Benxue Gong

Xinran Liu, Zhenyu Zhao, Benxue Gong

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.

Benxue Gong, Zhenyu Zhao, Chenyang Wang

We propose a local Legendre frame (LLF) method for function approximation from equispaced data on a finite interval. Motivated by the difficulty of stable high-order polynomial approximation at equispaced points, especially in the presence of the Runge phenomenon, the method partitions the interval into subintervals, maps each subinterval to a common reference interval, and computes local coefficients by a truncated singular value decomposition (TSVD) regularization. Since all subintervals share the same local sampling matrix, the method admits a natural offline--online implementation. We establish a quasi-optimal estimate for the regularized reconstruction and discuss practical parameter selection. Numerical results show that LLF attains high accuracy for relatively smooth and moderately oscillatory functions, while it remains applicable to highly oscillatory functions, although comparable accuracy generally requires more sampling points. For continuous piecewise smooth functions with derivative singularities, the method also provides an effective detect--localize--correct strategy based on one-sided coefficient-energy indicators. These results indicate that LLF provides a stable and flexible local approximation framework for equispaced data.