University of Leeds, School of Mathematics
Analysis (wavelets and operators): Some useful results and formulae (1)
J.R. Partington
March 2003
Please let me know of any errors, omissions or obscurities!
R∞
Lp spaces: functions in Lp (R) are measurable and satisfy kf kp = ( −∞ |f (t)|p dt)1/p <
∞, regarding two functions as the same if they are equal almost everywhere.
R ∞ These are
2
Banach spaces, and L (R) is a Hilbert space with inner product hf, gi = −∞ f (t)g(t) dt.
For L∞ (R), take the norm to be the (essential) supremum of |f (t)|.
FourierP
series: L2 (0, T ) has an orthonormal basis, en (t) = T −1/2 exp(2πint/T ), n ∈ Z,
∞
so f = n=−∞ hf, en ien , converging in L2 norm.
R∞
Fourier transforms: for f ∈ L1 (R) define fˆ(w) = −∞ f (t) exp(−2πiwt) dt, for w ∈ R,
and write F f = fˆ. This extends to a bounded linear operator F from L2 (R) onto
L2 (R), and kfˆk2 = kf k2 (Plancherel). Moreover, (fˆ)ˆ(t) = f (−t), that is,
R∞
f (t) = −∞ fˆ(w) exp(2πiwt) dw, a.e., at least if fˆ ∈ L1 (R) ∩ L2 (R). We get as a
corollary that hf, gi = hfˆ, ĝi, for f , g ∈ L2 (R). Write F̌ for the inverse Fourier
transform of F .
Haar wavelets: let V0 ⊂ L2 (R) be the closed subspace of all functions f which are
constant on all intervals (k, k + 1), k ∈ Z. Let φ(t) = χ(0,1) (t) and φk (t) = φ(t − k) for
k ∈ Z. Then (φk )k∈Z is an orthonormal basis of V0 . Now for j ∈ Z let Vj be the space
of functions constant on all intervals (k/2j , (k + 1)/2j ), k ∈ Z.
S
T
Then . . . ⊂ V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ . . .. Also Vj = L2 (R) and Vj = {0}. Now
f (t) ∈ Vj ⇐⇒ f (2−j t) ∈ V0 , and Vj has o.n.b. consisting of 2j/2 φ(2j t − k), k ∈ Z.
Any chain of subspaces with these properties is called a multiresolution approximation
or multiresolution analysis (MRA) of L2 (R).
Define the Haar wavelet ψ(t) = φ(2t) − φ(2t − 1). The functions ψk (t) = ψ(t − k),
k ∈ Z form an o.n.b. for a space W0 such that V0 ⊕ W0 = V1 (orthogonal direct sum).
Then Vj ⊕ Wj = Vj+1 , where Wj has o.n.b. ψj,k (t) = 2j/2 ψ(2j t − k), k ∈ Z. Finally
L2 (R) = . .P
. ⊕ W−2P
⊕ W−1 ⊕ W0 ⊕ W1 ⊕ . . . and has orthonormal basis (ψj,k )j,k∈Z .
∞
∞
Hence f = j=−∞ k=−∞ hf, ψj,k iψj,k , converging in L2 norm.
Band-limited functions: f ∈ L2 (R) is time-limited, if ∃T > 0 s.t. f (t) = 0 a.e. for
|t| > T ; and f is band-limited if fˆ is time-limited.
Paley–Wiener space: P W (b) = {f ∈ L2 (R) : fˆ(w) = 0 a.e. for |w| > b}. Such
functions are restrictions to R of entire functions satisfying |f (z)| ≤ kfˆk1 e2πby , for
z = x + iy ∈ C. Hence the only function both time- and band-limited is 0 a.e.
Whittaker–Kotel’nikov–Shannon
sampling theorem: for f ∈ P W (b), we have f (t) =
P∞
(1/2b) n=−∞ f (n/2b)kn/2b (t), where ks (t) = sin 2πb(t − s)/(π(t − s)), the reproducing
√
kernel function. Moreover, the functions (1/ 2b)kn/2b (t) form an o.n.b. of P W (b)
and hf, ks i = f (s) for all s ∈ R: this is the reproducing kernel Hilbert space (r.k.H.s.)
property.
Other r.k.H.s. include the Hardy space H 2 on the open unit disc D, where kw (z) =
1/(1 − wz); also the space Cn on the set {1, 2, . . . , n}. In general ks (t) = kt (s) and
ks (s) > 0 unless all functions vanish at s.
Now define V0 = P W (1/2) and Vj = P W (2j−1 ) for j ∈ Z. These give a multiresolution
approximation, as defined above, and an orthonormal basis of Vj is given by
2j/2 φ(2j t − k), k ∈ Z, where φ(t) = sin πt/(πt) = k0 (t).
Then the Littlewood–Paley wavelet is ψ(t) where ψ̂(w) = χ(1/2,1) (w), i.e.,
1
ψ(t) = (sin 2πt − sin πt)/(πt). As before we obtain an o.n.b. of L2 (R), consisting of
wavelets 2j/2 ψ(2j t − k), j, k ∈ Z, where for a fixed j they form an o.n.b. of a space
Wj such that Vj ⊕ Wj = Vj+1 .
A Riesz basis (φk ) in a Hilbert spacePH satisfies (i) P
H is the closedPlinear span of
2 1/2
the (φk ), and (ii) ∃A, B > 0 s.t. A( k |ak | )
≤ k ak φk k ≤ B( k |ak |2 )1/2 for
all (ak ) ∈ ℓ2 (finite Riesz bases in finite-dimensional spaces are also allowed, indeed
any algebraic basis is a Riesz basis in this case). Then (φk ) is a Riesz basis ⇐⇒
∃U : H → H bounded, linear, invertible, such that (U φk ) is an orthonormal basis.
P
A frame (φk ) in H satisfies ∃A, B > 0 s.t. Akφk2 ≤ k |hφ, φk i|2 ≤ Bkφk2 for all
φ ∈ H. Then o.n.b. =⇒ R.b. =⇒ frame. The numbers hφ, φk i are called the frame
coefficients of φ. In a finite-dimensional space, a frame is just any finite spanning set.
Now (φk ) is an infinite
linear map T : ℓ2 → H (ONTO)
P frame iff there is a bounded
such that T ((ak )) =
ak φk for all (ak ) ∈ ℓ2 . Define the mapping S = T T ∗ : H → H
P
by Sψ = hψ, φk iφk . Then S is a positive, bounded, invertible operator, and (φek ) =
(S −1 φk ) is also a frame, the dual frame to (φk ). The dual frame to (φek ) is (φk ) once
P
more. Also ψ = hψ,
φk iφek . We can invert S using
P∞
−1
S = (2/(B + A)) n=0 (I − 2S/(B + A))n , which converges rapidly.
Windowed Fourier transforms: take g ∈ L2 (R) of norm 1, often real and positive; write
ft (u) = f (u)g(u − t), for f ∈ L2 (R) and t ∈ R. Define fe(w, t) = fˆt (w). This also
ê
equals e−2πiwt f (−t, w), where the RHS is formed using the window ĝ (time-frequency
R∞ R∞
localization). Inversion formula: f (u) = −∞ −∞ fe(w, t)g(u − t)e2πiwu dw dt, when all
integrals converge absolutely. Also kfek2 = kf k2 , i.e., we get a norm-preserving linear
operator W : L2 (R) → L2 (R2 ) (not onto). This can be discretized: fe(mw0 , nt0 ) =
hf, gmni, where gmn (u) = e2πimw0 u g(u − nt0 ). For suitable w0 , t0 > 0, these form a
frame, and we can invert the discrete transform.
Heisenberg’s inequality: for f ∈ L2 (R), ktf (t)k2 kwfˆ(w)k2 ≥ kf k22 /(4π), with equality
for the Gaussian function.
R∞
Wavelet transform: let ψ ∈ L2 (R) satisfy Cψ = −∞ (|ψ̂(w)|2 /|w|) dw < ∞, and write
ψ x,y (t) = |x|−1/2 ψ((t−y)/x), for x, y ∈ R withRxR 6= 0. Define f ◦ (x, y) = hf, ψ x,y i. Better ‘zooming’ properties than the W.F.T. Also
f ◦ (x, y)g ◦(x, y) dx dy/x2 = Cψ hf, gi,
giving a weak inversion formula. The transform maps L2 (R) isometrically into a subspace H of L2 (R2 , µ), where dµ = dx dy/(Cψ x2 ), which is a r.k.H.s. on R2 with kernel
k(s,t) (x, y) = hψ x,y , ψ s,ti.
Discretising: write ψj,k (t) = aj/2 ψ(aj t − kb) for some fixed a > 1 and b > 0 and
consider inner products hf, ψj,k i. Again we would like a frame (e.g. for the Mexican
2
hat function C(1 − x2 )e−x /2 , the choice a = 2 and b = 1 will do).
Other wavelets can be constructed using a MRA (Vj )j∈Z , a father wavelet φ and a
mother wavelet ψ, as in the two examples above.
Wavelet
Properties of ψ(t)
Properties of ψ̂(w)
Haar
Compact support, discontinuous
O(1/w), C ∞
∞
Littlewood–Paley
O(1/t), C
Compact support, discontinuous
∞
Meyer
Rapidly-decreasing, C
Compact support, can be C ∞
k
Battle–Lemarié
Rapidly-decreasing, C
O(1/wk ), C ∞
Daubechies
Compact support, C k
O(1/wk ), C ∞
Compact support of both ψ and ψ̂ is always impossible; exponential decay and smoothness for both isn’t possible for o.n.b. of wavelets, but is for frames (e.g. Mexican hat).
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