De la Vallée Poussin type approximation for solving some Fredholm integral equations

Technical Summary: De la Vallée Poussin Type Approximation for Solving Some Fredholm Integral Equations

Problem Statement
The paper addresses the numerical solution of second-kind Fredholm integral equations (FIEs) of the form:
$$f(y) - \nu \int_{-1}^{1} f(x)k(x, y)w(x) dx = g(y), \quad y \in (-1, 1),$$
where $w(x)$ is a Jacobi weight, $g$ and $k$ are known functions, and $f$ is the unknown solution. The study focuses on scenarios where the known functions may exhibit endpoint singularities controlled by a Jacobi weight $u$, potentially distinct from the weight $w$ defining the integral operator. The goal is to solve these equations in the space $C_u$ of locally continuous functions endowed with weighted uniform norms, handling cases involving weakly singular kernels (algebraic or logarithmic) and highly oscillatory kernels.

Methodology
The authors propose a numerical method, termed the VP-method, based on de la Vallée Poussin (VP) polynomial approximations at Jacobi zeros. Unlike classical Lagrange interpolation, which uses a single set of nodes to construct a unique interpolating polynomial, the VP operator $V_n^m(v_{\alpha,\beta})$ is a quasi-projector that depends on an additional parameter $m$ (where $1 < m < n$).

The method proceeds as follows:

1. Approximation: The integral operator $K$ and the source function $g$ are approximated using the VP operator $V_n^m$ associated with a Jacobi weight $v_{\alpha,\beta}$. This weight $v_{\alpha,\beta}$ is chosen independently of the integral weight $w$ to optimize approximation properties.
2. Discretization: The continuous equation is replaced by a finite-dimensional equation $(I - \nu K_n^m)f_n^m = g_n^m$, where $f_n^m$ is a polynomial in a specific subspace $S_n^m$ of dimension $n$.
3. Linear System: The solution is reduced to solving a linear system of order $n$. The system matrix involves "modified moments" (integrals of the kernel against Jacobi polynomials). The authors provide recurrence relations to compute these moments efficiently for weakly singular, logarithmic, and highly oscillatory kernels.
4. Comparison: The method is compared against the L-method, a projection method based on Lagrange interpolation at the same nodes, as proposed in prior literature [5].

Key Contributions

• Theoretical Stability and Convergence: The paper proves that under suitable assumptions on the weights and the kernel, the VP-method is stable and converges in the weighted uniform space $C_u$. Crucially, the Lebesgue constants of the VP operator are uniformly bounded with respect to $n$, whereas those for Lagrange interpolation grow at least as $\log n$.
• Convergence Rates: The error estimate for the VP-method is shown to be of the order of the best polynomial approximation, $O(E_{n-m}(f)_u)$. In contrast, the L-method requires the solution to belong to Hölder–Zygmund subspaces and incurs an additional $\log n$ factor in the error estimates to ensure convergence.
• Generalized Applicability: The VP-method allows the approximation weight $v_{\alpha,\beta}$ to differ from the integral weight $w$. This flexibility significantly broadens the range of applicable weight functions and solution behaviors compared to the L-method, which often requires $v_{\alpha,\beta} = w$ and imposes stricter conditions on the weights to guarantee convergence.
• Handling Singularities and Oscillations: The authors establish sufficient conditions for the compactness of the integral operator in weighted spaces for weakly singular and highly oscillatory kernels. They derive specific recurrence relations for the modified moments required by the algorithm in these challenging cases.
• Conditioning: The linear systems arising from the VP-method are shown to be well-conditioned, with condition numbers that remain bounded as the discretization dimension $n$ increases.

Results
Numerical experiments were conducted on various test cases involving:

• Weakly singular kernels (algebraic and logarithmic).
• Highly oscillatory kernels.
• Functions with endpoint singularities and steep variations.

The results confirm the theoretical error estimates. Key findings include:

• Global Accuracy: The maximum absolute errors of the VP-method and the L-method are often comparable when both are applicable.
• Local Accuracy: The VP-method consistently achieves higher local (pointwise) accuracy than the L-method. This is attributed to the VP operator's ability to mitigate the Gibbs phenomenon, particularly near steep function variations or singularities.
• Parameter Sensitivity: The performance of the VP-method depends on the choice of the parameter $\theta = m/n$. While no universal theoretical criterion for the optimal $\theta$ is provided, numerical tests suggest that values closer to 1 (approaching the Fejér operator) tend to degrade performance, while intermediate values often yield the best results.
• Robustness: In cases where the L-method fails to satisfy convergence assumptions (e.g., specific combinations of weights $u$ and $w$), the VP-method successfully converges.

Significance and Claims
The paper claims that the VP-method offers a robust alternative to classical Lagrange-based projection methods for solving Fredholm integral equations. Its primary significance lies in:

1. Eliminating Logarithmic Growth: By utilizing uniformly bounded Lebesgue constants, the method avoids the $\log n$ degradation in error estimates and stability often associated with Lagrange interpolation.
2. Flexibility: The ability to decouple the approximation weight from the integral weight allows the method to handle a wider class of problems, including those with singularities where classical methods may fail.
3. Superior Pointwise Behavior: The method provides more accurate reconstructions of the solution in critical regions (e.g., near singularities or rapid variations) by effectively damping Gibbs-type oscillations.
4. Computational Efficiency: The method leads to well-conditioned linear systems and utilizes efficient recurrence relations for moment computation, making it suitable for large-scale computations.

The authors conclude that the VP-method combines favorable approximation properties, stability, and computational efficiency, making it a valuable tool for solving second-kind Fredholm integral equations, particularly in the presence of singularities and oscillatory behavior.