High-Precision Integral-Value-Based MQ Quasi-Interpolation

本文针对积分值条件下的拟插值问题，提出了一种基于Multiquadric (MQ)函数的新型高精度数值逼近方法。作为一类条件正定径向基函数，MQ函数凭借其指数级收敛特性在拟插值理论中具有重要的应用价值。现有的MQ拟插值方法主要基于函数值，在实际应用中，函数信息经常以连续区间上的积分值形式呈现，本文重点解决仅知积分值条件下的构造问题。具体地，首先基于积分值的线性组合实现对节点处函数值及二阶导数值的逼近，进而结合利用函数值与二阶导数信息的拟插值方法，构造出新型的高精度积分值型MQ拟插值算子并推导了相应的误差估计表达式。数值实验结果表明，该方法有较好的逼近效果且其数值收敛阶与理论分析是吻合的，验证了所提算法的有效性。This paper proposes a novel high-precision numerical approximation method for quasi-interpolation problems under integral value conditions, utilizing Multiquadric (MQ) functions. As a class of conditionally positive definite radial basis functions, MQ functions hold significant application value in quasi-interpolation theory due to their exponential convergence properties. Existing MQ quasi-interpolation methods primarily rely on function values; however, in practical scenarios, functional information is often presented in the form of integral values over continuous intervals. This work focuses on addressing the construction of quasi-interpolation operators under the condition of known integral values. Specifically, we first approximate the function values and second-order derivative values at nodes through linear combinations of integral values. Subsequently, by integrating a quasi-interpolation framework that incorporates both function values and second-order derivative information, a novel high-precision integral-value-based MQ quasi-interpolation operator is constructed, accompanied by derived error estimation formulas. Numerical experiments demonstrate the favorable approximation performance of the proposed method, with the numerical convergence order aligning well with theoretical analyses, thereby validating the effectiveness of the algorithm.