# MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π

Save this PDF as:

Size: px
Start display at page:

Download "MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 4: Fourier Series and L 2 ([ π, π], µ) ( 1 π"

## Transcription

1 MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 4: Fourier Series and L ([, π], µ) Square Integrable Functions Definition. Let f : [, π] R be measurable. We say that f is square integrable if f is integrable, i.e., if For future use, we shall write f = f dµ < +. ( 1 π ) 1/ f dµ π and we shall call this the L -norm of f (pronounced L-two, not L-squared ). Remarks. (i) Of course, as f is real valued, f = f, so we could just write f dµ. However, the definition can be extended to complex valued functions when we really do want f. (ii) The factor 1/π is not essential but is introduced to be convenient for the calculations with Fourier series which occur later in the chapter. We d like to use to give a distance (more formally, a metric) on the set of square integrable functions by dist(f, g) = f g = ( 1 π 1/ f g dµ). π However, if f = g µ-a.e. then f g dµ = 0, so we would have distinct functions lying distance zero apart. To avoid this, we consider equivalence classes of square integrable functions under the equivalence relation Thus an equivalence class has the form f g f = g µ-a.e.. [f] = {g : [, π] R : f = g µ-a.e.}, for some square integrable function f : [, π] R. 1 Typeset by AMS-TEX

2 Definition. We write L ([, π], µ, R) for the set of equivalence classes of square integrable functions on [, π]. After having made this definition, we shall proceed more informally and write f L ([, π], µ, R) (rather than the pedantic [f] L ([, π], µ, R)) whenever f : [, π] R is square integrable. However, bear in mind that two functions which are equal almost everywhere are now considered to be the same. We may also write f L ([, π], µ, R) when f : [, π] R is such that f dµ < + but note that such an f must be finite µ-a.e. and hence equal µ-a.e. to a function with values in R. Now we shall show that L ([, π], µ, R) is a vector space and that d(f, g) = f g is a metric on L ([, π], µ, R). First we need a technical result. Lemma 4.1. If f, g L ([, π], µ, R) then π fg dµ f + g π, with equality if and only if f = g µ-a.e. (i.e. if f and g are the same element of L ([, π], µ, R)). Proof. We have so 0 (f(x) g(x)) = f(x) f(x)g(x) + g(x), f(x)g(x) f(x) + g(x) Note that fg is integrable since, as above but for ( f g ), fg f +g. Multiplying (*) by 1/π and integrating gives the first statement. Clearly, we get equality if and only if (f g) dµ = 0. By Proposition 3.1, this holds if and only if (f g) = 0 µ-a.e, i.e., f = g µ-a.e. Corollary. If f, g L ([, π], µ, R) then π fg dµ f + g π, with equality if and only if f = g µ-a.e. Lemma 4.. If f, g L ([, π], µ, R) then (i) Hölder Inequality: 1 π fg dµ f g, π with equality if and only if for some c R, f = c g µ-a.e. or f g = 0. (ii) Cauchy-Schwarz Inequality: 1 π fg dµ π f g, with equality if and only if for some c R, f = cg µ-a.e. or f g = 0. Proof. Exercise. ( )

3 Lemma 4.3. If f, g L ([, π], µ, R) then so is f + g and f + g f + g. Proof. Clearly f + g is measurable. Also 1 π + g) π (f dµ = 1 (f + fg + g ) dµ π 1 π (f + fg + g ) dµ f + f g + g = ( f + g ) (where we have used Hölder s inequality for the second inequality). Hence f + g L ([, π], µ, R) and taking square roots gives the inequality (called Minkowski s Inequality). Corollary. L ([, π], µ, R) is a vector space over R. Proof. It is trivial that if f L ([, π], µ, R) and c R then cf L ([, π], µ, R) (and cf = c f ). By the above lemma, if f, g L ([, π], µ, R), so is f + g L ([, π], µ, R). Definition. A metric on a set X is a function d : X X R such that, for x, y, z X, (1) d(x, y) 0 and d(x, y) = 0 if and only if x = y; () d(x, y) = d(y, x); (3) d(x, z) d(x, y) + d(y, z). Theorem 4.4. d(f, g) = f g is a metric on L ([, π], µ, R). Proof. It is immediate from its definition that f g 0. Suppose that f g = 1 π f g dµ = 0. Applying Proposition 3.1, we see that f g = 0 µ-a.e., i.e., f = g µ-a.e., so f and g represent the same element of L ([, π], µ, R). This shows (1). Condition () follows immediately from the definition. For (3), assume that f, g, h L ([, π], µ, R). By Minkowski s Inequality, d(f, h) = f h = (f g) + (g h) f g + g h = d(f, g) + d(g, h) as required. The next important result says that square integrable functions may be approximated arbitrarily well by continuous functions with respect to. 3

4 Theorem 4.5. Continuous functions are -dense in L ([, π], µ, R). In other words, given f L ([, π], µ, R) and ϵ > 0, we can find a continuous function g : [, π] R such that f g < ϵ. The proof is given in an appendix and is not examinable. The following slight strengthening will be needed later on. Corollary 4.6. Given f L ([, π], µ, R) and ϵ > 0, we can find a continuous function g : [, π] R such that g() = g(π) and f g < ϵ. Proof. Exercise. (Hint: use Theorem 4.5 to find a continuous function h : [, π] R such that f h < ϵ/. Modify h on [, + δ] [π δ, π], for an appropriately small δ > 0 to obtain a continuous function g : [, π] R with the required properties.) Definition. We say that a sequence of functions f n L ([, π], µ, R) converges to f L ([, π], µ, R) if f n f = 0. Definition. Recall that a metric space (X, d) is said to be complete if every Cauchy sequence converges to a point in X. (A sequence x n X is a Cauchy sequence if, for all ϵ > 0, there exists N 1 such that if n, m N then d(x n, x m ) < ϵ. Theorem 4.7. L ([, π], µ, R) is complete. Proof. Let f n, n 1, be a Cauchy sequence in L ([, π], µ, R) with respect to the metric d(f, g) = f g determined by the norm. By definition, this means that, given ϵ > 0, we can find an integer N 1 such that n, m N = f n f m < ϵ. Applying this definition with ϵ = i, i = 1,,..., we can find a increasing sequence of positive integers N i such that n, m N i = f n f m < 1 i. Define functions g 0 = 0 and g i = f Ni, i 1. Then g i+1 g i = f Ni+1 f Ni < 1 i for i 1. Thus, by the Comparison Test, the series g i+1 g i converges. Let the sum be denoted by S. 4

5 Now consider a new sequence of functions h n, n 1, defined by h n (x) = g i+1 (x) g i (x). For any fixed x, the sequence of numbers h n (x) is increasing, so we can defined h(x) := h n(x) R {+ }. Thus we have a measurable function h : [, π] R. We want to show that h L ([, π], µ, R). First note that so that h n g i+1 g i S, h n dµ = π h n πs. Since h n is an increasing sequence of non-negative measurable functions converging pointwise to h, the Monotone Convergence Theorem tells us that h dµ = h n dµ πs, so h = h is integrable. Thus, h L ([, π], µ, R), as we claimed. Since h is integrable it is finite µ-a.e. Thus, h is finite µ-a.e. For each x [, π] for which h(x) is finite, the series of real numbers (g i+1 (x) g i (x)) converges absolutely and hence converges. We will denote its sum by g(x). For x with h(x) = +, we set g(x) = 0. Note that (g i+1 (x) g i (x)) = g n (x) g 0 (x) = g n (x). Hence, for µ-a.e. x. Moreover, g n(x) = g(x) = (g i+1 (x) g i (x)) = g(x), g n(x) g i+1 (x) g i (x) = h n(x) = h(x), 5

6 for µ-a.e. x. Thus g(x) h(x) for µ-a.e. x and so g is integrable, giving g L ([, π], µ, R). We also observe that g(x) g n (x) ( g(x) + g n (x) ) (h(x)). Since g(x) g n (x) = 0 for µ-a.e. x, the Dominated Convergence Theorem tells us that This implies that g g n dµ = 0. g g n = 0. Hence, given ϵ > 0, we can choose an i sufficiently large that g g i < ϵ/ and i < ϵ/. Recall that g i = f Ni. Thus, whenever n N i, we have g f n g g i + g i f n < ϵ + ϵ = ϵ. This shows that g f n L ([, π], µ, R). = 0, i.e., the sequence f n converges in the space Inner Products and Hilbert Spaces Definition. Let V be a vector space over R. A map, : V V R is called an inner product if, for all u, v, w V and a, b R, (1) u, v = v, u ; () au + bv, w = a u, w + b v, w ; (3) u, u 0 and u, u = 0 if and only if u = 0. Lemma 4.8. The formula f, g = 1 π fg dµ defines an inner product on the vector space L ([, π], µ, R) and f = f, f 1/. Proof. Parts (1) and () of the definition of inner product are easy to check. Part (3) is equivalent to the statement that f = 0 if and only if f represents the 0 element in L ([, π], µ, R), which follows from Proposition 3.1. Thus the metric on L ([, π], µ, R) is obtained from this inner product by d(f, g) = f g = f g, f g 1/. (In fact, any inner product defines a metric in this way.) Definition A vector space with an inner product which is complete with respect to the associated metric is called a Hilbert space. Thus we have already proved: Theorem 4.9. L ([, π], µ, R) is a Hilbert space. 6

7 Orthogonality Definition. Let V be a vector space with an inner product,. We shall write for the associated norm v = v, v 1/. We say that a collection of vectors {v n } in V is orthogonal if v n, v m = 0 whenever n m. We say they are orthonormal if, in addition, v n = 1 for all n. We will use couple of standard results about orthogonal/orthonormal vectors. Lemma Let {v k } n c 1,..., c n R. Then be a finite orthogonal family in a vector space V and let c k v k = c k v k. Proof. Note that, by the definition of inner product and orthogonality, c 1 v 1 + c v = c 1 v 1 + c v, c 1 v 1 + c v = c 1 v 1, v 1 + c 1 c v 1, v + c v, v = c 1 v 1 + c v. It is left as an exercise to complete the proof by induction. Lemma Let {v k } n be a finite orthonormal family in a vector space V. Then, for w V, the minimum value of n w c k v k over all choices of c 1,..., c n R occurs when c k = w, v k. Proof. Let c 1,..., c n be arbitrary real numbers and set a k = w, v k. Write By the preceding lemma, u = u = a k v k and v = c k v k. a k and v = c k. Also w, v = w, c k v k = c k w, v k = c k a k. 7

8 Thus It follows that w v = w v, w v = w w, v + v = w c k a k + = w a k + = w u + c k (a k c k ) (a k c k ). w v w u with equality if and only if n (a k c k ) = 0, i.e. if and only if c k = a k = w, v k for all k = 1,..., n. Lemma 4.1. The family of functions F = { } 1, cos(nx), sin(nx) : n 1, is orthonormal in L ([, π], µ, R). Proof. We have 1 π { 0 if k 0 e ikx dµ = π if k = 0. Use the formulae cos(nx) = (e inx + e inx )/ and sin(nx) = (e inx e inx )/i to obtain the result. Fourier Series As in Chapter 1, the Fourier series of an integrable function f is where a 0 = 1 π a n = 1 π a (a n cos(nx) + b n sin(nx)), n=1 f(x)dµ and, for n 1, f(x) cos(nx)dµ, b n = 1 π f(x) sin(nx)dµ, n 1, where we have written the integrals with respect to µ as we do not assume f is Riemann integrable. (We have writtten the first term a 0 / as (a 0 / )(1/ ) for a reason.) 8

9 In terms of the inner product, we have and, for n 1, f, cos(nx) = 1 π 1 f, = 1 f(x) dµ = a 0, π f(x) cos(nx)dµ = a n, f, sin(nx) = 1 π the Fourier coefficients of f. Thus the Fourier series may be expressed as 1 1 f, + f, cos(nx) cos(nx) + n=1 f, sin(nx) sin(nx). n=1 f(x) sin(nx)dµ = b n, Define sin( nx) if n < 0 φ n (x) = 1 if n = 0 cos(nx) if n > 0. Then the Fourier series has the succinct expression Also, and S n (f, x) = = 1 1 f, + n= f, φ n φ n (x). f, cos(kx) cos(kx) + f, φ k φ k (x). f, sin(kx) sin(kx) σ n (f, x) = 1 n (S 0(f, x) + S 1 (f, x) + + S (f, x)) = k= () n k n f, φ k φ k (x). Theorem 4.13 (Riesz-Fischer Theorem). Let f L ([, π], µ, R). Then S n (f, ) converges to f in L ([, π], µ, R), i.e, S n (f, ) f = ( 1 π S n (f, ) f dµ) 1/ 0, as n +. 9

10 Before we prove this, we recall Fejér s Theorem (Theorem 1.3) from Chapter 1. Here is a slightly specialized version: Suppose that g : [, π] R is continuous and g() = g(π). Then the sequence of functions σ n (g, ) converges uniformly to g, as n +. If we define σ n (g, ) g = sup x [,π] σ n (g, x) g(x) then uniform convergence is equivalent to σ n (g, ) g = 0. Also note that σ n (g, ) g σ n (g, ) g. Proof. Suppose f L ([, π], µ, R) and let ϵ > 0 be given. By Theorem 4.5, we can find a continuous function g : [, π] R such that f g < ϵ/. By Fejér s Theorem, we can choose N sufficiently large that Thus, if n N then n N = σ n (g, ) g < ϵ. σ n (g, ) g σ n (g, ) g < ϵ. Combing the two estimates, if n N then f σ n (g, ) f g + g σ n (g, ) < ϵ + ϵ = ϵ. We may write σ n (g, x) = k= () d k φ k (x), for some d k R. By Lemma 4.11, f k= () f, φ k φ k (x) f k= () d k φ k (x). Thus, if n N then f S n (f, ) < ϵ, as required. Having proved the theorem, we are now entitled to say that for f L ([, π], µ, R), f = f, φ n φ n n= ( ) in L ([, π], µ, R). 10

11 Theorem F = { } 1, cos(nx), sin(nx) : n 1 = {φ n : n Z} is a (Schauder) basis for the vector space L ([, π], µ, R). In other words, for each f L ([, π], µ, R) there is a unique sequence {c n, n Z} such that f = n= c nφ n in L ([, π], µ, R), that is, f n n c k φ k = 0. Consequently, the Fourier series of a function is that function written in terms of the basis. Proof. The existence of {c n : n Z} follows from ( ) above. It remains to show that the representation is unique. Suppose that for some f L ([, π], µ, R) we have {c n, n Z} and {d n, n Z} such that n n f c k φ k = 0, n f n d k φ k = 0. Then n d k φ k c k φ k = n k c k ) φ k = 0. (d Consider any m Z and choose n m. Then φ m, (d k c k ) φ k = (d k c k ) φ m, φ k = d m c m. On the other hand, n φ m, (d k c k ) φ k φ m k c k ) φ k 0, as n + (d (using the Cauchy-Schwarz Inequality). Taking these together, we see that c m = d m, for all m Z, so the uniqueness of the representation follows. Carleson s Theorem The Riesz-Fischer Theorem was proved in As it deals with convergence with respect to, it leaves open the question of pointwise convergence, of S n (f, x) to f(x). We have already seen (Theorem 1.4) that even for continuous f, we do not necessarily have convergence at every point. But what about almost every point? It turns out that the answer to this is yes for square integrable functions. This was proved by Lennart Carleson in 1966 and is regarded as one of the high points of mathematical analysis in the twentieth century. 11

12 Theorem 4.15 (Carleson s Theorem). Let f L ([, π], µ, R). Then S n (f, x) converges to f(x) for µ-a.e. x [, π], as n +. Remark. In contrast, the result is false if one only assumes that f is integrable. Indeed, there is an example of Kolmogorov (193) which shows that there is an integrable function f : [, π] R for which S n (f, x) does not converge at any point. 1

### I. Pointwise convergence

MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### 1 Inner Products and Norms on Real Vector Spaces

Math 373: Principles Techniques of Applied Mathematics Spring 29 The 2 Inner Product 1 Inner Products Norms on Real Vector Spaces Recall that an inner product on a real vector space V is a function from

### Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

### Sine and Cosine Series; Odd and Even Functions

Sine and Cosine Series; Odd and Even Functions A sine series on the interval [, ] is a trigonometric series of the form k = 1 b k sin πkx. All of the terms in a series of this type have values vanishing

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.

MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

### Our goal first will be to define a product measure on A 1 A 2.

1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ -algebra A 1 A 2 with the unit element is the σ -algebra generated by the

### {f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### 1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### x a x 2 (1 + x 2 ) n.

Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### THE BANACH CONTRACTION PRINCIPLE. Contents

THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### 1. Let X and Y be normed spaces and let T B(X, Y ).

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 2005-03-14 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

### Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:

Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### MEASURE AND INTEGRATION. Dietmar A. Salamon ETH Zürich

MEASURE AND INTEGRATION Dietmar A. Salamon ETH Zürich 12 May 2016 ii Preface This book is based on notes for the lecture course Measure and Integration held at ETH Zürich in the spring semester 2014. Prerequisites

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

### THE PRIME NUMBER THEOREM

THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Theta Functions. Lukas Lewark. Seminar on Modular Forms, 31. Januar 2007

Theta Functions Lukas Lewark Seminar on Modular Forms, 31. Januar 007 Abstract Theta functions are introduced, associated to lattices or quadratic forms. Their transformation property is proven and the

### Absolute continuity of measures and preservation of Randomness

Absolute continuity of measures and preservation of Randomness Mathieu Hoyrup 1 and Cristóbal Rojas 2 1 LORIA - 615, rue du jardin botanique, BP 239 54506 Vandœuvre-lès-Nancy, FRANCE, mathieu.hoyrup@loria.fr.

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### 12. Inner Product Spaces

1. Inner roduct Spaces 1.1. Vector spaces A real vector space is a set of objects that you can do to things ith: you can add to of them together to get another such object, and you can multiply one of

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### Note on some explicit formulae for twin prime counting function

Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Section 4.4 Inner Product Spaces

Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer

### Fixed Point Theorems

Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation

### Lecture 14: Section 3.3

Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Ri and. i=1. S i N. and. R R i

The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

### MA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity

MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### and s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space

RAL ANALYSIS A survey of MA 641-643, UAB 1999-2000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σ-algebras. A σ-algebra in X is a non-empty collection of subsets

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### Introduction to Monte-Carlo Methods

Introduction to Monte-Carlo Methods Bernard Lapeyre Halmstad January 2007 Monte-Carlo methods are extensively used in financial institutions to compute European options prices to evaluate sensitivities

### Introduction to Sturm-Liouville Theory

Introduction to Ryan C. Trinity University Partial Differential Equations April 10, 2012 Inner products with weight functions Suppose that w(x) is a nonnegative function on [a,b]. If f (x) and g(x) are

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### Differential Operators and their Adjoint Operators

Differential Operators and their Adjoint Operators Differential Operators inear functions from E n to E m may be described, once bases have been selected in both spaces ordinarily one uses the standard

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### MINIMAL GENERATOR SETS FOR FINITELY GENERATED SHIFT-INVARIANT SUBSPACES OF L 2 (R n )

MINIMAL GENERATOR SETS FOR FINITELY GENERATED SHIFT-INVARIANT SUBSPACES OF L 2 (R n ) MARCIN BOWNIK AND NORBERT KAIBLINGER Abstract. Let S be a shift-invariant subspace of L 2 (R n ) defined by N generators

### 33 Cauchy Integral Formula

33 AUHY INTEGRAL FORMULA October 27, 2006 PROOF Use the theorem to write f ()d + 1 f ()d + 1 f ()d = 0 2 f ()d = f ()d f ()d. 2 1 2 This is the deformation principle; if you can continuously deform 1 to

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### The Fourier Series of a Periodic Function

1 Chapter 1 he Fourier Series of a Periodic Function 1.1 Introduction Notation 1.1. We use the letter with a double meaning: a) [, 1) b) In the notations L p (), C(), C n () and C () we use the letter

### The Delta Method and Applications

Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

### FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Homework # 3 Solutions

Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8