7 Million Study Materials
From students who've taken these classes before
From students who've taken these classes before
24/7 Access to Tutors
Personal attention for all your questions
Personal attention for all your questions
Learn
93% of our members earn better grades
93% of our members earn better grades
George Mason | MATH 671
Fourier Analysis
Professors
- Napoletani, D
24 sample documents related to MATH 671
-
George Mason MATH 671
MATH 675 HOMEWORK #3 SOLUTIONS Exercise 1. Exercise 11, page 61 in Stein and Shakarchi. Note that |fk (n) f (n)| = 1 2 1 = 2 1 2 fk (x) einx dx f (x) einx dx (fk (x) f (x) einx dx |fk (x) f (x)| dx. Given > 0 choose K so large that -
George Mason MATH 671
Exercise 4. Prove the following variant of the Poisson Summation Formula. If f S, > 0 and R, then f n= x + n 2in/ e = f (m + ) e2ix(m+)/ . m= (Hint: Find a function f1 (x) such that f1 (m) = f (m + ).) -
George Mason MATH 671
MATH 671 SPRING 2005 MIDTERM EXAM Instructions. 1. This exam is due by 7:20pm Monday, April 4, 2005. 2. This exam is closed book and closed note. You may not consult with any outside source while you are taking this exam. However, you may have with -
George Mason MATH 671
MATH 675 HOMEWORK #5 SOLUTIONS Exercise 1. Exercise 8, page 89 in Stein and Shakarchi. SOLUTION: We know from Problem 6 of Chapter 2 that f (n) = 0 2 n2 2 n=0 n even, n = 0 n odd Therefore by Plancherels formula 1 2 Now, 1 2 Therefore, -
George Mason MATH 671
MATH 675 HOMEWORK #1 SOLUTIONS Exercise 1. Given Eulers formula eix = cos(x) + i sin(x) derive the trig identities 2 sin() sin() = cos( ) cos( + ), 2 sin() cos() = sin( ) + sin( + ). Solution: 2 sin() sin() = 2 ei ei 2i ei ei 2i 1 = (ei(+) e -
George Mason MATH 671
MATH 675 HOMEWORK #6 SOLUTIONS Exercise 1. Exercise 2, page 161 in Stein and Shakarchi. SOLUTION: If f (x) = f () = = 1 |x| 1 then 0 otherwise 1 1 e2ix d 1 (e2i e2i ) 2i sin(2) 1 e2i e2i = = . 2i For the second part, we take an indirect appr -
George Mason MATH 671
MATH 675 HOMEWORK #4 SOLUTIONS Exercise 1. Exercise 10, page 27 in Stein and Shakarchi. SOLUTION: Since x = r cos() and y = r sin() we have x y = + = cos() + sin() r x r y r x y and x y = + = r sin() + r cos() . x y x y Moreover, 2 2 2 -
George Mason MATH 671
MATH 675 HOMEWORK #5 DUE 21 MARCH 2007 Exercise 1. Exercise 8, page 89 in Stein and Shakarchi. Exercise 2. For each pair of integers (j, k) dene the dyadic interval Ij,k by Ij,k = [2j k, 2j (k + 1)], and the Haar function hj,k (x) by hj,k = 2j/2 (Ij -
George Mason MATH 671
Math 671001 (Fourier Analysis) Spring 2007 Instructor: David Walnut Oce: Science and Technology I, room 261 Phone: 703 993 1478 (voice); 703 993 1491 (fax) Email: dwalnut@gmu.edu Oce hours: TR 1:302:30 and by appointment. Text: Elias M. Stein and Ram -
George Mason MATH 671
MATH 675 HOMEWORK #7 SOLUTIONS Exercise 1. Exercise 9, page 163 in Stein and Shakarchi. SOLUTION: (a) We know that FR () = 1 | [R,R] (). R This follows by direct calculation from Exercise 2, p. 161 (followed by dilation by R1 ), or from equation -
George Mason MATH 671
MATH 675 HOMEWORK #2 SOLUTIONS Exercise 1. Exercise 4, page 59 in Stein and Shakarchi. SOLUTION: For part (b) note that since f () is odd, f (n) = 1 i i f () ein d = f () sin(n) d = () sin(n) d. 2 0 0 Continuing and using a standard integral -
-
-
-
George Mason MATH 671
%!PS-Adobe-2.0 %Creator: dvipsk 5.499a Copyright 1986, 1992 Radical Eye Software %Title: 671s01sb.dvi %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips -o 671s01sb.ps 671s01sb %DVIPSSource: TeX output 2001.0 -
-
George Mason MATH 671
MATH 675 HOMEWORK #4 DUE 28 MARCH 2005 Exercise 1. Exercise 1.32 (a), Kammler. Exercise 2. Exercise 1.33 (b), (c), Kammler. Exercise 3. Exercise 4.11, Kammler. Exercise 4. Exercise 4.14, Kammler. Exercise 5. Exercise 4.16, Kammler. Exercise 6. Prove -
-
-
-
George Mason MATH 671
Math 671001 (Fourier Analysis) Spring 2005 Instructor: David Walnut Office: Science and Technology I, room 261 Phone: 703 993 1478 (voice); 703 993 1491 (fax) Email: dwalnut@gmu.edu Office hours: TR 2:003:00 and by appointment. Text: David W. Kammler -
George Mason MATH 671
MATH 675 HOMEWORK #1 DUE 31 JANUARY 2007 Exercise 1. Given Euler\'s formula eix = cos(x) + i sin(x) derive the trig identities 2 sin() sin() = cos( - ) - cos( + ), 2 sin() cos() = sin( - ) + sin( + ). Exercise 2. Exercise 5, page 26 in Stein and Shak -
George Mason MATH 671
MATH 675 HOMEWORK #2 DUE 7 FEBRUARY 2007 Exercise 1. Exercise 4, page 59 in Stein and Shakarchi. Exercise 2. Exercise 5, page 59 in Stein and Shakarchi. Exercise 3. Exercise 6, page 59 in Stein and Shakarchi. Exercise 4. Exercise 9(a), page 60, Stei -
7,000,000 study materials • 24/7 tutors • earn better grades