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Möbius-Transformed Trapezoidal Rule

Yuya Suzuki, Nuutti Hyvönen, Toni Karvonen

TL;DR

This work introduces a Möbius-transformed trapezoidal rule for numerical integration over the real line with weights from a broad class of monotonic Schwartz functions. By mapping the real line to a periodic domain via a Möbius change of variables, the method reduces to fast, optimal-rate trapezoidal quadrature for periodic functions, achieving the best possible worst-case convergence rate for $f$ in $W^{\alpha,2}_{\rho}(\mathbb{R})$. It provides both deterministic and randomized variants, as well as an $L^p_{\rho}$-approximation framework that leverages trig interpolation on the transformed domain; the results extend to multivariate settings through componentwise transforms. The approach is notable for requiring only evaluations of the weight at quadrature nodes and for handling weights decaying at or slower than exponential rates, with practical advantages such as nested quadrature and FFT-based acceleration. Overall, the paper establishes optimal convergence guarantees for a broad, robust class of weighted integration problems on the real line and lays groundwork for high-dimensional extensions.

Abstract

We study numerical integration by combining the trapezoidal rule with a Möbius transformation that maps the unit circle onto the real line. We prove that the resulting transformed trapezoidal rule attains the optimal rate of convergence if the integrand function lives in a weighted Sobolev space with a weight that is only assumed to be a positive Schwartz function decaying monotonically to zero close to infinity. Our algorithm only requires the ability to evaluate the weight at the selected nodes, and it does not require sampling from a probability measure defined by the weight nor information on its derivatives. In particular, we show that the Möbius transformation, as a change of variables between the real line and the unit circle, sends a function in the weighted Sobolev space to a periodic Sobolev space with the same smoothness. Since there are various results available for integrating and approximating periodic functions, we also describe several extensions of the Möbius-transformed trapezoidal rule, including function approximation via trigonometric interpolation, integration with randomized algorithms, and multivariate integration.

Möbius-Transformed Trapezoidal Rule

TL;DR

This work introduces a Möbius-transformed trapezoidal rule for numerical integration over the real line with weights from a broad class of monotonic Schwartz functions. By mapping the real line to a periodic domain via a Möbius change of variables, the method reduces to fast, optimal-rate trapezoidal quadrature for periodic functions, achieving the best possible worst-case convergence rate for in . It provides both deterministic and randomized variants, as well as an -approximation framework that leverages trig interpolation on the transformed domain; the results extend to multivariate settings through componentwise transforms. The approach is notable for requiring only evaluations of the weight at quadrature nodes and for handling weights decaying at or slower than exponential rates, with practical advantages such as nested quadrature and FFT-based acceleration. Overall, the paper establishes optimal convergence guarantees for a broad, robust class of weighted integration problems on the real line and lays groundwork for high-dimensional extensions.

Abstract

We study numerical integration by combining the trapezoidal rule with a Möbius transformation that maps the unit circle onto the real line. We prove that the resulting transformed trapezoidal rule attains the optimal rate of convergence if the integrand function lives in a weighted Sobolev space with a weight that is only assumed to be a positive Schwartz function decaying monotonically to zero close to infinity. Our algorithm only requires the ability to evaluate the weight at the selected nodes, and it does not require sampling from a probability measure defined by the weight nor information on its derivatives. In particular, we show that the Möbius transformation, as a change of variables between the real line and the unit circle, sends a function in the weighted Sobolev space to a periodic Sobolev space with the same smoothness. Since there are various results available for integrating and approximating periodic functions, we also describe several extensions of the Möbius-transformed trapezoidal rule, including function approximation via trigonometric interpolation, integration with randomized algorithms, and multivariate integration.
Paper Structure (11 sections, 11 theorems, 97 equations, 2 figures, 1 table)

This paper contains 11 sections, 11 theorems, 97 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Monotonic Schwartz weights
  4. Möbius transformations
  5. Sobolev spaces
  6. Möbius-Transformed Trapezoidal Rule
  7. Randomized trapezoidal rule
  8. $L^p_{\rho}$ approximation
  9. Multivariate extension by componentwise transforms
  10. Concluding remarks
  11. Induction proofs

Key Result

Lemma 2.1

Let $\omega \in \mathcal{S}_+^{\rm mon}$ and $\alpha, \beta \in \mathbb{N}_0$. For any $r > 1$, i.e., $\frac{|\omega^{(\beta)}|^r}{\omega}$ is rapidly decreasing.

Figures (2)

  • Figure 1: Absolute integration error for the Gaussian weight and $f(x)=|x|^p$, which corresponds to $I_\rho(f) = (2^p/\pi)^{1/2}\Gamma((p+1)/2)$ where $\Gamma$ is the Gamma function. The blue line shows the error for the Gauss--Hermite quadrature and the red line for the trapezoidal rule with a cut-off from KSG2023. The Möbius-transformed trapezoidal rule (green) achieves the fastest convergence of the error.
  • Figure 2: Absolute integration error for the logistic weight and $f(x)=|x|^p$, which corresponds to $I_\rho(f) = -2 \, p! \, {\rm Li}_p(-1)$ where ${\rm Li}$ denotes the polylogarithm Jodra2014. The blue line shows the error for the Gauss--Logistic quadrature from G2020. The Möbius-transformed trapezoidal rule (green) exhibits much faster convergence.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 3.1: Upper bound on integration error
  • Lemma 3.2
  • proof
  • Remark 3.3: Implementation
  • ...and 12 more