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Math 212a Lecture 1 Shlomo Sternberg Outline Some history. Irrationality of π. Transcendental numbers. What is the definition of a function? Some dubious 18th century sums. Toeplitz’s theorem. Cesaro summability. Abel summability Fourier and Fourier series. The Gibbs phenomenon. Math 212a Lecture 1 Introduction. Shlomo Sternberg September 2, 2014

Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

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Page 1: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Math 212a Lecture 1Introduction.

Shlomo Sternberg

September 2, 2014

Page 2: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

The official title of this course is

“Theory of functions of a real variable”.

I assume that you know what the real numbers are - that theyare the completion of the rational numbers, and that they arecomplete in the sense that any Cauchy sequence of realnumbers converges to a real number limit.

In the notes to this course I review the concept of “completion”in the more general context of metric spaces. But thetreatment depends on existence of the real number system.

Page 3: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

1 Some history.Irrationality of π.Transcendental numbers.

2 What is the definition of a function?

3 Some dubious 18th century sums.Toeplitz’s theorem.Cesaro summability.Abel summability

4 Fourier and Fourier series.

5 The Gibbs phenomenon.

6 Appendix: Proof that e is transcendental.

Page 4: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

History: irrational numbers.

The Greeks know that√

2 is irrational. A proof appears in ascholium to Euclid’s elements. To the best of my knowledge,no such statement was made about π in Greek mathematics.

Page 5: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

al Biruni, Maimonides.

The great Islamic mathematician, astronomer, and historian, alBiruni (937-1018) states in al-Qanun al-Masudi (3 vols.Hyderabad 1954-56) vol., Part III ch. 5 p.303 “the perimeter ofthe circle to its diameter is a certain ratio; its number(meaning the measure of the perimeter) to its number (that ofthe diameter) is a ratio which is surd (Arabic: samm)”. I amindebted to Prof. Tony Levy for this reference. So al Biruni“knew” that π is irrational.

The great Jewish jurist and philosopher, Maimonides(1135-1205), also states in his commentary to the Mishnah(Eruvin I-5) that π is irrational. Neither of these authors offer aproof of this fact.

The first proof was given by Lambert in 1761. Lambert’s proofdepended on the integral calculus.

Page 6: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Ivan Niven’s version of the proof.

Lambert’s proof was chewed over and over for severalcenturies. The proof I am about to present due to Niven seemsto have reduced the argument to a bare minimum.

To prove: that π = ab where a and b are (positive) integers is

impossible.

Suppose the contrary. For any positive integer n define f = fnby

f (x) =xn(a− bx)n

n!so 0 < f (x) <

πnan

n!

on 0 < x < π. Define

F := f − f ′′ + f iv − f vi + · · ·+ (−1)nf (2n).

Page 7: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

f (x) = xn(a−bx)n

n! so f (k)(0) = 0 for k < n and = the(k − n)th derivative of (a− bx)n at zero for k ≥ n which is aninteger. Hence F (0) is an integer. Also

f (π − x) =1

n!

(a

b− x)n

(a− a + bx)n = f (x)

so F (π) is an integer.

Page 8: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

We have shown that F (0) and F (π) are integers. NowF = f − f ′′ + f iv − f vi + · · ·+ (−1)nf (2n) so

F ′′ + F = f .

We have (F ′ sin−F cos)′ = F ′′ sin +F ′ cos−F ′ cos +F sin =(F ′′ + F ) sin = f sin so by the fundamental theorem of thecalculus,∫ π

0f (x) sin(x)dx = (F ′ sin−F cos)

∣∣π0

= F (π) + F (0)

is a positive integer as the integrand is positive on (0, π). Butby choosing n large enough we can make the integral as smallas we like. This contradiction shows that π = a

b is impossible.

Page 9: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Liouville and Cantor.

The first person to show that there exist numbers which are“transcendental”, i.e. not solutions of an algebraic equationwith rational coefficients, was Liouville, who showed in 1844that the number

∞∑n=1

10−n!

is transcendental.

In 1874 Cantor showed the “most” real numbers aretranscendental by proving the the algebraic numbers arecountable and that the real numbers are uncountable.

Page 10: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Hermite and Lindemann.

Hermite proved in 1873 that e is transcendental, andLindemann proved in 1884 that π is transcendental using themethods developed by Hermite. A consequence of Lindemann’sresult is that it is impossible to “square the circle” by means ofstraightedge and compass, for such a construction would leadto an algebraic number θ with θ2 = π, and if θ is algebraic so isθ2.

I will present a proof that e is transcendental in the appendixto this lecture.

Page 11: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Definition of the real numbers. Dedekind andCantor.

It is therefore somewhat surprising that it was not until 1872that a definition was given of the real numbers. In fact twodefinitions, one via ”Dedekind cuts” given by Dedekind andone by Cantor via equivalence classes of Cauchy sequences ofrational numbers.

Page 12: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

The meaning of the word “function” however will not besettled in this course. Here is an example of the kind of troublewe will have to face: The first person to give a correct proof ofthe convergence of Fourier series to the function it shouldrepresent (for a reasonable class of functions) was Dirichlet in1829. In the course of his analysis of what Fourier series meanhe introduced the so-called “function” 1Q which takes thevalue 1 on the rational numbers and the value 0 on theirrational numbers.

Page 13: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Is 1Q a function?

It can not be evaluated or graphed on any computer sincecomputers use floating point arithmetic and see only rationalnumbers. From the point of view of a computer 1Q isidentically one.

From the point of view of Lebesgue integration theory whichwe will study later on this semester, two functions which agreeexcept on a set of “measure zero” are to be regarded as thesame. The rational numbers have measure zero in theLebesgue theory. So from the point of view of Lebesgue’stheory 1Q is identically zero.

Page 14: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Can we actually apply the definition?

Can we actually apply the definition? For example, Eulerintroduced the number γ defined as the limit as n→∞ of∑n

k=11k − log n. To this day we do not know if γ is rational or

irrational. So what is 1Q(γ)?

Conclusion: We should not admit Dirichlet’s 1Q into ourmenagerie of functions.

Page 15: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

What about Dirac’s “δ-function”?

This was defined by Dirac to be the “function” δ such thatδ(x) = 0 for x 6= 0 and is “so infinite at 0” that∫ ∞

−∞δ(x)g(x)dx = g(0)

for any continuous function g . From the point of view ofLebesgue’s (or Riemann’s) integration theory, this statement isnonsense: the value of a function at a single point (even ifinfinite) can not affect the value of an integral. When I was astudent learning about Lebesgue integration, we would laugh atthe physicists for believing in the existence of such a function.

Page 16: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Yet Dirac’s δ function is not only important in physics, we willbe making a lot of use of it in this course. It is easilyimplemented on the computer. It is the rule which assigns toeach continuous function g the value g(0). In other words, it isthe map

g 7→ g(0).

It is not a function of a real variable but rather a function of afunction of a real variable. We will definitely include the Diracδ function into our menagerie.

Page 17: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Since many of the “functions” we will study involve infiniteseries, we start by looking at some infinite series from the viewpoint of 18th century mathematicians:

Let a := 1− 1 + 1− 1 + · · · . Then

a = 1− (1− 1 + 1− 1 + · · · ) = 1− a.

So a = 12 .

Let s := 1− 2 + 3− 4 + 5− 6 + · · · . Then

s = 1 + (−2 + 3− 4 + · · · )= 1− (1− 1 + 1− 1 + · · · )− (1− 2 + 3− 4 + 5− 6 + · · · )= 1− a− s

so s = 14 .

Page 18: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Let Z := 1 + 2 + 3 + 4 + · · · . Then

Z − s = 0 + 4 + 0 + 8 + · · · = 4Z .

So 3Z = −s or

Z = − 1

12.

Page 19: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

18th century philosophy.

Euler: “ich glaube, dass jede series einem bestimmten Werthaben musse. Um aber allen Schwieregkeiten, welche dagegengemacht worden, zu begegnen, so sollte dieser Wert nicht mitdem Namen der Summe belegt werden, weil man mit dieserWort gemeiniglich eine solchen Begriff zu verknipfen pflegt, alswenn die Summe durch eine wirkliche Summierungherausgebracht wurde: welche Idee bei den seribusdivergentibus nicht stattfindet...”

Page 20: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Born: 15 April 1707 in Basel, Switzerland

Died: 18 Sept 1783 in St Petersburg, Russia

Leonhard Euler

Page 21: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Hardy Divergent series (p.5) “ it does not occur to a modernmathematician that a collection of symbols should have a‘meaning’ until one has been assigned to it by definition. [This]was not a triviality even to the greatest mathematicians of theeighteenth century. They had not the habit of definition: it wasnot natural to them to say, in so many words, “by X we meanY”. ... mathematicians before Cauchy asked not ‘How shall wedefine 1− 1 + 1− 1 + · · · ?’ but ‘What is 1− 1 + 1− 1 + · · ·?’, and this habit led them into unnecessary perplexities andcontroversies which were often really verbal.”

Page 22: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Some justifications.

The geometric series

1

1− z= 1 + z + z2 + z3 + z4 + · · ·

converges for |z | < 1. The function z 7→ 11−z exists and is

holomorphic for all z 6= 1, and its value at −1 is 12 .

So if we define the Abel sum of a series∑

an to be the limit(if it exists)

limx→1

∑anxn

then we can say that the Abel sum of 1− 1 + 1− 1 + · · · is 12 .

Page 23: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Differentiating the above geometric series for |z | < 1 gives

1

(1− z)2= 1 + 2z + 3z2 + 4z3 + · · · .

Again, the function z 7→ 1(1−z)2 is holomorphic for all z 6= 1 and

its value at −1 is 14 . So the Abel sum of 1− 2 + 3− 4 + · · · is

14 .

Page 24: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Justification of the formula 1 + 2 + 3 + · · · = − 112 .

This lies a bit deeper: The series

1 +1

2s+

1

3s+

1

4s+ · · ·

converges for <s > 1 and so defines a holomorphic functions 7→ ζ(s) there. It turns out that ζ has a singularity at s = 1but can be analytically continued to all z 6= 1. Its value at −1is − 1

12 .

In fact, Euler computes this value by means of what is knowntoday as the Euler-MacLaurin summation formula.

We shall get these results in lecture 6 after we develop theFourier transform - especially the Mellin transform.

Page 25: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Toeplitz’s theorem.

Let M = (mnk) be an infinite matrix (indexed by the positiveintegers N) which satisfies the following three conditions:

1 limn→∞mnk = 0 for all k (the terms in every column→ 0).

2 There is an H such that∑

k |mnk | ≤ H for every n. (Therows are uniformly absolutely summable.)

3 limn→∞∑

k mnk = 1. (The row sums tend to 1.)

Let sk → s be a convergent sequence of real numbers. (Forexample sk might be the partial sums of a convergent series.)Since the sk are bounded, condition 2) implies that the series

σn :=∑k

mnksk converges for each n.

Page 26: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Theorem

[Toeplitz.] σn → s.

Write sk − s = rk so σn = s∑

k mnk +∑

k mnk rk . Byhypothesis 3), s

∑k mnk → s. Since rk → 0 we are reduced to

proving the theorem for the case s = 0.

So we need to prove the following:

Page 27: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Hypotheses:

1 limn→∞mnk = 0 for all k (the terms in every column→ 0).

2 There is an H such that∑

k |mnk | ≤ H for every n. (Therows are uniformly absolutely summable.)

3 limn→∞∑

k mnk = 1. (The row sums tend to 1.)

4 sk → 0.

Letσn :=

∑k

mnksk .

To prove:σn → 0.

Page 28: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Proof.

For any ε > 0 choose N = N(ε) so large that |sk | < ε/2H fork > N. By 1) we may then choose P = P(ε) so large that

|mnk | <ε

2∑N

k=1 |sk |for k = 1, 2, . . . ,N, for n > P.

Then for n > P we have

|σn| ≤N∑

k=1

|mnksk |+ε

2H

∑k>N

|mnk |

≤ ε

2+

ε

2H

∞∑k=1

|mnk | ≤ ε.

Page 29: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

M summability.

Given any matrix M satisfying the conditions of Toeplitz’stheorem, it is possible that the sequence M · s makes sense forsome sequence s without s being convergent. That is, it isconceivable that the series σn =

∑k mnksk all converge. It is

also possible that the σn converge to some limit s. Then wewill say that

snM→ s,

or that “∑

aj is M - summable” to s, if sk =∑k

1 aj is thepartial sum of a series

∑aj .

Page 30: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Cesaro summability.

For example, the matrix

C :=

1 0 0 0 · · ·

1/2 1/2 0 0 · · ·1/3 1/3 1/3 0 · · ·

......

...... · · ·

clearly satisfies the hypotheses of Toeplitz’s theorem. Then

σn =1

n(s1 + s2 + · · ·+ sn) .

If s1 = 1, s2 = 0, s3 = 1 etc., then σn → 12 . These values of sk

are the partial sums of the series 1− 1 + 1− 1 + · · · . We saythat this series is Cesaro summable to 1

2 .

Page 31: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Abel summability.

Let 0 ≤ rn < 1, rn → 1. Let ank := (1− rn)rkn . (Here we areindexing over the non-negative integers.) Clearlylimn→∞ ank = 0 for each fixed k , and

∑k |ank | =

∑k ank = 1

by the geometric series. So

A = (ank) =(

(1− rn)rkn

)satisfies the hypotheses of Toeplitz’s theorem.

For this choice of matrix we have:

Page 32: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

σn = (1− rn)∞∑k=0

sk rkn

= s0 + (s0 − s1)rn + (s2 − s1)r 2n + · · · .

So if the sn are partial sums:s0 = c0, s1 = c0 + c1, s2 = c0 + c1 + c2 etc., then

σn =∑k≥0

ck rkn .

If the σn approach a limit we have “Abel summability relativeto this subsequence”. If the limit as r → 1 of

∑k ck rk exists

(not merely for a subsequence) then we have we have what isknown as Abel summability. ( It is easy to check that Cesarosummability implies Abel summability.)

Page 33: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Some history.

Daniel Bernoulli(1753) proposed a trigonometric series as asolution for the problem of vibrating strings and Euler foundthe formula for the “Fourier coefficients”

f (x) =∑

ane inx , an =1

∫ π

−πf (x)e−inxdx

in 1757.

Page 34: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

The problem of how ”heat diffuses” in a continuous mediumdefied the mathematics of early 1800’s. This problem wassolved by Fourier using “intuitive methods”. He submitted hismanuscript Theorie de la propagation de la chaleur to theInstitut de France on 21 December 1807. The prize committeeconsisted of Laplace, Lagrange, Lacroix and Monge, withPoisson acting as secretary. They rejected the manuscript.Lagrange was particularly firm, questioning the meaning of theequality sign in

f (x)“ = ”∑

ane inx , an =1

∫ π

−πf (x)e−inxdx .

Page 35: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

For example for the “square wave”

f (x) :=

{−1 −π ≤ x < 0,1 0 ≤ x < π

the series is4

π

∑n≥1

sin(2n − 1)x

2n − 1.

How could this series converge when the series of its coefficients

4

π

(1 +

1

3+

1

5+

1

7+ · · ·

)diverges?

Page 36: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Poisson summarized the rejection in 5 pages saying essentiallythat the paper was rejected on the grounds that it containednothing new or interesting. Fourier submitted a revised versionto essentially the same group of judges in 1811 as a candidatefor the Grand prix de mathematiques for 1812. Although theyawarded him the prize, they refused to publish the paper in theMemoires de l’Academie des Sciences. In 1824 Fourier becamethe Secretary of the Academy and he then had his 1811 paperpublished.

Page 37: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Fourier’s influence.

In addition to stimulating the development of mathematics andphysics for the next two centuries, the work of Fourier changedthe notion of a function. Previously a function had been anatural object, such as a polynomial or exponential ortrigonometric function. In short, a function was somethinggiven by a closed expression. They were like biological speciesto be discovered and classified. Now, after Fourier, a functioncould be invented and be quite arbitrary. Also, there was somecontroversy as to the meaning of the definite integral, say as itappears in the formula above for the Fourier coefficients. In the18th century, an integral was thought of as an anti-derivative.This is the way an operation like “int” is handled by a symbolicmanipulation program like Maple today.

Page 38: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

The fact that the integral is the difference of theanti-derivatives at the end-points was a definition, not atheorem. Since the anti-derivatives of some very importantfunctions, such as e−x

2/2 can not be expressed in terms ofelementary known functions, what meaning does an expressionsuch as

∫ ba e−x

2/2dx have? It was Fourier’s important idea tobypass the anti-derivative, and define the definite integral asthe area under the curve. Thus the definite integral becomesprimary, and the indefinite integral or anti-derivative becomessecondary.

Page 39: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Jean Baptiste Joseph Fourier

Born: 21 March 1768 in Auxerre, Bourgogne, France

Died: 16 May 1830 in Paris, France

Page 40: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Dirichlet.

Getting back to Fourier series, the problem of establishing theirconvergence was open. Poisson published an erroneous proof in1820, and Cauchy published a flawed proof in 1826 - the errorpointed out by Dirichlet in 1829 where he, Dirichlet, gave thefirst correct proof of sufficient conditions for convergence - thata piecewise smooth integrable function converges at everypoint to the average of its right and left hand limits. Thisresult was extended by Jordan in 1881 to all functions ofbounded variation.

Dirichlet also realized the once we extend the notion of afunction, there will be certain functions for which “the areaunder the curve” definition of the definite integral makes nosense, such as his “function” 1Q.

Page 41: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Born: 13 Feb 1805 in Düren, French Empire (now Germany)

Died: 5 May 1859 in Göttingen, Hanover (now Germany)

Johann Peter Gustav Lejeune Dirichlet

Page 42: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Riemann.

But the “area under the curve” definition did make sense forcontinuous functions, or functions which were continuous withonly finitely many jumps. This was greatly extended byRiemann in his Habilitationschrift entitled On therepresentability of a function by trigonometric series written in1854, but not published until after his death in 1867. He usedhis new concept of integration (today known as the Riemannintegral) to conclude that the Fourier coefficients of anyRiemann integrable function tend to zero as n→∞. This wasgeneralized by Lebesgue, to what is today known as theRiemann-Lebesgue lemma.

Page 43: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Lebesgue.

Riemann’s notion of the integral prevailed until the begining ofthe twentieth century with the arrival of the Lebesgue integral.Lebesgue was also very interested in Fourier series, and the firstapplication his new integral were to this subject. Lebesgue’snotion of integral extended to a much wider class of functions -Riemann integrability is equivalent to continuity almosteverywhere. There was much more freedom in taking limitsunder the integral sign. The theory of Lebesgue involves theconcept of the “measure” of a set and the associated notion ofan integral of a function.

Page 44: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Caratheodory and Hausdorff.

Lebesgue’s notion of measure was extended by Caratheodory in1905 to account for the area of k -dimensional subsets ofn-dimensional space, where k is an integer, and this wasgeneralized by Hausdorff in 1917 to allow the possibility of kbeing any real number, thus introducing the notion of“fractional dimension”. This line of work was continued by asmall group of researchers (mainly Besicovitch and hiscoworkers) but exploded into the public consciousness by thework of Mandelbrot in the 1980’s with the advent of computergraphics. “Fractals” are now part of the everyday language.

Page 45: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Daniel and Stone.

The integration aspect of Lebesgue’s theory, starting with theidea that an integral is a linear function on a certain class offunctions, and that this notion was primary, with measuretheory secondary was introduced by Daniell in 1911 andbrought to perfection by Stone in 1942. In the meanwhile,Hilbert spaces such as the space L2 of square integrablefunctions, and more generally the Lp spaces consisting offunctions for which

∫R |f |

p <∞, p ≥ 1 were being studied. Akey theorem proved by F. Riesz in the 1930’s says thatLp, p > 1 is the dual space of Lq where p and q are related by

1

p+

1

q= 1.

Page 46: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Riesz.

In other words, an element of Lp can be thought of (and is themost general) bounded linear function on Lq where we use thenorm

‖g‖q :=

(∫R|g |q

)1/q

on Lq. In other words, instead of thinking of an element of Lp

as a function (on its domain of definition, for example the realnumbers, so assigning a complex number to every realnumber), we can think of it as a functional, a rule whichassigns numbers to functions.

Page 47: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

For example, an element of L2 can be thought of as assigning anumber to any element of L2 in a way which is continuous withrespect to L2 convergence. We will exploit this “duality”, andthe Riesz representation theorem in its various guises will be akey ingredient in this course.

Page 48: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Back to the square wave.

Recall that the “square wave” is defined as

f (x) :=

{−1 −π ≤ x < 0,1 0 ≤ x < π

,

Its Fourier series is

4

π

∑n≥1

sin(2n − 1)x

2n − 1.

Indeed:

Page 49: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

an =1

[∫ π

0e−inxdx −

∫ 0

−πe−inxdx

]=

1

2

−in

[e−inπ − 1

].

This last expression in brackets vanishes when n is even, andequals −2 if n is odd. So the even terms of the Fourier seriesvanish, while the sum of the terms involving e inx for odd n is

1

4

i |n|[e inx − e−inx

]=

4

π|n|sin |n|x .

Page 50: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Let sn denote the n-th partial sum. Here are the graphs of snfor some values of n: Notice that there is blip overshooting thesquare wave whose distance from the horizontal (of about .18)does not decrease as n increases. Rather the width of the blipappears to go to zero.

Page 51: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Page 52: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Verification.

Let us verify these facts: To locate the maxima or minima wecompute

dsndx

=4

π

n∑k=1

cos(2k − 1)x

which we can sum as a geometric sum (for x 6= nπ) as

4

π

1

2

(e ix

n−1∑0

e2kix + e−ixn−1∑

0

e−2kix

)

=4

π

1

2

(e ix

1− e2inx

1− e2ix+ e−ix

1− e−2inx

1− e2ix

)=

2

π

sin 2nx

sin x.

Page 53: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

We have shown that dsndx = 2

πsin 2nx

sin x for x 6= nπ. But thisextends by continuity to all values of x . This function vanishesat x = ± π

2n , so these are the extrema closest to the origin.

Differentiating one more time we check that we have amaximun to the right and a minimum to the left of the origin.Evaluating sn at its maximum near the origin gives:

sn( π

2n

)=

4

π

n∑k=1

sin[(2k − 1) π2n ]

2k − 1

=2

π

n∑k=1

sin[(2k − 1) π2n ]

(2k − 1) π2n

π

n→ 2

π

∫ π

0

sin t

tdt.

Page 54: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Evaluating the integral gives approximately 1.18. Indeed, wehave the power series expansion

sin t

t=∞∑r=0

(−1)r t2r

(2r + 1)!

which is valid for all t We may integrate term by term. We get∫ π

0

sin t

tdt = π

(1− π2

3!3+π4

5!5− π6

6!6+ · · ·

).

As the series is an oscillating series with decreasing terms, theerror involved in truncating the series is at most the first termneglected, and stopping at an odd term overshoots the markwhile stopping at an even term undershoots the mark.Summing the first five terms gives 1.179384 while adding thenext term gives 1.178957.

Page 55: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

So this overshoot never disappears. This is the Gibbsphenomenon, Gibbs, 1899, first discovered by Wilbraham in1848 and proved as a general phenomenon - that the overshootat the jump is about 9 percent of the total jump by Bocher in1906. So naturally it is called the Gibbs phenomenon.It appears from the figure that we have uniform convergence tothe square wave as long as we stay a postive distance awayfrom the jump. This is an illustration of Dirichlet’s theoremwhich we will discuss in the next lecture. We also will learnthat the square wave belongs to L2 and that its Fourier seriesconverges to it in the L2 norm. So the Gibbs phenomenonshows something about the subtlety of L2 convergence.

In the next two slides I will reproduce the original Gibbs paperwhich appeared in NATURE on April 27, 1899. Gibbs workedwith the zigzag function rather than the square wave function.

Page 56: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Page 57: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Page 58: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Josiah Willard Gibbs

Born: 11 Feb 1839 in New Haven, Connecticut, USA

Died: 28 April 1903 in New Haven, Connecticut, USA

Page 59: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Proof that e is transcendental.

The proof given here is taken from Topics in Algebra by I.N.Herstein, pages 216 -219. Hermite’s proof was simplified byHilbert, and further streamlined by Hurwitz. Herstein followsHurwtiz’s proof.

Page 60: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Let f be a polynomial of degree r with real coefficients. Let

F := f + f ′ + f ′′ + · · ·+ f (r).

For any differentiable function g we haveddx (e−xg(x)) = e−x(g ′ − g). So

d

dx(e−xF (x)) = −e−x f (x).

The mean value theorem asserts that if g is continuouslydifferentiable on an interval [x1, x2] then

g(x2)− g(x1)

x2 − x1= g ′(x1 + θ(x2 − x1))

for some θ with 0 < θ < 1.We apply this to the function e−xF (x) and the interval [0, k],to obtain

Page 61: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

e−kF (k)− F (0) = −ke−θkk f (kθk)

where 0 < θk < 1. Multiply this by ek to obtain

F (k)− ekF (0) = −ke(1−θk )k f (kθk) (Eqk).

Call the right hand side of this equation εk . It depends, ofcourse, on f . Now suppose that e satisfies an algebraicequation

cnen + cn−1en−1 + · · · c0 = 0 (∗)where the cj are integers and c0 6= 0. Multiply (Eqk) by ck andsum from k = 1 to n, to obtain

c1F (1) + c2F (2) + · · · cnF (n)− F (0)[c1e + c2e2 + · · · cnen]

= c1ε1 + · · · cnεn.In view of the supposed equation (∗), the expression in bracketsequals −c0. So we obtain

Page 62: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

c0F (0) + c1F (1) + · · ·+ cnF (n) = c1ε1 + · · ·+ cnεn. (1)

This is true for any polynomial f (where, recall, the εk dependon f ). We will derive a contradiction by showing that we canfind a prime number p > n, p > c0, and a polynomial f = fpsuch that

1 F (i) is an integer divisible by p for i = 1, . . . , n.2 F (0) is an integer not divisible by p. Since p 6 | c0, this

implies that3 c0F (0) + c1F (1) + · · · cnF (n) is an integer not divisible by

p.4 |c1ε1 + · · ·+ cnεn| < 1.

The last two items contradict one another in view of (1).Indeed, the left hand side of (1) is an integer, and the onlyinteger less than one in absolute value is 0. But 0 is divisible byp.

Page 63: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

So we want to show that we can choose a large p and apolynomial fp so that items 1, 2, and 4 hold.

The polynomial (used by Hermite) is f = fp where

f (x) :=1

(p − 1)!xp−1(1− x)p(2− x)p · · · (n − x)p.

For i < p the polynomial f and its first p − 1 derivatives vanishat 1, 2, . . . , n. If we multiply out we get the expansion

f (x) =1

(p − 1)!

[(n!)pxp−1 + a0xp + a1xp+1 + · · ·

]where the aj are integers.

Page 64: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

Now if g is any polynomial with integer coefficients then, fori ≥ p, the i-th derivative of 1

(p−1)! g is a polynomial with

integer coefficients divisible by p, since the i-th derivative of xk

vanishes for k < i and is k(k − 1) · · · (k − i + 1)xk−i for k ≥ i .So for the f = fp above, and i ≥ p, the i-th derivative of f is apolynomial with integer coefficients. Hence F when evaluatedat 1, 2, . . . , n is an integer divisible by p. This establishes item1).

Page 65: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

As for item 2), the first p − 2 derivatives of f vanish at 0, andfor i ≥ p, the i-th derivative of f is an integer divisible by p.But f (p−1)(0) = (n!)p and since p > n this is not divisible by p.This establishes item 2) and hence item 3).

Page 66: Math 212a Lecture 1 - Introduction.people.math.harvard.edu/~shlomo/212a/01.pdfHistory: irrational numbers. The Greeks know that p 2 is irrational. A proof appears in a scholium to

Math 212aLecture 1

ShlomoSternberg

Outline

Some history.

Irrationality ofπ.

Transcendentalnumbers.

What is thedefinition of afunction?

Some dubious18th centurysums.

Toeplitz’stheorem.

Cesarosummability.

Abelsummability

Fourier andFourier series.

The Gibbsphenomenon.

Appendix:Proof that e istranscenden-tal.

As to item 4), we have

εk = −ke(1−θk )k f (kθk) = −e(1−θk )k(1− θk)p · · · (n − kθk)p(kθk)p−1k

(p − 1)!.

Since 0 < θk < 1 and k ≤ n we see that

|εk | ≤ennp(n!)p

(p − 1)!

which we can make as small as we like by choosing psufficiently large. This establishes 4) and with it proves that eis transcendental.