7 Basic Concepts of Signal Approximation
In this chapter, we study basic concepts of mathematical signal analysis. To
this end, we first introduce the continuous Fourier transform F,
(Ff )(ω) = f (x) · e−ixω dω for f ∈ L1 (R), (7.1)
R
as a linear integral transform on the Banach space L1 (R) of absolutely
Lebesgue-integrable functions. We motivate the transfer from Fourier series
C
of periodic functions f ∈ C2π to Fourier transforms of non-periodic functions
f ∈ L (R). In particular, we provide a heuristic account to the Fourier trans-
1
formation Ff in (7.1), where we depart from Fourier partial sums Fn f , for
f ∈ C2π . Then, we analyze the following relevant questions.
(1) Is the Fourier transform F invertible?
(2) Can F be transferred to the Hilbert space L2 (R)?
(3) Can F be applied to multivariate functions f ∈ Lp (Rd ), for p = 1, 2?
We give positive answers to all questions (1)-(3). The answer to (1) leads
us, for f ∈ L1 (R), with Ff ∈ L1 (R), to the Fourier inversion formula
1
f (x) = (Ff )(ω)eixω dω for almost every x ∈ R.
2π R
To analyze (2), we study the spectral properties of F, where we identify the
Hermite functions hn in (4.55) as eigenfunctions of F. As we show, the Her-
mite functions (hn )n∈N0 form a complete orthogonal system in the Hilbert
space L2 (R). This result leads us to the Plancherel theorem, Theorem 7.30,
providing the continuous extension of F to an automorphism on L2 (R). The
basic properties of the Fourier operator F can be generalized from the uni-
variate case to the multivariate case, and this gives an answer to (3).
Finally, we formulate and prove the celebrated Shannon sampling theorem,
Theorem 7.34 (in Section 7.3), giving a fundamental result of mathematical
signal processing. According to the Shannon sampling theorem, a signal, i.e.,
a function f ∈ L2 (R), with bounded frequency density can be reconstructed
exactly from its samples (i.e., function values) on an infinite uniform grid at a
sufficiently small sampling rate. Our proof of the Shannon sampling theorem
serves to demonstrate the relevance and the significance of the introduced
Fourier methods.
© Springer Nature Switzerland AG 2018 237
A. Iske, Approximation Theory and Algorithms for Data Analysis, Texts
in Applied Mathematics 68, https://doi.org/10.1007/978-3-030-05228-7_7
238 7 Basic Concepts of Signal Approximation
The second half of this chapter is devoted to wavelets. Wavelets are popu-
lar and powerful tools of modern mathematical signal processing, in particular
for the approximation of functions f ∈ L2 (R). A wavelet approximation to f
is essentially based on a multiresolution of L2 (R), i.e., on a nested sequence
· · · ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ · · · ⊂ Vj−1 ⊂ Vj ⊂ · · · ⊂ L2 (R) (7.2)
of closed scale spaces Vj ⊂ L2 (R). The nested sequence in (7.2) leads us to
stable approximation methods, where f is represented on different frequency
bands by orthogonal projectors Πj : L2 (R) −→ Vj . More precisely, for a fixed
scaling function ϕ ∈ L2 (R), the scale spaces Vj ⊂ L2 (R) in (7.2) are generated
by dilations and translations of basis functions ϕjk (x) := 2j/2 ϕ(2j x − k), for
j, k ∈ Z, so that
Vj = span{ϕjk : k ∈ Z} ⊂ L2 (R) for j ∈ Z.
Likewise, for a corresponding wavelet function ψ ∈ L2 (R), the orthogonal
complement Wj ⊂ Vj+1 of Vj in Vj+1 ,
Vj+1 = Wj ⊕ Vj for j ∈ Z,
is generated by basis functions ψkj (x) := 2j/2 ψ(2j x − k), for j, k ∈ Z, so that
Wj = span{ψkj | k ∈ Z} for j ∈ Z.
The basic construction of wavelet approximations to f ∈ L2 (R) is based
on refinement equations of the form
ϕ(x) = hk ϕ(2x − k) and ψ(x) = gk ϕ(2x − k),
k∈Z k∈Z
for specific coefficient masks (hk )k∈Z , (gk )k∈Z ⊂ 2
.
The development of wavelet methods, dating back to the early 1980s, has
since then gained enormous popularity in applications of information techno-
logy, especially in image and signal processing. Inspired by a wide range of
applications in science and engineering, this has led to rapid progress concer-
ning both computational methods and the mathematical theory of wavelets.
Therefore, it is by no means possible for us to give a complete overview
over the multiple facets of wavelet methods. Instead, we have decided to
present selected basic principles of wavelet approximation. To this end, we
restrict the discussion of this chapter to the rather simple Haar wavelet
⎧
⎨ 1 for x ∈ [0, 1/2),
ψ(x) = χ[0,1/2) (x) − χ[1/2,1) (x) = −1 for x ∈ [1/2, 1),
⎩
0 otherwise,
and its corresponding scaling function ϕ = χ[0,1) . For a more comprehensive
account to the mathematical theory of wavelets, we recommend the classical
textbooks [14, 18, 49] and, moreover, the more recent textbooks [29, 31, 69]
for a more pronounced connection to Fourier analysis.
