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The Non-Periodic Problem -- The Chebyshev Expansion


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
and we obtain the two sets of eigensystems

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 tex2html_wrap_inline11159 for k = 0 we may also write this as
Thus, we go from the interval [-1,1] into the tex2html_wrap_inline11001-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 up previous contents
Next: Spectral accuracy Up: SPECTRAL APPROXIMATIONS Previous: The (Pseudo)Spectral Fourier Expansions

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998