable approximation error and convergence, demonstrated for cosine and Bessel functions. The appendices provide explicit cosine and Bessel expansion formulas and supporting coefficient calculations.

5 Conclusion

  • The method produces barycentric rational function expansions that approximate analytic functions accurately in a region from a few evaluations at surrounding nodes, with error and convergence readily quantified.The demonstrations cover cosine and Bessel functions of the first kind; the authors focus on natural nodes such as extrema and zeros.Source: §5 · pp. 24–25

A Barycentric rational expansions for cosine

B Barycentric rational expansions for the Bessel functions

B.1.1 Calculating the Laurent series principal parts

  • For the Bessel denominator's pole at z0 = 0 when q ≥ 2, the appendix obtains Laurent coefficients by solving a matrix equation using coefficients from the Bessel power series.At nonzero nodes the poles are simple, so the corresponding Laurent-coefficient equation is analytically trivial to solve.Source: §B.1.1 · p. 28

B.1.2 Calculating the analytic remainders

The contour-integral analysis shows that the analytic remainders vanish as the contour covers the complex plane, leaving the Laurent principal parts. The Bessel expansions then specify their poles and coefficients through the stated matrix equations.

B.1.2 Calculating the analytic remainders

  • As the contour expands to cover the complex plane and include all poles, the analytic remainders vanish, leaving only the Laurent principal parts.The remainder integrals either vanish in the limit or cancel one another.Source: §B.1.2 · p. 30

B.2 Bessel functions weighted at extrema with p = 2

  • The Bessel expansion gives an expression using Bessel-function zeros and Laurent principal-part coefficients obtained from a matrix equation.The zeros exclude z0 = 0, and the coefficients are specified by Eqs. (95) and (93).Source: §B.2 · pp. 30–31

B.2.1 Calculating the Laurent series principal parts

  • The Laurent principal-part coefficients are derived by solving a matrix equation separately for the numerator and denominator, with coefficients determined by the stated index and parameter choices.The passage specifies the coefficient setup and notes that other coefficients vanish.Source: §B.2.1 · p. 31

B.2.2 Calculating the analytic remainders

For the denominator remainder, contour integrals with j ≥ 1 vanish in the large-contour limit, while the j = 0 term remains. The numerator integrals vanish in that limit.

B.2.2 Calculating the analytic remainder

  • Denominator contour integrals vanish as m → ∞ for j ≥ 1, but the j = 0 integral does not vanish.The surviving term contributes to the denominator's analytic remainder.Source: §B.2.2 · p. 32
  • The top-side contour integrals vanish as m → ∞ for every j because of exponential factors.Source: §B.2.2 · p. 33
  • In the large-contour limit, the only nonvanishing denominator remainder integrals are the j = 0 top- and bottom-side integrals, which are equal by parity.These integrals determine the denominator analytic remainder.Source: §B.2.2 · p. 33

B.3 Bessel functions weighted at zeros with p = 1

The expansion for Bessel functions weighted at zeros with p = 1 is expressed through the Bessel zeros and their Laurent principal parts. The principal-part coefficients are obtained from the specified matrix equation.

B.3 Bessel functions weighted at zeros with p = 1

  • The section gives the resulting expansion for Bessel functions weighted at zeros with p = 1.Its poles include the Bessel zeros other than z0 = 0, with principal parts specified for those zeros and the origin.Source: §B.3 · p. 33
  • The Laurent coefficients for the denominator are calculated by solving the stated matrix equation with the parameters specified for this case.Source: §B.3 · p. 33

B.3.1 Calculating the Laurent series principal part

  • The pole at z0 = 0 uses a separate matrix setup, while the other poles are simple.The passage gives the coefficient pattern for the origin and identifies the remaining poles as simple.Source: §B.3.1 · p. 34

B.3.2 Calculating the analytic remainder

The analytic remainder of 1/Jq(z) is shown to equal that of 1/J′q(z), and hence is zero. The equality follows from matching contour-integral expansions.

B.3.2 Calculating the analytic remainder

  • The denominator analytic remainder is evaluated using contour integrals of the stated forms.The calculation begins with the right side of the contour.Source: §B.3.2 · p. 34
  • The contour-integral expansions for 1/Jq(z) match those for 1/J′q(z) on both the right and top sides.This makes their analytic remainders equal.Source: §B.3.2 · p. 34
  • The analytic remainders of 1/Jq(z) and 1/J′q(z) are equal, so the former is zero because the latter is zero.Source: §B.3.2 · p. 34

B.4 Bessel functions weighted at zeros with p = 2

The section gives the barycentric expansion for Bessel functions weighted at their zeros with p = 2 and describes how its Laurent principal parts are calculated.

B.4 Bessel functions weighted at zeros with p = 2

  • The numerator’s earlier calculations are reused from Sec. B.3; the new work is the denominator’s Laurent expansion and analytic remainder.The Laurent coefficients are calculated using matrix equations, including Eqs. (77) and (93).Source: §B.4 · p. 35
  • The resulting expansion uses the Bessel-function zeros other than z0 = 0, with nonzero-node Laurent coefficients given by Eq. (119).The Laurent principal part at z0 = 0 is specified separately.Source: §B.4 · p. 35

B.4.1 Calculating the Laurent series principal part

  • For the denominator’s pole at z0 = 0, the calculation uses Eq. (93) with parameters from Sec. B.3.1 and coefficients from the Bessel-function power series.The specified series coefficients apply for j ≥ 0, with all other terms vanishing.Source: §B.4.1 · p. 35

B.4.2 Calculating the analytic remainder

The section reports analytic-remainder relationships for Bessel-function expressions, then gives an expansion weighted at both extrema and zeros with p = 1 and describes its principal-part calculations.

B.4.2 Calculating the analytic remainder

  • The analytic remainders for 1/Jq(z) and 1/(Jq(z))^2 are related to those for derivative-based expressions, with the latter remainder stated as 1.The passage identifies the remainder for 1/Jq(z) as the same as for 1/(dJq(z)/dz).Source: §B.4.2 · p. 35

B.5 Bessel functions weighted at both extrema and zeros with p = 1

  • The resulting p = 1 expansion uses Bessel-function extrema and zeros other than z0 = 0, with nonzero-node principal-part coefficients specified in Eq. (125).The principal part at z0 = 0 is given separately.Source: §B.5 · p. 36

B.5.1 Calculating the Laurent series principal part

  • At z0 = 0, the principal-part calculation sets j0 = 0, j*0 = q, and j*1 = q − 1, using the stated Bessel-series coefficients.The other poles are simple, and their Laurent coefficients depend on whether a node is an extremum or a zero of Jq(z).Source: §B.5.1 · pp. 36–37

B.5.2 Calculating the analytic remainder

The analytic-remainder calculation combines contour-integral limits and parity cancellations; the p = 2 weighting at both extrema and zeros is reported as unsuitable under the paper’s method.

B.5.2 Calculating the analytic remainder

  • For the j = 0 numerator term, parity makes the right- and left-side contour contributions cancel, so that integral never contributes to the remainder.Source: §B.5.2 · p. 37
  • The contour contributions yield the analytic remainder by vanishing as m → ∞ or being zero by parity.For j ≥ 1, the specified contribution vanishes in the limit; for j = 0, parity cancels the right- and left-side numerator terms.Source: §B.5.2 · pp. 37–38

B.6 Bessel functions weighted at both extrema and zeros with p = 2

  • Weighting at both extrema and zeros with p = 2 is unsuitable because the numerator and denominator analytic-remainder integrals do not converge as m → ∞.This violates the conditions of Corollary 1.1.Source: §B.6 · p. 38
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