On a filter for exponentially localized kernels based on Jacobi polynomials
The problem of detection of singularities of a function from spectral data and the closely related problem of spectral approximation of piecewise smooth functions arise in many applications. The locations of singularities are found using an appropriate filter, and then pseudo-spectral methods are used on the maximum intervals of smoothness to compute the approximation on these intervals. A filter is a bi-infinite matrix \(H,\) and the corresponding mollifier is given by \[ \Phi_n(H, \theta)=\sum_{k\in \mathbb Z}H_{k,n}\exp(ik\theta), \quad \sigma_n(H, f, \theta)=\frac{1}{2\pi}\int_{-\pi}^\pi f(\phi) \Phi_n(H, \theta -\phi)d\phi , \theta\in {\mathbb{R}} . \] The purpose of this paper is to demonstrate the construction of a filter in the case of Jacobi polynomials. Let \(\alpha,\beta\geq-\frac{1} {2},\) and for \(k=0,1,\dots, p_{k}^{(\alpha,\beta)}\) denote the orthonormalized Jacobi polynomial of degree \(k,\) so that \[ \int_{-1}^1 p_{k}^{(\alpha,\beta)}(x) p_{m}^{(\alpha,\beta)}(x) w_{\alpha,\beta}(x)dx=\delta_{k,m}, \] where \(w_{\alpha,\beta}(x)=(1-x)^\alpha (1+x)^\beta, -1\leq x\leq 1.\) The authors introduce a translation operator \(\tau_{y}\) and use it to define a convolution operator by \[ \left( f\ast g\right) (x):=\int_{-1}^{1}f(y)\tau_{y}g(x)w_{\alpha,\beta }(y)dy,\;\;\;f,g\in L^{1}\left( \alpha,\beta\right) \] so that \[ \widehat{( f\ast g)}(k)=\left\{ p_{k}^{\left( \alpha ,\beta\right) }\left( 1)\right) \right\} ^{-1}\hat{f}(k)\hat{g}(k), \] where \[ \hat{f}(k)=\int_{-1}^1 f(x) p_{k}^{(\alpha,\beta)}(x)w_{\alpha,\beta}(x)dx. \] The authors construct a kernel localized at \(x=1\) and then extend the result by using translation theorems they developed and obtain a matrix \(H\) satisfying \[ \begin{aligned} \left| \sum_{k=0}^{\infty}H_{k,n}p_{k}^{(\alpha,b)}\left( \cos \theta\right) p_{k}^{(\alpha,\beta)}(\cos\varphi)\right| & \leq c_{1}n^{2\max(\alpha,\beta)+2}\exp(-cn\left( \theta-\varphi\right) ^{2}),\\ & \theta,\varphi\in[0,\pi],n=1,2,\dots \end{aligned} \] Specializing to the case of \ Chebyshev polynomials, \(\alpha=\beta=-\frac {1}{2},\) the authors construct an exponentially localized polynomial basis for the corresponding Hilbert space \(L^{2}\left( -\frac{1}{2},-\frac{1}{2}\right) .\) The Riesz basis consists of translates of a sequence of polynomial kernels, such that each element of this basis is localized exponentially near the point used to define the translate. Numerical computations of the filter and demonstration of the superiority of the exponentially localized kernels over another kernel are discussed.
scientific article; zbMATH DE number 5641735
scientific article