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 [0, 2π] and expansion in Fourier modes,
pj(x) := √1--eijx for integer j ∈ ℤ. (9.1 ) 2π
These are orthonormal under the standard complex scalar product in 2 L ([0,2π]),
∫2π----- ⟨f, g⟩ := f(x)g(x )dx, f, g ∈ L2([0, 2π]), (9.2 ) 0
⟨pj,pj′⟩ = δjj′. (9.3 )
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,
∑∞ f(x) = fˆjpj(x), (9.4 ) j=−∞
converges to f in the 2 L norm if
∑∞ |fˆj|2 < ∞. (9.5 ) j=− ∞

The Fourier coefficients ˆfj can be computed from the orthonormality condition (9.3View Equation) of the basis elements defined in Eq. (9.1View Equation),

∫2π ˆ --1-- − ijx fj = ⟨pj,f⟩ = √ --- f(x)e dx. (9.6 ) 2π 0
The truncated expansion of f is (assuming N to be even)
N ∕2 ∑ 𝒫N [f](x ) = fˆjpj(x), (9.7 ) j= −N∕2
where the notation is motivated by the fact that the 𝒫N operator can also be seen as the orthogonal projection under the above scalar product to the space spanned by {pj : j = − N ∕2 ...N ∕2}, see also Section 9.2 below. The error of the truncated expansion, using the orthonormality of the basis functions (Parseval’s property) is
2 ∑ 2 ∥f − 𝒫N [f ]∥ = |fˆj| , (9.8 ) |j|>N ∕2
from which it can be seen that a fast decay in the error relies on a fast decay of the high frequency Fourier coefficients ˆ |fj| as j → ∞. Using the explicit definition of the basis elements pj in Eq. (9.1View Equation) and the scalar product in (9.2View Equation), we have
|2π | 1 ||∫ || |fˆj| = |⟨pj,f⟩| = √----|| f(x )e− ijxdx|| . (9.9 ) 2π |0 |
Integrating by parts multiple times,
| | | | |∫2π | |∫2π | |fˆ| = √--1--- || f′(x)e−ijxdx ||= ...= √--1----|| f(s)(x )e−ijxdx ||= -1-||⟨f(s),p⟩|| j 2π |j| || || 2 π|j|s || || |j|s j 0 0 -1--∥∥ (s)∥∥ -1--∥∥ (s)∥∥ ≤ |j|s f ∥pj ∥ = |j|s f ,
and the process can be repeated for increasing s as long as the s-derivative f (s) remains bounded in the L2 norm. In particular, if f ∈ C ∞, then the Fourier coefficients decay to zero
ˆ |fj| → 0 as j → ∞ (9.10 )
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 (9.8View Equation). More precisely, the following result can be shown (see, for example, [237Jump To The Next Citation Point]):

Theorem 16. For any f ∈ Hs [0,2 π] p (p standing for periodic) there exists a constant C > 0 independent of N such that

∥ ∥ −s∥ dsf ∥ ∥f − 𝒫N [f]∥ ≤ CN ∥∥ dxs-∥∥ (9.11 )
for all N ≥ 1.

In fact, an estimate for the difference between u and its projection similar to (9.11View Equation) but on the infinity norm can also be obtained [237Jump To The Next Citation Point].

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

𝒟 = − ∂2x (9.12 )
is seen to be self-adjoint under the standard scalar product (9.2View Equation),
⟨f, 𝒟g ⟩ = ⟨𝒟f, g ⟩ (9.13 )
for periodic, twice–continuously-differentiable functions f and g. Therefore, the eigenfunctions pj of the problem
𝒟pj(x) = λjpj(x ) (9.14 )
are orthogonal (and can be chosen orthonormal) – they turn out to be the Fourier modes (9.1View Equation) – represent an orthonormal complete set for periodic functions in 2 L, the expansion (9.4View Equation) converges, and the error in the truncated expansion (9.7View Equation) is given by the decay of high-order coefficients; see Eq. (9.8View Equation). Assuming f is smooth enough, the fast decay of such modes is a consequence of 𝒟 being self-adjoint, the basis elements p j being solutions to the problem (9.14View Equation),
-1- pj = λ 𝒟pj, (9.15 ) j
and the eigenvalues satisfying λj ≃ j2 for large j (in the Fourier case, λj = j2 holds exactly). Combining these properties,
ˆ -1-- -1-- --1--|| (2) || -1---|| (s) || |fj| = |⟨f,pj⟩| = |λ | |⟨f,𝒟pj ⟩| = |λ | |⟨𝒟f, pj⟩| = |λ |2 ⟨𝒟 f,pj⟩ = ...= |λ |s ⟨𝒟 f,pj⟩ ∥ j∥ j j j ≤ --1--∥ 𝒟(s)f ∥, (9.16 ) |λj|s
where (s) 𝒟 denotes the application of 𝒟s times (in this case, (s) 𝒟 is equal to s 2s (− 1) ∂x).

The main property that leads to spectral convergence is then the fast decay of the Fourier coefficients; see Eq. (9.16View Equation), provided the norm of 𝒟 (s) remains bounded for large s.

Before moving to the non-periodic case we notice that in either the full or truncated expansions, the integrals (9.6View Equation) 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 (a,b) (which can actually be unbounded; for example, (0,∞ ) 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 ω on the open interval (a,b), we define

∫b ⟨h,g⟩ = h(x)g(x)ω (x)dx, (9.17 ) ω a
and its induced norm, ∘ ------- ∥h ∥ω := ⟨h,h ⟩ω, on the Hilbert space L2(a,b) ω of all real-valued, measurable functions h,g on the interval (a,b) for which ∥h∥ ω and ∥g∥ ω are finite.

Consider now the Sturm–Liouville problem

𝒟p (x ) = ω(x)λ p (x), (9.18 ) j j j
where 𝒟 is a second-order linear-differential operator on (a,b) along with appropriate boundary conditions so that it is self-adjoint under the non-weighted scalar product,
⟨f,𝒟g ⟩ω=1 = ⟨𝒟f, g⟩ω=1, (9.19 )
for all twice–continuously-differentiable functions f,g on (a,b), 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,
∑∞ f(x) = ˆfjpj(x ), (9.20 ) j=0
∑N 𝒫 [f](x) = ˆf p (x), (9.21 ) N j j j=0
with coefficients
ˆ fj = ⟨pj,f ⟩ω. (9.22 )
The truncation error is similarly given by
2 ∑ ˆ2 ∥f − 𝒫N [f]∥ω = fj (9.23 ) j>N
and spectral convergence is again obtained if the coefficients ˆfj decay to zero as j → ∞ faster than any power law. Consider then, the singular Sturm–Liouville problem
𝒟pj (x) = − ∂x [m (x)∂xpj(x)] + n (x )pj(x ) = ω(x)λjpj(x) (9.24 )
with the functions m, n : (a,b) → ℝ being continuous and bounded and such that n(x ) ≥ 0 and m (x) > 0 for all x ∈ (a,b) and – thus the singular part of the problem –,
m (a) = m (b) = 0. (9.25 )
For twice–continuously-differentiable functions with bounded derivatives, the boundary terms arising from integration by parts of the expression ⟨f,𝒟g ⟩ω=1 cancel due to Eq. (9.25View Equation), and it follows that the operator 𝒟 is self-adjoint; see Eq. (9.19View Equation). Therefore, one can proceed as in the Fourier case and arrive to
∥ ∥ 1 ∥∥( 𝒟 ) (s) ∥∥ |fˆj| ≤ ---s-∥ -- f ∥ (9.26 ) |λj| ∥ ω ∥ω
with spectral convergence if ∞ f ∈ C and, for example,
|λj| ≃ j2, for large enough j > jmin. (9.27 )

Theorem 17. The solutions to the singular Sturm–Liouville problem (9.24View Equation) with (a,b) = (− 1,1) and

m (x) = (1 − x)α+1(1 + x)1+β, (9.28 ) α β ω(x) = (1 − x) (1 + x) , (9.29 ) n(x) = 0, (9.30 )
where α,β > − 1, are the Jacobi polynomials (α,β) Pj (x). Here, (α,β) P j (x) has degree j, and it corresponds to the eigenvalue
λ = j(j + α + β + 1 ). (9.31 ) j

Notice that the eigenvalues satisfy the asymptotic condition (9.27View Equation) 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 𝒫N [f] of a function f in Jacobi polynomials:

Theorem 18. For any f ∈ Hs [0,2 π] ω (ω refers to the weight function) there exists a constant C > 0 independent of N such that

∥∥ dsf ∥∥ ∥f − 𝒫N [f]∥ω ≤ CN −s∥∥ (1 − x2 )s∕2---s∥∥ (9.32 ) dx ω
for all N > s.

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

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