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Material Type: Notes; Professor: Laugesen; Class: Harmonic Analysis; Subject: Mathematics; University: University of Illinois - Urbana-Champaign; Term: Fall 2009;
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*Copyright c© 2009, Richard S. Laugesen ([email protected]). This work is
licensed under the Creative Commons Attribution–Noncommercial–Share Alike 3. Unported License. To view a copy of this license, visit http://creativecommons. org/licenses/by-nc-sa/3.0/.
Preface
A textbook presents more than any professor can cover in class. In contrast, these lecture notes present exactly*^ what I covered in Harmonic Analysis (Math 545) at the University of Illinois, Urbana–Champaign, in Fall 2008. The first part of the course emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle, for example, where it is used to prove Lp^ convergence of Fourier series. Maximal functions and Calder´on– Zygmund decompositions are treated in Rd, so that they can be applied again in the second part of the course, where the Fourier transform is studied. Real methods are used throughout. In particular, complex methods such as Poisson integrals and conjugate functions are not used to prove bounded- ness of the Hilbert transform. Distribution functions and interpolation are covered in the Appendices. I inserted these topics at the appropriate places in my lectures (after Chap- ters 4 and 12, respectively). The references at the beginning of each chapter provide guidance to stu- dents who wish to delve more deeply, or roam more widely, in the subject. Those references do not necessarily contain all the material in the chapter. Finally, a word on personal taste: while I appreciate a good counterex- ample, I prefer spending class time on positive results. Thus I do not supply proofs of some prominent counterexamples (such as Kolmogorov’s integrable function whose Fourier series diverges at every point). I am grateful to Noel DeJarnette, Eunmi Kim, Aleksandra Kwiatkowska, Kostya Slutsky, Khang Tran and Ping Xu for TEXing parts of the document.
Please email me with corrections, and with suggested improvements of any kind.
Richard S. Laugesen Email: [email protected] Department of Mathematics University of Illinois Urbana, IL 61801 U.S.A.
*modulo some improvements after the fact
Definition 1.1. For f ∈ L^1 (T) and n ∈ Z, define
f̂ (n) = n-th Fourier coefficient of f
=
2 π
T
f (t)e−int^ dt. (1.1)
The formal series S[f ] =
f (n)eint^ is the Fourier series of f.
Aside. For f ∈ L^2 (T), note f̂ (n) = 〈f, eint〉 where 〈f, g〉 = (^21) π
T f^ (t)g(t)^ dt^ is that L^2 inner product. Thus f̂ (n) =amplitude of f in direction of eint. See Chapter 5. Theorem 1.2 (Basic properties). Let f, g ∈ L^1 (T), j, n ∈ Z, c ∈ C, τ ∈ T. Linearity (̂f + g)(n) = f̂ (n) + ̂g(n) and (̂cf )(n) = c f̂ (n) Conjugation ̂f (n) = f̂ (−n) Trigonometric polynomial P (t) =
n=−N ane
int (^) has P̂ (n) = an for |n| ≤ N and P̂ (n) = 0 for |n| > N
̂ takes translation to modulation, ̂fτ (n) = e−inτ^ f̂ (n)
̂ takes modulation to translation, [f (t)eijt]̂ (n) = f̂ (n − j)
̂ : L^1 (T) → `∞(Z) is bounded, with | f̂ (n)| ≤ ‖f ‖L (^1) (T)
Hence if fm → f in L^1 (T) then f̂m(n) → f̂ (n) (uniformly in n) as m → ∞. Proof. Exercise.
Lemma 1.3 (Difference formula). For n 6 = 0,
f̂ (n) = 1 4 π
T
[f (t) − f (t − π/n)] e−int^ dt.
Proof.
f̂ (n) = − 1 2 π
T
f (t)e−in(t+π/n)^ dt since e−iπ^ = − 1
2 π
T
f (t − π/n)e−int^ dt (1.2)
by t 7 → t − π/n and periodicity. By (1.2) and the definition (1.1),
f̂ (n) =^1 2
f (n) +
f (n) =
4 π
T
[f (t) − f (t − π/n)] e−int^ dt.
Lemma 1.4 (Continuity of translation). Fix f ∈ Lp(T), 1 ≤ p < ∞. The map
φ : T → Lp(T) τ 7 → fτ
is continuous.
Proof. Let τ 0 ∈ T. Take g ∈ C(T) and observe
‖fτ − fτ 0 ‖Lp(T) ≤ ‖fτ − gτ ‖Lp(T) + ‖gτ − gτ 0 ‖Lp(T) + ‖gτ 0 − fτ 0 ‖Lp(T) = 2‖f − g‖Lp(T) + ‖gτ − gτ 0 ‖Lp(T) → 2 ‖f − g‖Lp(T)
as τ → τ 0 , by uniform continuity of g. By density of continuous functions in Lp(T), 1 ≤ p < ∞, the difference f − g can be made arbitrarily small. Hence lim supτ →τ 0 ‖fτ − fτ 0 ‖Lp(T) = 0, as desired.
Corollary 1.5 (Riemann–Lebesgue lemma). f̂ (n) → 0 as |n| → ∞.
Proof. Lemma 1.3 implies
| f̂ (n)| ≤
‖f − fπ/n‖L (^1) (T),
which tends to zero as |n| → ∞ by the L^1 -continuity of translation in Lemma 1.4, since f = f 0.
Smoothness and decay
The Riemann–Lebesgue lemma says f̂ (n) = o(1), with f̂ (n) = O(1) explicitly by Theorem 1.2. We show the smoother f is, the faster its Fourier coefficients decay.
Theorem 1.6 (Less than one derivative). If f ∈ Cα(T), 0 < α ≤ 1 , then f̂ (n) = O(1/|n|α).
Here Cα(T) denotes the H¨older continuous functions: f ∈ Cα(T) if f ∈ C(T) and there exists A > 0 such that |f (t) − f (τ )| ≤ A|t − τ |α^ whenever |t − τ | ≤ 2 π.
Convolution
Definition 1.10. Given f, g ∈ L^1 (T), define their convolution
(f ∗ g)(t) =
2 π
T
f (t − τ )g(τ ) dτ, t ∈ T.
Theorem 1.11 (Convolution and Fourier coefficients). If f ∈ Lp(T), 1 ≤ p ≤ ∞, and g ∈ L^1 (T), then f ∗ g ∈ Lp(T) with
‖f ∗ g‖Lp(T) ≤ ‖f ‖Lp(T)‖g‖L (^1) (T) (1.3)
and (̂f ∗ g)(n) = f̂ (n)̂g(n), n ∈ Z.
Further, if f ∈ C(T) and g ∈ L^1 (T) then f ∗ g ∈ C(T).
Thus ̂ takes convolution to multiplication.
Proof. That f ∗ g ∈ Lp(T) satisfies (1.3) is exactly Young’s Theorem A.3. Then by Fubini’s theorem,
(̂f ∗ g)(n) = 1 2 π
T
2 π
T
f (t − τ )g(τ ) dτ
e−int^ dt
2 π
T
2 π
T
f (t − τ )e−in(t−τ^ )^ dt
g(τ )e−inτ^ dτ
= f̂ (n)̂ g(n).
Finally, if f ∈ C(T) and g ∈ L^1 (T) then f ∗ g is continuous because (f ∗ g)(t + δ) → (f ∗ g)(t) as δ → 0 by uniform continuity of f.
Convolution facts [Katznelson, Section I.1.8]
(f ∗ g)(t) =
2 π
T
f (t − τ )g(τ ) dτ
2 π
T
f (θ)g(t − θ) dθ where τ = t − θ, dτ = −dθ
= (g ∗ f )(t).
Convolution is also associative, and linear with respect to f and g.
∑n j=−n aj^ e
ijt (^) then
(P ∗ f )(t) =
∑^ n
j=−n
aj f̂ (j)eijt. (1.4)
Proof.
(P ∗ f )(t) =
∑^ n
j=−n
aj
2 π
T
eij(t−τ^ )f (τ ) dτ
∑^ n
j=−n
aj eijt^ f̂ (j).
[Sanity check: (̂P ∗ f )(j) = aj f̂ (j) = P̂ (j)f̂ (j) as expected.]
More generally, (1.4) holds for P (t) =
j=−∞ aj^ e
ijt (^) provided {aj } ∈ ` (^1) (Z).
Definition 2.1. A summability kernel is a sequence {kn} in L^1 (T) satisfying:
1 2 π
T
kn(t) dt = 1 (Normalization) (S1)
sup n
2 π
T
|kn(t)| dt < ∞ (L^1 bound) (S2)
lim n→∞
{δ<|t|<π}
|kn(t)| dt = 0 (L^1 concentration) (S3)
for each δ ∈ (0, π).
Some kernels satisfy a stronger concentration property:
lim n→∞ sup δ<|t|<π
|kn(t)| = 0 (L∞^ concentration) (S4)
for each δ ∈ (0, π).
Call the kernel positive if kn ≥ 0 for each n.
Example 2.2. Define the Dirichlet kernel
Dn(t) =
∑^ n
j=−n
eijt^ (2.1)
ei(n+1)t^ − e−int eit^ − 1
by geometric series (2.2)
sin
(n + 12 )t
sin
2 t
(S1) holds by (2.1). You can show (optional exercise) that ‖Dn‖L^1 (T) ∼ (const.) log n as n → ∞, so that (S2) fails. ∴ {Dn} is not a summability kernel.
Example 2.3. Define the Fej´er kernel
Fn(t) =
D 0 (t) + · · · + Dn(t) n + 1
∑^ n
j=−n
|j| n + 1
eijt^ by (2.4) and (2.1) (2.5)
n + 1
sin
(n+ 2 t
sin
2 t
by (2.4), (2.2) and geometric series (2.6)
-Π Π
10
20
Figure 2.1: Dirichlet kernel with n = 10
-Π Π
10
Figure 2.2: Fej´er kernel with n = 10
-Π Π
10
20
Figure 2.3: Poisson kernel with r = 0. 9
-Π Π
10
20
Figure 2.4: Gauss kernel with s = 0. 01
The Gauss kernel is indexed by s ∈ (0, ∞), with limiting process s ↘ 0. The analogue of (S1) holds by (2.10), and Gs ≥ 0 by (2.11) so that (S2) holds also. For (S4),
sup δ<|t|<π
|Gs(t)| ≤
2 π √ 4 πs
e−δ
(^2) / 4 s
n 6 =
e−(πn)
(^2) / 4 s
by (2.11)
→ 0 as s ↘ 0.
∴ {Gs} is a positive summability kernel.
Connection to Fourier series
Sn(f ) = Dn ∗ f
Proof. Dn(t)
∑n j=−n 1 e ijt (^) implies
(Dn ∗ f )(t) =
∑^ n
j=−n
1 f̂ (j)eijt^ = Sn(f )
by Convolution Fact (1.4).
σn(f ) = Fn ∗ f
Proof. Fn(t)
∑n j=−n(1^ −^
|j| n+1 )e
ijt (^) implies
(Fn ∗ f )(t) =
∑^ n
j=−n
|j| n + 1
f (j)eijt^ = σn(f )
by Convolution Fact (1.4). Alternatively, use that σn(f ) = [S 0 (f ) + · · · + Sn(f )]/(n + 1) and Fn = [D 0 + · · · + Dn]/(n + 1).
Thus for summability of Fourier series, we want Fn ∗ f → f.
Abel mean of S[f ] = Pr ∗ f
