Fourier Anaysis, Lecture Notes- Maths, Study notes of Mathematics

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More real analysis

Week 6 Lecture 11

Further problems raised in the nineteenth century by:

• investigations of Fourier series
• rising standards of proof

Joseph Fourier Th´eorie analytique de la chaleur (1822)

Suppose φ(x) = a sin x+b sin 2x+c sin 3x+...

and φ(x) = xφ′(0) + x 2 2 φ

′′(0) + x^3 6 φ

Then the coefficient of sin nx is

π

∫ (^) π 0

φ(x) sin nxdx

Fourier’s results gave rise to many important questions concerning:

• convergence
• uniform convergence
• integration
• existence of limits

A theorem of Cauchy (1821)

When the various terms of a series are con- tinuous with respect to that variable... the sum s of the series is also a continuous func- tion of x.

Another theorem of Cauchy (1823)

Suppose that in the interval [x 0 , X] we have A < f ′(x) < B;

then we also have

A < f (X) − f (x 0 ) X − x 0

< B

The need for ‘infinitely slow’ or ‘uniform’ convergence was only gradually recognised:

• Phillip Seidel (Berlin), 1848 (who began by refuting Cauchy’s theorem)
• Gabriel Stokes (Cambridge) 1849 (in ‘On the critical values of the sums of periodic series’)

Integration

• Leonhard Euler (1768) Institutiones calculi integralis (1768)

Definition 1 Clearly just as in Analysis two operations are always contrary to each other, as subtrac- tion to addition, division to multiplication, extraction of roots to raising of powers, so also by similar reasoning integral calculus is contrary to differential calculus.

Definition 2 Since the differentiation of any function of any function of x has a form of this kind: Xdx, when a differential Xdx of such a form is proposed, in which X is any function of x, the function whose differential = Xdx is called its integral, and is usually indicated by the prefix

∫ : so that

∫ Xdx denotes that variable quantity whose differential is Xdx.

Cauchy’s integral (1823)

Define:

S = (x 1 − x 0 )f (x 0 ) + (x 2 − x 1 )f (x 1 ) +...

+(X − xn− 1 )f (xn− 1 )

Cauchy’s proof of the fundamental theorem (1823)

Suppose F =

∫ (^) X x 0 f^ (x)dx

From Lesson 22: F (x) = (x − x 0 )f [(x 0 ) + θ(x − x 0 )]

From Lesson 23: ∫ x+α x 0 f^ (x)^ −^

∫ (^) x x 0 f^ (x) =^

∫ (^) x+α x f^ (x) = αxf (x + θα)

So that F (x + α) − F (x) = αf (x + θα)

Riemann integration (1853 )

What are we to understand by ∫ (^) b a f^ (x)dx^?

Define

S = δ 1 f (a+δ 1 ǫ 1 )+δ 2 f (x 1 +δ 2 ǫ 2 )+.. .+δnf (xn− 1 +δnǫn)

Does a limit A exist however the δ and ǫ are chosen?

Lebesgue integration (1901)