### 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]):

Theorem 16. For any (p standing for periodic) there exists a constant independent of such that

for all .

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,

Theorem 17. The solutions to the singular Sturm–Liouville problem (9.24) with and

where , are the Jacobi polynomials . Here, has degree , and it corresponds to the eigenvalue

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:

Theorem 18. For any ( refers to the weight function) there exists a constant independent of such that

for all .

Sturm–Liouville problems are discussed in, for example, [431]. Below we discuss some properties of general orthogonal polynomials.