Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

When one solves differential equations by a spectral method,it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients anto modified basis functions that incorporate the boundary conditions.For homogeneous Dirichlet boundary conditions,u(±1)=0,popular choices include the "Chebyshev difference basis" ζn(x)≡Tn+2(x)-Tn(x) with coefficients here denoted by bnand the "quadratic factor basis" Qn(x)≡(1-x2)Tn(x) with coefficients cn.If u(x) is weakly singular at the boundary,then the coefficients andecrease proportionally to O(A(n)/nκ) for some positive constant κ,where A(n) is a logarithm or a constant.We prove that the Chebyshev difference coefficients bndecrease more slowly by a factor of 1/n while the quadratic factor coefficients cndecrease more slowly still as O(A(n)/nκ-2).The error for the unconstrained Chebyshev series,truncated at degree n=N,is O(|A(N)|/Nκ) in the interior,but is worse by one power of N in narrow boundary layers near each of the endpoints.Despite having nearly identical error norms in interpolation,the error in the Chebyshev basis is concentrated in boundary layers near both endpoints,whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x.Meanwhile,for Chebyshev polynomials,the values of their derivatives at the endpoints are O(n2),but only O(n) for the difference basis.Furthermore,we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases,solved by the least squares method.We also find an interesting fact that on the face of it,the aliasing error is regarded as a bad thing;actually,the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation.But the premise is under the same basis,and when involving different bases,it may not be established yet.

The Well-Posed Operators with Their Spectra in Lpw-Spaces

In this paper, we have considered the general ordinary quasi-differential operators generated by a general quasi-differential expression τp,q in Lpw-spaces of order n with complex coefficients and its formal adjoint τ+q',p' in Lpw-spaces for arbitrary p,q∈[1,∞). We have proved in the case of one singular end-point that all well-posed extensions of the minimal operator T0 (τp,q) generated by such expression τp,q and their formal adjoint on the interval [a,b) with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions can be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while others are new.