Nivel cognitivo sobre la Ley de Régimen Tributario Interno

Prerequisites: §6C(i),§6E,§6F,§0C. Recommended: §8A.

Let f, g:X−→C be complex-valued functions. Recall from §6C(i) that we define theirinner product:

hf, gi := 1

M

Z

X

f(x)·g(x) dx,

whereM is the length/area/volume of domain X. Once again,

kfk₂ := hf, fi1/2 = 1 M Z X f(x)f(x)dx 1/2 = 1 M Z X |f(x)|2 dx 1/2 .

The concepts of orthogonality, L2 distance, and L2 convergence are exactly the same as before. Let L2([−L, L]; C) be the set of all complex-valued functions

f : [−L, L]−→Cwithkfk₂<∞. For all n∈Z, let

En(x) := exp πinx L .

(thus, E0 =11 is the constant unit function). For alln >0, notice that Euler’s Formula (see page 551) implies:

En(x) = Cn(x) + i·Sn(x)

and E−n(x) = Cn(x) − i·Sn(x)

(8D.1)

Also, note that hEn,Emi = 0 if n 6= m, and kEnk₂ = 1 (Exercise 8D.1), so

E

these functions form an orthonormal set.

If f : [−L, L] −→ C is any function with kfk₂ < ∞, then we define the

(complex) Fourier coefficientsof f:

b fn := hf,Eni = 1 2L Z L −L f(x)·exp −πinx L dx. (8D.2)

The(complex) Fourier Seriesoff is then the infinite summation of functions: ∞

X

n=−∞

b

fn·En. (8D.3)

Theorem 8D.1. Complex Fourier Convergence

(a) The set {. . . ,E−1,E0,E1, . . .} is anorthonormal basis for L2([−L, L]; C).

Thus, if f ∈L2([−L, L]; C), then the complex Fourier series (8D.3) con-

verges tof inL2-norm.

Furthermore, {fb_n}∞_n=−∞ is the unique sequence of coefficients with this

property.

(b) Iff is continuously differentiable1on[−π, π], then the Fourier series(8D.3)

convergespointwiseon (−π, π).

More generally, iff is piecewiseC1_{, then the complex Fourier series(8D.3)}

converges to f pointwise on each C1 _{interval for f. In other words, if}

{j1, . . . , jm} is the set of discontinuity points of f and/or f0 in [−L, L], andjm < x < jm+1, thenf(x) = lim

N→∞ N X n=−N b fnEn(x). (c) If ∞ X n=−∞ fbn

< ∞, then the series (8D.3) converges to f uniformly on

[−π, π].

(d) Supposef : [−π, π]−→ Ris continuous and piecewise differentiable, f0 ∈

L2[−π, π], and f(−π) = f(π). Then the series (8D.3) converges to f

uniformlyon[−π, π].

(e) Iff is piecewiseC1_{, andK⊂(j}

m, jm+1)is any closed subset of aC1 interval

off, then the series(8D.3) convergesuniformlytof onK.

Proof. For(a) is Exercise 8D.2 (Hint: Use Theorem 8A.1(a) on page 162 and E

Proposition 8D.2 below).

For a direct proof of (a), see [Kat76, §I.5.5, p.29-30].

(b)isExercise 8D.3 (Hint: (i) use Theorem 8A.1(b) on page 162 and Proposition E

8D.2 below. (ii) For a second proof, derive(b)from from(e).)

(c) is Exercise 8D.4 (Hint: Use the Weierstrass M-test, Proposition 6E.13 on E

page 129.)

(d) is Exercise 8D.5 (Hint: use Theorem 8A.1(d) on page 162 and Proposition E

8D.2 below).

For a direct proof of (d) see [WZ77, Theorem 12.20, p.219].

For(e)see [Fol84, Theorem 8.43, p.256] or [Kat76, Corollary on p.53 of§II.2.2]. 2

1_{This means thatf(x) =f}

r(x) +ifi(x), wherefr: [−L, L]−→Randfi: [−L, L]−→Rare

Proposition 8D.2. Relation between Real and Complex Fourier Series

Letf : [−π, π]−→R be a real-valued function, and let{A_n}∞_n=0 and {B_n}∞_n=1

be its real Fourier coefficients, as defined on page 161. We can also regardf as a complex-valued function; let{fb_n}∞_n=−∞ be the complex Fourier coefficients of

f, as defined by equation(8D.2)on page 172. Let n∈N₊. Then

(a) fb_n= 1₂(A_n−iB_n), and fb_−n = fb_n = 1₂(A_n+iB_n).

(b) Thus,An=fb_n+fb_−n, and B_n=i(fb_n−fb_−n).

(c) fb₀=A₀.

Proof. Exercise 8D.6 Hint: use the equations (8D.1). 2 E

Exercise 8D.7. Show that Theorem 8D.1(a) and Theorem 8A.1(a) are equivalent,

E

using the Proposition 8D.2.

Remark 8D.3: Further remarks on Fourier convergence

(a) In Theorems 7A.1(b), 7A.4(b), 8A.1(b) and 8D.1(b), ifxis a discontinuity point off, then the Fourier (co)sine series converges to the average of the ‘left-hand’ and ‘right-hand’ limits off atx, namely:

f(x−) +f(x+)

2 , where f(x−) := limy%x f(y) and f(x+) := limy&x f(y). (b) If the hypothesis of Theorems 7A.1(c), 7A.4(c), 8A.1(c) or 8D.1(c) is sat- isfied, then we say that the Fourier series (real, complex, sine or cosine) converges absolutely. (In fact, Theorems 7A.1(d)[i], 7A.4(d)[i], 8A.1(d) or 8D.1(d) can be strengthened to yield absolute convergence). Absolute convergence is stronger than uniform convergence, and functions with ab- solutely convergent Fourier series form a special class; see [Kat76, §I.6, p.31-33] for more information.

(c) In Theorems 7A.1(e), 7A.4(e), 8A.1(e) and 8D.1(e), we don’t quite needf

to bedifferentiableto guarantee uniform convergence of the Fourier (co)sine series. Let α >0 be a constant; we say that f is α-H¨older continuous on [−π, π] if there is someM <∞ such that,

For allx, y∈[0, π], |f(x)−f(y)|

|x−y|α ≤ M.

Bernstein’s Theoremsays: Iff isα-H¨older continuous for someα > 1₂, then the Fourier series (real, complex, sine or cosine) off will converge uniformly

(indeed, absolutely) tof; see [Fol84, Theorem 8.39] or [Kat76, Thm 6.3 on p.32]. (Iff was differentiable, then f would be α-H¨older continuous with

α= 1, so Bernstein’s Theorem immediately implies Theorems 7A.1(e) and 7A.4(e).)

(d) Thetotal variation off is defined

var(f) := sup N∈N sup −π≤x0<···<xN≤π N X n=1 f(xn)−f(xn−1) (∗) Z π −π f0(x) dx.

Here, the supremum is taken over all finite increasing sequences {−π ≤

x0 < x1 < · · · < xN ≤ π} (for any N ∈ N), and equality (∗) is true if and only if f is continuously differentiable. Zygmund’s Theorem says: if var(f)<∞(i.e. f hasbounded variation) andf isα-H¨older continuous for someα >0, then the Fourier series of f will converge uniformly (indeed, absolutely) tof on [−π, π]; see [Kat76, Thm 6.4 on p.33].

(e) However, merely beingcontinuousisnotsufficient for uniform Fourier con- vergence, or even pointwise convergence. There exists a continuous func- tion f : [0, π]−→ R whose Fourier series does not converge pointwise on (0, π) —i.e. the series diverges at some points in (0, π); see [WZ77, Theo- rem 12.35, p.227] or [Kat76, Theorem 2.1, p.51]. Thus, Theorems 7A.1(b), 7A.4(b), 8A.1(b) and 8D.1(b) are false if we replace ‘differentiable’ with ‘continuous’.

(f ) Fixp∈[1,∞). For anyf : [−π, π]−→C, we define theLp-norm off: kfk_p = Z π −π |f(x)|p dx 1/p .

(Thus, if p = 2, we get the familiar L2-norm kfk₂). Let Lp[−π, π] be the set of all integrable functions f : [−π, π]−→ C such that kfk_p < ∞. Theorem 8D.1(a) say that, iff ∈L2[−π, π], then the complex Fourier series of f converges to f in L2-norm. The Fourier series of f also converges in

Lp-norm for any other p ∈ (1,∞). That is, for any p ∈ (1,∞) and any

f ∈Lp[−π, π], we have lim N→∞ f − N X n=−N b fnEn p = 0.

See [Kat76, Theorem 1.5, p.50]. Iff ∈Lp[−π, π] is purely real-valued, then the same statement holds for the real Fourier series:

lim N→∞ f − A0+ N X n=1 AnCn+ N X n=1 BnSn ! p = 0.

To understand the significance of Lp-convergence, we remark that if p is very large, then Lp convergence is ‘almost’ the same as uniform conver- gence. Also:

• Ifp > q, thenLp[−π, π]⊂Lq[−π, π]. (Exercise 8D.8). E

For example, iff ∈L3[−π, π], then it follows thatf ∈L2[−π, π] (but not vice versa). Iff ∈L2[−π, π], then it follows that f ∈L3/2[−π, π] (but not vice versa).

• Ifp > q, and the Fourier series off converges to f inLp-norm, then it also converges to f in Lq-norm; see e.g. [Fol84, Proposition 6.12, p.178].

For example, iff ∈L2[π, π], then Theorem 8D.1(a) implies that the Fourier series off converges to f inLq-norm for allq ∈[1,2]. (How- ever, ifq <2, then there are functions inLq[−π, π] to which Theorem 8D.1(a) does not apply).

Finally, similar Lp-convergence statements hold for the Fourier (co)sine series of real-valued functions inLp[0, π].

(g) The pointwise convergence of a Fourier series is a somewhat subtle and complicated business, once you depart from the realm ofC1 _{functions. In}

particular, the Fourier series of continuous (but non-differentiable) func- tions can be badly behaved. This is perplexing, because we know that Fourier series converge inL2 norm for any function inL2[−π, π] (which in- cludes all sorts of strange functions which are not differentiable anywhere). To bridge the gap betweenL2_{and pointwise convergence, a variety of other}

‘summation schemes’ have been introduced for Fourier coefficients. These include:

• TheCes´aro mean lim

N→∞ 1 N N X n=1 SN(f), whereSN(f) := N X n=−N b fnEn is

theNth partial sum of the complex Fourier series (8D.3). • TheAbel mean lim

r%1

∞

X

n=−∞

r|n|fb_nE_n.

These sums have somewhat nicer convergence properties than the ‘stan- dard’ Fourier series (8D.3). (See § 18F on page 461 for further discussion of the Abel mean.)

(h) There is a close relationship between the Fourier series of complex-valued functions on [−π, π], and the Laurent series of complex-analytic functions defined near the unit circle; see§ 18E on page 454.

(i) Remark (h) and the periodic boundary conditions required for Theorem 8D.1(d) both suggest that the Fourier series ‘wants’ us to identify the interval (−π, π] with the unit circleSin the complex plane, via the bijection

φ: (−π, π]−→ Sdefined by φ(x) =eix. Now,S is anabelian group under the complex multiplication operator. That is: ifs, t∈S, then their product

s·t is also in S, the multiplicative inverse s−1 is in S, and the identity element 1 is an element ofS. Furthermore,Sis a compact subset ofC, and the multiplication operation is continuous with respect to the topology of S. In summary, S is a compact abelian topological group. The functions {En}∞n=−∞ are then continuous homomorphismsfrom S into S (these are called thecharactersof the group).

The existence of the Fourier series (8D.3) and the convergence properties enumerated in Theorem 8D.1 are actually aconsequenceof these facts. In fact, ifG is anycompact abelian topological group, then one can develop a version of Fourier analysis on G. The characters of G are the continu- ous homomorphisms from G into the unit circle group S. The set of all characters ofGforms an orthonormal basis for L2(G), so that almost any ‘reasonable’ function f : G −→ C can be expressed as a complex-linear combination of these characters.

The study of Fourier series, their summability, and their generalizations to other compact abelian groups is calledharmonic analysis, and is a crucial tool in many areas of mathematics, including the ergodic theory of dy- namical systems and the representation theory of Lie groups. See [Fol84, Ch.8], [WZ77, Ch.12] or the book [Kat76] to learn more about this vast and fascinating area of mathematics.