9.1 Spectral convergence
9.1.1 Periodic functions
An intuition about the expansion of smooth functions into orthogonal
polynomials and spectral convergence can be obtained by first considering the periodic case in
and
expansion in Fourier modes,
These are orthonormal under the standard complex scalar product in
,
Furthermore, they form a complete set of orthonormal functions under the norm induced by the above
scalar product. More explicitly, the expansion of a continuous, periodic function in these modes,
converges to
in the
norm if
The Fourier coefficients
can be computed from the orthonormality condition (9.3) of the basis
elements defined in Eq. (9.1),
The truncated expansion of
is (assuming
to be even)
where the notation is motivated by the fact that the
operator can also be seen as the orthogonal
projection under the above scalar product to the space spanned by
, see also
Section 9.2 below. The error of the truncated expansion, using the orthonormality of the basis functions
(Parseval’s property) is
from which it can be seen that a fast decay in the error relies on a fast decay of the high frequency Fourier
coefficients
as
. Using the explicit definition of the basis elements
in Eq. (9.1) and the
scalar product in (9.2), we have
Integrating by parts multiple times,
and the process can be repeated for increasing
as long as the s-derivative
remains
bounded in the
norm. In particular, if
, then the Fourier coefficients decay to zero
faster than any power law, which is usually referred to as spectral convergence. The spectral
denomination comes from the property that the decay rate of the error is dominated by the spectrum of an
associated Sturm–Liouville problem, as discussed below. The convergence rate for each Fourier mode in the
remainder can be extended to the whole sum (9.8). More precisely, the following result can be shown (see,
for example, [237
]):
In fact, an estimate for the difference between
and its projection similar to (9.11) but on the infinity
norm can also be obtained [237
].
In preparation for the discussion below for non-periodic functions, we rephrase and re-derive
the previous results in the following way. Integrating by parts twice, the differential operator
is seen to be self-adjoint under the standard scalar product (9.2),
for periodic, twice–continuously-differentiable functions
and
. Therefore, the eigenfunctions
of
the problem
are orthogonal (and can be chosen orthonormal) – they turn out to be the Fourier modes (9.1)
– represent an orthonormal complete set for periodic functions in
, the expansion (9.4)
converges, and the error in the truncated expansion (9.7) is given by the decay of high-order
coefficients; see Eq. (9.8). Assuming
is smooth enough, the fast decay of such modes is a
consequence of
being self-adjoint, the basis elements
being solutions to the problem (9.14),
and the eigenvalues satisfying
for large
(in the Fourier case,
holds exactly).
Combining these properties,
where
denotes the application of 
times (in this case,
is equal to
).
The main property that leads to spectral convergence is then the fast decay of the Fourier coefficients;
see Eq. (9.16), provided the norm of
remains bounded for large
.
Before moving to the non-periodic case we notice that in either the full or truncated expansions, the
integrals (9.6) need to be computed. Numerically approximating the latter leads to discrete expansions
and an appropriate choice of quadratures for doing so leads to a powerful connection between the
discrete expansion and interpolation. We discuss this in Section 9.4, directly for the non-periodic
case.
9.1.2 Singular Sturm–Liouville problems
Next, consider non-periodic domains
(which can actually be unbounded; for example,
as in the case of Laguerre polynomials) in the real axis. We discuss how bases of orthogonal
polynomials with spectral convergence properties arise as solutions to singular Sturm–Liouville
problems.
For this we need to consider more general scalar products. For a continuous, strictly-positive weight
function
on the open interval
, we define
and its induced norm,
, on the Hilbert space
of all real-valued, measurable
functions
on the interval
for which
and
are finite.
Consider now the Sturm–Liouville problem
where
is a second-order linear-differential operator on
along with appropriate boundary
conditions so that it is self-adjoint under the non-weighted scalar product,
for all twice–continuously-differentiable functions
on
, which are subject to the
boundary conditions. Then, the set of eigenfunctions is also complete, and orthonormal under the
weighted scalar product, and there is again a full and truncated expansion as in the Fourier case,
with coefficients
The truncation error is similarly given by
and spectral convergence is again obtained if the coefficients
decay to zero as
faster than any
power law. Consider then, the singular Sturm–Liouville problem
with the functions
being continuous and bounded and such that
and
for all
and – thus the singular part of the problem –,
For twice–continuously-differentiable functions with bounded derivatives, the boundary terms arising from
integration by parts of the expression
cancel due to Eq. (9.25), and it follows that the
operator
is self-adjoint; see Eq. (9.19). Therefore, one can proceed as in the Fourier case and arrive to
with spectral convergence if
and, for example,
Notice that the eigenvalues satisfy the asymptotic condition (9.27) and, roughly speaking, guarantees
spectral convergence. More precisely, the following holds (see, for example, [197]) – in analogy
with Theorem 16 for the Fourier case – for the expansion
of a function
in Jacobi
polynomials:
Sturm–Liouville problems are discussed in, for example, [431]. Below we discuss some properties of
general orthogonal polynomials.