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The Non-Periodic Problem -- The Chebyshev Expansion

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We start by considering the second order Chebyshev ODE

This is a special case of the general Sturm-Liouville (SL) problem

Noting the Green identity

we find that L is (formally) self-adjoint provided certain auxiliary conditions are satisfied. In the nonsingular case where , we augment (app_cheb.2) with homogeneous boundary conditions,

Then L is self-adjoint in this case with a complete eigensystem : each
has the ``generalized'' Fourier expansion

with Fourier coefficients

The decay rate of the coefficients is algebraic: indeed

The asymptotic behavior of the eigenvalues for nonsingular SL problem is

and hence, unless w(x) satisfies an infinite set of boundary restrictions, we end with algebraic decay of

This leads to algebraic convergence of the corresponding spectral and pseudospectral projections.

In contrast, the singular case is characterized by, p(a) = p(b) = 0; in this case L is self-adjoint independent of the boundary conditions (since the Poisson brackets [ , ] drop), and we end up with the spectral decay estimate -- compare (app_fourier.22)

Thus, the decay of is as rapid as the smoothness of w(x) permits.

As a primary example for this category of singular SL problems we consider the Jacobi equation associated with weights of the form ,

We now focus our attention on the Chebyshev-SL problem (app_cheb.1) corresponding to . The transformation

yields

and we obtain the two sets of eigensystems

and

The second set violates the boundedness requirement which we now impose

and so we are left with

The trigonometric identity

yields the recurrence relation

hence, are polynomials of degree k - these are the Chebyshev polynomials

which are orthonormal w.r.t. Chebyshev weight ,

In analogy with what we had done before, we consider now the Chebyshev-Fourier expansion

To get rid of the factor for k = 0 we may also write this as

Thus, we go from the interval [-1,1] into the -periodic circle by even extension, with Fourier expansion of , compare (app_fourier.9),

Another way of writing this employs a symmetric doubly infinite Fourier-like summation, where

with and

The Parseval identity reflects the completeness of this system

which yields the error estimate    Next: Spectral accuracy Up: SPECTRAL APPROXIMATIONS Previous: The (Pseudo)Spectral Fourier Expansions