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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 to study functions defined on a discrete set of points. This is of course how engineers actually use Fourier analysis. We begin with the discrete set (group) Z of all integers. This is certainly a group with respect to the binary operation of +. What should be the definition of the Fourier transform f of a function f on this set? To begin with, what should the domain of the function f be? By analogy with what we did earlier, let us determine the set of homomorphisms of the group Z into the circle group T. Remember, this is the way we decided what the domain of the Fourier transform on the real line should be. THEOREM Let φ be a function on Z that maps into the circle group T and satisfies the law of exponents φ(k + l) = φ(k)φ(l). (φ is a homomorphism of the additive group Z into the multiplicative group T.) Then there exists a unique element x [, 1) such that φ(k) = e 2πikx for all integers k. PROOF. Notice first that φ() = φ( + ) = φ()φ() = φ() 2, so that φ() must be equal to 1. (Why couldn t φ() be?) Now let λ be the element of T for which λ = φ(1), and let x be the unique point in the interval [, 1) such that λ = e 2πix. (Why is there only one such number x?) We have then that φ(1) = λ = e 2πix. Then, φ(2) = φ(1 + 1) = φ(1) φ(1) = λ 2 = e 2πi2x, φ(3) = λ 3 = e 2πi3x, φ(4) = λ 4 = e 2πi4x,.... That is, for any k, we must have φ(k) = λ k = e 2πikx. Now, for k <, we know that 1 = φ() = φ(k k) = φ(k)φ( k) = λ k φ( k), showing that φ( k) = 1 λ k = λ k = e 2πi( k)x. This proves the theorem. The preceding theorem shows that the set of homomorphisms of the group Z into the circle BbbT are parameterized by the numbers x in the interval [, 1). That is, for each element x [, 1), there is a homomorphism φ x of Z into T defined by φ x (k) = e 2πikx, and these are all the homomorphisms of Z into T. In other words, the set of homomorphisms of the group Z into the group T is parameterized by the points in the interval [, 1). So, following our earlier development, we should define the Fourier transform of a function f on Z to be the function f on [, 1) given by f(x) = f(k)φ x (k) = 1 f(k)e kx.

2 2 REMARK. Notice that, just as in the case of the real line, we are only able to define the Fourier transform of a function f on Z if it is absolutely summable. that is, only if f(k) <. EXERCISE (a) Let f be the function on Z given by f(k) = unless k = ±1, and f(1) = f( 1) = 1. What is the Fourier transform of f? (b) Fix a positive integer N, and let f be the function on Z given by f(k) = 1 if N k N and otherwise. Find the Fourier transform of f. What should Fourier s Theorem be in this case? that is, how can we recover the function f on Z from the function f on [, 1). Before giving the answer, we make the following observations: EXERCISE Let f be an absolutely summable function on the group Z of integers, and write f for the function on [, 1) defined by f(x) = f(k)e kx. (a) Use the Weierstrass M-Test to show that f is a continuous function on [, 1). The Weierstrass M-Test says the following: Suppose {u k } is an infinite sequence of continuous functions, and suppose that {m k } is a corresponding sequence of nonnegative numbers for which u k (x) m k for all x, and for which the infinite series m k converges. Then the infinite series u k (x) converges to a continuous function. (b) Conclude that f is a square-integrable function on [, 1). Here is Fourier s Theorem in this context: THEOREM Let f be an absolutely summable function on Z. Then, for each integer n, we have f(n) = PROOF. 1 f(x)e 2πinx dx = which proves the theorem. = = = 1 1 f(x)e 2πinx dx. 1 f(k) f(k)δ n,k = f(n), f(k)e kx e 2πinx dx f(k)e 2πi(n k)x dx 1 e 2πi(n k)x dx REMARK. This whole thing looks awfully familiar. It surely must be related to the Fourier transform on the circle. Indeed, is this just the inverse of that transform?

3 EXERCISE (a) If f is a square-integrable function on [, 1), what is the formula for its Fourier transform, and what is the formula for the inverse transform? (b) If g is a function on Z, what is the formula for its Fourier transform, and what is the formula for the inverse transform? (c) Exactly how are parts (a) and (b) related? Of course, we could now try to do all the same kinds of things we did with the other Fourier transforms. It may be a little less interesting this time, primarily because functions on the discrete set Z are not continuous, and they don t have derivatives. The kinds of problems one encounters with these functions are not differential equations. Perhaps we could work on difference equations. One thing that can be defined in this case is convolution. DEFINITION. Let f and g be two absolutely summable functions on Z. Define the convolution f g of f and g to be the function on Z given by f g(n) = f(n k)g(k). 3 EXERCISE (a) Show that, if f and g are absolutely summable, then so is f g. (b) Prove the convolution theorem in this case: f g(x) = f(x)ĝ(x). (c) Show that, in this case, there is an identity for convolution. That is, there is a function e on Z for which f = f e for every f. (d) Show that f g(n) = f(k)g(l). k,l:k+l=n The Finite Fourier Transform Fix a positive integer N, and let G be the finite set, 1, 2,..., N 1. The set G is a group (actually a ring), where addition and multiplication are computed mod N. It is frequently denoted by Z N. EXERCISE Let N = 64. (a) Compute in this group Z 64. (b) Compute in this ring. We wish to define a Fourier transform on this group. We follow our standard procedure. THEOREM Let φ be a function on G = Z N that maps into the circle group T and satisfies the law of exponents φ(k + l) = φ(k)φ(l). (φ is a homomorphism of the additive group Z N into the multiplicative group T.) Then there exists a unique integer j N 1 such that φ(k) = e 2πi jk N

4 4 for all k G. That is, these homomorphisms are parameterized by the numbers j =, 1, 2,..., N 1. PROOF. Notice first that φ() = φ( + ) = φ() 2, so that φ() must be equal to 1. Now, let z = φ(1). Then, φ(2) = φ(1 + 1) = φ(1) φ(1) = z 2, φ(3) = z 3, φ(4) = z 4,.... That is, for any k N 1 < we must have φ(k) = z k. Now, because N is the same as in the group G, we must have that z N = φ(n) = φ() = 1. That is, z N must equal 1. Write z = e 2πix for some x [, 1). We must have that e 2πiNx = (e 2πix ) N = φ(1) N = φ(n) = 1, implying that xn is equal to some integer j between and N 1. (Why?) Hence, x = j/n, and φ(k) = z k = e 2πixk for all k G, as desired. EXERCISE For each integer j N 1, define a function φ j on G = Z N by φ j (k) = e 2πi jk N. Prove that φ j is a homomorphism of the group G into the circle group T, and use Theorem 11.3 to conclude that these are all such homomorphisms of G. Now we proceed just as we did in the case of Z. Having determined the set of homomorphisms of our group into the circle group T, and having seen that they are parameterized by the numbers j N 1, we define the Fourier transform of a function on Z N. This is a finite set, so we won t need any kind of absolute summability assumption. We can define the Fourier transform for any function on this group. DEFINITION. Let V be the set (vector space) of all complex-valued functions defined on the set G = Z N. If f is an element of the vector space V, we define the finite Fourier transform f of f to be the function, also defined on the set, 1, 2,..., N 1, by f(j) = = k= k= f(k)φ j (k) f(k)e jk N. EXERCISE (a) Let f be define on G by f(k) = 1/2 k. Find the finite Fourier transform f of f. Is this anything like the Fourier transform of the function 1/2 x on the circle? (b) Suppose N > 8, and let f(k) = 1 for k 7, and f(k) = otherwise. Find f.

5 5 EXERCISE For f and g in V, define f g = k= f(k)g(k). (a) Prove that V is a complex inner product space with respect to the above definition. That is, show that (1) c 1 f 1 + c 2 f 2 g = c 1 f 1 g + c 2 f 2 g for all elements f, g V and all complex numbers c 1 and c 2. (2) f g = g f for all f, g V. (3) f f for all f V, and f f = if and only if f is the element of V. (b) Let f, f 1,..., f be the functions in V defined as follows: f n (k) = δ n,k. That is, f n is the function that equals 1 on the integer n and equals on all other integers k. Prove that the collection f,..., f is an orthonormal set in V. That is, show that f n f k = δ n,k. (c) Prove that the functions f,..., f is a basis for the vector space V. That is, show that each f V can be written in a unique way as a linear combination of the vectors (functions) f n. Conclude then that f,..., f is an orthonormal basis for V. (d) If T is a linear transformation of V into itself, recall from Linear Algebra that the entries of the matrix A that represents the transformation T with respect to this orthonormal basis are given by A j,k = T (f k ) f j, where both k and j run from to N 1. THEOREM The Fourier transform on V is a linear transformation from the N-dimensional vector space V into itself. If f,..., f is the orthonormal basis of the preceding exercise, then the entries of the matrix A that represents the Fourier transform with respect to this basis are given by j and k both running from to N 1. A j,k = e jk N, EXERCISE (a) Use Exercise 11.8 to prove this theorem. Of course, you will need to compute the Fourier transforms of the functions f,..., f. (b) Suppose f is a function (thought of as a column vector) in the vector space V. Verify that the jth component of the column vector A f is given by the formula (A f) j = f(j). Here is Fourier s Theorem in this case. It should be easier than the other cases, for this time the Fourier transform is a linear transformation of a finite dimensional vector space into itself. If it is 1-1, there should be an inverse given by the inverse of the matrix A.

6 6 THEOREM The finite Fourier transform on V is invertible, and in fact we can recover f from f as follows: f(k) j= f(j)e 2πi jk N. PROOF. Let B be the N N matrix whose entries are given by B l,m e2πi lm N. We claim that B is the inverse of the matrix A that represents the finite Fourier transform on V. Thus let us show that BA is the identity matrix. If j = k, then (AB) j,j = A j,m B m,j e jm N e 2πi jm N = 1. Therefore, down the diagonal of AB we have 1 s. Now, if j k, we have (AB) j,k = 1 A j,m B m,k e jm N = 1 (e N =, m(k j) 2πi e N k j 2πi N ) m (k j)n 2πi 1 e N k j 1 e2πi N e 2πi mk N where we have used the formula for the sum of a geometric series at the end of this calculation. Hence, AB has s off the diagonal, and so AB is the identity matrix, and B is A 1.

7 7 Therefore, f = B(A(f), or as claimed. f(k) = (B A f) k j= j= e 2πi jk N (A f)j e 2πi jk N f(j), EXERCISE Let f be a function in V. (a) Suppose N is even, say N = 2M. Show that where and for 1 j M, and f(k) = a + M (a j cos(2π jk N ) + b j sin(2π jk N )), j=1 a = 1 2M ( f() f(m)), a j = 1 2M ( f(j) + f(n j)) b j = i 2M ( f(j) f(n j))). (b) Deduce that, from the point of view of frequency analysis, the highest frequencies present in a signal (function) f on a finite set of N = 2M elements is M 1. Just as in the other cases we have studied, there is a notion of convolution on the space V. DEFINITION. If f and g are functions in V, define a function f g on G by f g(k) = f(l)g(k l). THEOREM Let f and g be elements of V. Then f g(j) = f(j)ĝ(j) for all j N 1. EXERCISE Prove the preceding theorem The Fast Fourier Transform

8 8 Computer designers tell us that all a computer really does is binary additions. Multiplications are just enormous addition calculations. Therefore, when estimating the cost of any computer computation, the designers count the number of multiplications and ignore any simple additions. If f is an element of V, how many multiplications must be carried out to compute the finite Fourier transform f, i.e., f(j) for all j N 1? The formula for f(j) is f(j) = k= f(k)e jk N. So, for each j, we must compute N products. Hence, to compute the entire transform f it appears that we must compute N N = N 2 products. Remarkably, and marvelously, this number N 2 multiplications can be reduced to N log N multiplications if we choose N to be a power of 2. (Here, the logarithm is meant to be taken with respect to the base 2.) This reduction is enormous if N is large. For example if N = 124 = 2 1, then N 2 is on the order of 1 6, a million, while N log N is on the order of 1 4. THEOREM Suppose N = 2 p. Then the finite Fourier transform of any element f in V can be computed using at most N log N = p2 p multiplications. PROOF. We must compute all products of the form jk f(k)e 2 p for both j and k ranging from to 2 p 1. The trick is that some of these products are automatically equal to others, so that a single multiplication can suffice to compute two different products, thereby cutting our work roughly in half. Let s explore this more carefully. Note the following two things: 1. If k is an even integer, say k = 2l, then e (j+2p 1 )k 2 p = e (jk+2p l 2 p jk = e 2 p, so that jk f(k)e 2 p = f(k)e (j+2p 1 )k 2 p, jk and therefore, for even numbers k, we need only compute the product f(k)e 2 p for j 2 p 1 1, i.e., for approximately half the j s. Each of the remaining products equals one of these. Next, if k is odd, say k = 2l + 1, then e (j+2p 1 )k 2 p jk = e 2 p e 2p 1 (2l+1) 2 p jk = e 2 p e πi jk = e 2 p, so that, if k = 2l + 1 is an odd number, again we need only compute the product jk f(k)e 2 p for j 2 p 1 1. The remaining half of the products are just the negatives of these, and the computer guys tell us that it costs nothing to change the sign of a number.

9 Moreover, for even numbers k = 2l, the sums of products we must compute to find the finite Fourier transform of f are of the form 2 p p 1 2lj 1 f(2l)e 2 p = f(2l)e jl 2 p 1, so that if a function g is defined on the set, 1,..., 2 p 1 1 by g(l) = f(2l), then computing the part of f having to do with even numbers k = 2l is exactly the same as if we were computing the discrete Fourier transform on the set, 1, 2,..., 2 p 1 1 of g. (You can probably see a mathematical induction argument coming up here.) In the case when k is odd, the sums of products we must compute look like 2 p p 1 j(2l+1) 1 f(2l + 1)e 2 p = f(2l + 1)e 2 p 1 1 = ( f(2l + 1)e jl 2 p 1 e j 2 p jl 2 p 1 )e j 2 p, and, unlike the case when k was even, this is a bit more complicated than just computing the discrete Fourier transform of a function on the set, 1,..., 2 p 1 1. We again could let h be the function on the set, 1, 2,..., 2 p 1 1 defined by h(l) = f(2l+1). What we are doing first is computing the discrete Fourier transform of the function h, and then we are multiplying each value ĥ(j) by the number e j 2 p, which would be an additional 2 p 1 multiplications. Finally, let M p be the number of multiplications required to compute the discrete Fourier transform when N = 2 p. The theorem asserts that M p p2 p. We have seen above that M p = M p 1 + M p p 1. So, by mathematical induction, if we assume that M p 1 2 p 1 (p 1), then as desired. M p m p 1 + m p p 1 (p 1)2 p 1 + (p 1)2 p p 1 = (2p 2 + 1)2 p 1 < 2p2 p 1 = p2 p, 9

### 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

### 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

### 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)

### 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

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### 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

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### 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

### A first introduction to p-adic numbers

A first introduction to p-adic numbers David A. Madore Revised 7th december 2000 In all that follows, p will stand for a prime number. N, Z, Q, R and C are the sets of respectively the natural numbers

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### 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

### Fourier series. Jan Philip Solovej. English summary of notes for Analysis 1. May 8, 2012

Fourier series Jan Philip Solovej English summary of notes for Analysis 1 May 8, 2012 1 JPS, Fourier series 2 Contents 1 Introduction 2 2 Fourier series 3 2.1 Periodic functions, trigonometric polynomials

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### 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

### 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

### Worksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation

Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other

### 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

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### 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

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### 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

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Congruences. Robert Friedman

Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

### Student Outcomes. Lesson Notes. Classwork. Discussion (10 minutes)

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 5 8 Student Outcomes Students know the definition of a number raised to a negative exponent. Students simplify and write equivalent expressions that contain

### Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

### ARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION

ARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION PETE L. CLARK 1. Sums over divisors, convolution and Möbius Inversion The proof of the multiplicativity of the functions σ k, easy though it was, actually

### 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

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### 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.

### Outline of Complex Fourier Analysis. 1 Complex Fourier Series

Outline of Complex Fourier Analysis This handout is a summary of three types of Fourier analysis that use complex numbers: Complex Fourier Series, the Discrete Fourier Transform, and the (continuous) Fourier

### 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

### 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.,

### The Geometric Series

The Geometric Series Professor Jeff Stuart Pacific Lutheran University c 8 The Geometric Series Finite Number of Summands The geometric series is a sum in which each summand is obtained as a common multiple

### The Mean Value Theorem

The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers

### 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

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Chapter 7. Induction and Recursion. Part 1. Mathematical Induction

Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply

### Introducing Functions

Functions 1 Introducing Functions A function f from a set A to a set B, written f : A B, is a relation f A B such that every element of A is related to one element of B; in logical notation 1. (a, b 1

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### Practice Problems Solutions

Practice Problems Solutions The problems below have been carefully selected to illustrate common situations and the techniques and tricks to deal with these. Try to master them all; it is well worth it!

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### 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

### Basic Properties of Rings

Basic Properties of Rings A ring is an algebraic structure with an addition operation and a multiplication operation. These operations are required to satisfy many (but not all!) familiar properties. Some

### 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

### Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

### 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

### Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)

Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

.9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### {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

### ) + ˆf (n) sin( 2πnt. = 2 u x 2, t > 0, 0 < x < 1. u(0, t) = u(1, t) = 0, t 0. (x, 0) = 0 0 < x < 1.

Introduction to Fourier analysis This semester, we re going to study various aspects of Fourier analysis. In particular, we ll spend some time reviewing and strengthening the results from Math 425 on Fourier

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### 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

### Definition: Group A group is a set G together with a binary operation on G, satisfying the following axioms: a (b c) = (a b) c.

Algebraic Structures Abstract algebra is the study of algebraic structures. Such a structure consists of a set together with one or more binary operations, which are required to satisfy certain axioms.

### 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

### Z-TRANSFORMS. 1 Introduction. 2 Review of Sequences

Z-TRANSFORMS. Introduction The Z-transform is a tranform for sequences. Just like the Laplace transform takes a function of t and replaces it with another function of an auxillary variable s, well, the

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### PROVING STATEMENTS IN LINEAR ALGEBRA

Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements

### Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

### For the purposes of this course, the natural numbers are the positive integers. We denote by N, the set of natural numbers.

Lecture 1: Induction and the Natural numbers Math 1a is a somewhat unusual course. It is a proof-based treatment of Calculus, for all of you who have already demonstrated a strong grounding in Calculus

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### 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

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### 15 Markov Chains: Limiting Probabilities

MARKOV CHAINS: LIMITING PROBABILITIES 67 Markov Chains: Limiting Probabilities Example Assume that the transition matrix is given by 7 2 P = 6 Recall that the n-step transition probabilities are given

### Lecture 3. Mathematical Induction

Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### 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

### Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

### Sets and Cardinality Notes for C. F. Miller

Sets and Cardinality Notes for 620-111 C. F. Miller Semester 1, 2000 Abstract These lecture notes were compiled in the Department of Mathematics and Statistics in the University of Melbourne for the use

### Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

### Mathematical Induction

Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,