Hilbert空间中快速振荡函数近似积分的优化

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics Pub Date : 2025-04-05 DOI:10.1016/j.rinam.2025.100569

Abdullo Hayotov , Samandar Babaev , Abdimumin Kurbonnazarov

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引用次数: 0

摘要

在这项工作中，我们基于快速振荡函数的积分数值计算的泛函方法，构造了Sard意义上的最优正交公式。为了解决这个问题，我们将使用Sobolev的方法。要做到这一点，我们首先解决一个极值函数的边值问题。为了解决边值问题，我们使用傅里叶正变换和傅里叶反变换，求出给定微分算子的基本解。利用极值函数求出误差泛函的范数。对于给定的节点，我们沿着系数找到误差函数范数的最小值。这个正交公式对双曲函数sinh(x) cosh(x)和一个常数项是精确的。本文研究了Hilbert空间K2(3)（0,1）中ωh∈Z和ω∈R的情况。我们将构造的正交公式应用于计算机断层图像的重建。

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Optimization of approximate integrals of rapidly oscillating functions in the Hilbert space

In this work, we construct an optimal quadrature formula in the sense of Sard based on a functional approach for numerical calculation of integrals of rapidly oscillating functions. To solve this problem, we will use Sobolev’s method.

To do this, we first solve the boundary value problem for an extremal function. To solve the boundary value problem, we use direct and inverse Fourier transforms and find the fundamental solution of the given differential operator. Using the extremal function, we find the norm of the error functional. For the given nodes, we find the minimum value of the error functional norm along the coefficients.

This quadrature formula is exact for the hyperbolic functions