High-precision quadrature via local Fourier extension: analytic integration, uniform sampling, and correction for piecewise smooth integrands

Abstract

We propose a high-precision numerical quadrature framework based on local Fourier extension (LFE) approximations. The method constructs, on each subinterval, a truncated-SVD stabilized local Fourier continuation of the integrand on an extended periodic domain, and then evaluates the integral \emph{analytically} from the resulting Fourier coefficients. Under uniform sampling, the discrete LFE matrix and its TSVD factors are precomputed once and reused across all windows, yielding an efficient offline/online implementation that remains compatible with classical composite rules. We provide an error bound that reduces the quadrature error to the LFE approximation error and derive algebraic convergence rates for Sobolev-regular integrands. Numerical experiments demonstrate that, on smooth functions, the proposed quadrature reaches near machine precision with substantially fewer nodes than the composite Simpson rule. The advantage persists for oscillatory and variable-frequency integrands and becomes more pronounced for nonuniform phase structures. For continuous piecewise smooth integrands, we develop a correction strategy driven by coefficient-energy outliers to identify singularity-containing windows, followed by a localized procedure that brackets the singular point within one grid cell and corrects only the affected window contribution. The corrected quadrature restores near-spectral accuracy in the reported tests, including cases where the singularity is not aligned with the window endpoints.

Figures (5)

• Figure 1: Error decay versus the total number of uniform subintervals $M$ for the smooth test functions $f_1$--$f_3$.
• Figure 2: Error decay versus the total number of uniform subintervals $M$ for the oscillatory test function $f_4(x)=e^{-x}\sin(\omega x)$.
• Figure 3: Error decay versus the total number of uniform subintervals $M$ for the quadratic-phase oscillatory test function $f_5(x)=-2\kappa x\sin(\kappa x^2)$.
• Figure 4: Error decay versus the total number of uniform subintervals $M$ for the rapidly varying smooth test function $f_6(x)=\frac{2x}{(1+\alpha-x^2)^2}$.
• Figure 5: Distribution of the coefficient energy indicator $\eta_k=\|{\bf c}_k^\epsilon\|_2$ for the piecewise smooth test functions $f_7$ and $f_8$ with $M=160$. Each marker corresponds to one local window $I_k$. When the singular point lies inside a window (e.g. $\xi=0.3$, $\xi=\pi/5$, $\zeta=0.6$, $\zeta=0.73$), the corresponding $\eta_k$ exhibits a pronounced spike, indicating the presence of a singularity in that window. When the singular point coincides with a window endpoint (e.g. $\xi=0.5$ or $\zeta=0.25$), no window contains the singularity and therefore the coefficient energy remains uniformly bounded.

Theorems & Definitions (5)

• Lemma 2.1
• Theorem 2.2: Quadrature error is bounded by the LFE approximation error
• Theorem 2.3: Algebraic convergence
• Corollary 2.4: Uniform partition rate
• Remark 3.1: Scope