[Paper Review] Convergence rates of spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities

Shuhuang Xiang|arXiv (Cornell University)|Jun 2, 2020

Mathematical functions and polynomials23 references4 citations

TL;DR

This paper derives optimal convergence rates for spectral orthogonal projections (Jacobi, Gegenbauer, Chebyshev) of functions with algebraic and logarithmic singularities by leveraging Hilb-type formulas and van der Corput lemmas. It shows that for boundary singularities, increasing the Jacobi parameters $(\alpha,\beta)$ or Gegenbauer parameter $\lambda$ enhances convergence rates, with Legendre projections achieving half-order higher convergence than Chebyshev, while interior singularities yield convergence independent of these parameters.

ABSTRACT

Based on the Hilb type formula between Jacobi polynomials and Bessel functions, optimal decay rates on Jacobi expansion coefficients are derived, by applying van der Corput type lemmas, for functions of logarithmatic singularities, which leads to the optimal convergence rates on the Jacobi, Gegenbauer and Chebyshev orthogonal projections. It is interesting to see that for boundary singularities, one may get faster convergence rate on the Jacobi or Gegenbauer projection as $(α,β)$ and $λ$ increases. The larger values of parameter, the higher convergence rates can be achieved. In particular, the Legendre projection has one half order higher than Chebyshev. Moreover, if $\min\{α,β\}>0$ and $λ>\frac{1}{2}$, the Jacobi and Gegenbauer orthogonal projections have higher convergence orders compared with Legendre. While for interior singularity, the convergence order is independent of $(α,β)$ and $λ$.

Motivation & Objective

• To establish optimal convergence rates for spectral orthogonal projections of functions with algebraic and logarithmic singularities.
• To analyze how the convergence rate depends on Jacobi parameters $(\alpha,\beta)$ and Gegenbauer parameter $\lambda$ for boundary versus interior singularities.
• To resolve the suboptimal convergence observed in prior frameworks for functions with logarithmic or fractional singularities.
• To extend the analysis to generalized Gegenbauer functions of fractional degree and provide sharp bounds for expansion coefficients.

Proposed method

• Derives asymptotic estimates for Jacobi expansion coefficients using a Hilb-type formula connecting Jacobi polynomials and Bessel functions.
• Applies van der Corput-type lemmas to analyze oscillatory integrals arising from singular functions.
• Establishes decay rates for coefficients of functions with endpoint singularities $(1\pm x)^\gamma \ln^\mu(1\pm x)$ and interior singularities $|x-z_0|^s \ln^\mu|x-z_0|$.
• Translates coefficient decay into convergence rates in weighted Sobolev norms $H^{m,\alpha,\beta}(\Omega)$ for $\Omega = (-1,1)$.
• Derives optimal estimates for Gegenbauer and Chebyshev projections via parameter relations $\alpha = \beta = \lambda - 1/2$ and $\alpha = \beta = -1/2$, respectively.
• Extends results to general functions with multiple singularities using separation of singularities and min-max coefficient bounds.

Experimental results

Research questions

• RQ1 How do convergence rates of Jacobi, Gegenbauer, and Chebyshev projections behave for functions with logarithmic singularities at endpoints or interior points?
• RQ2 What is the dependence of convergence rates on the parameters $\alpha$, $\beta$, and $\lambda$ for boundary versus interior singularities?
• RQ3 Can the suboptimal convergence observed in prior frameworks for logarithmic singularities be improved via new asymptotic analysis?
• RQ4 What is the optimal decay rate of expansion coefficients for functions with $|x - z_0|^s \ln^\mu|x - z_0|$ singularities?
• RQ5 How do the convergence orders compare across Legendre, Chebyshev, and general Jacobi projections under varying regularity and singularity conditions?

Key findings

• For boundary singularities $f(x) = (1-x)^\gamma \ln^\mu(1-x)g(x)$ with $g \in C^\infty[-1,1]$, the Jacobi projection converges as $\mathcal{O}(N^{m - \alpha - 2\gamma - 1} \ln^\mu N)$, improving with larger $\alpha$.
• The Legendre projection ($\alpha = \beta = 0$) achieves a convergence rate half-order higher than Chebyshev, with $\mathcal{O}(N^{m - 1/2 - 2\gamma} \ln^\mu N)$.
• When $\min\{\alpha,\beta\} > -1/2$ and $\lambda > 1/2$, Jacobi and Gegenbauer projections outperform Legendre in convergence order.
• For interior singularities $f(x) = |x - z_0|^s \ln^\mu|x - z_0|g(x)$, the convergence rate is $\mathcal{O}(N^{m - s - 1/2} \ln^\mu N)$, independent of $\alpha$, $\beta$, and $\lambda$.
• When $\gamma, \delta$ are integers and $\mu \geq 1$, the logarithmic factor reduces to $\ln^{\mu-1} N$, yielding improved convergence for endpoint singularities.
• The results extend to multiple singularities via separation of singularities, yielding coefficient decay rates involving minima over all singularity contributions.