Math 133

Homework

Unless otherwise specified, all questions refer to the Stein & Shakarchi text.

You may use whatever resources you wish to complete the homework (textbook, notes, office hours, TA, other students, etc.); however it is strongly recommended that you try as much of the homework as you can on your own. If you find yourself relying heavily on external help for the homework, then this is a danger sign and there is a strong chance that you will not perform well in the midterms and finals. Conversely, if you can understand and solve all the homework problems well and by your own resources then the exams should not present much of a challenge.

Many of the questions will require short proofs or justification of your reasoning. We do not expect overly detailed, professional-quality proofs in this course, but we will ask that when necessary, your steps are written in a logical order and with appropriate logical connectives (for instance, you are strongly encouraged to use such English words and phrases as “since”, “because”, “if/then”, “let”, “define”, “suppose”, “on the other hand”, “by Theorem X.X”, “we need to show”, etc.), and are not just a string of equations with no explanation connecting them. Also you should be aware that certain theorems and results in this course are only valid under certain assumptions (e.g. one may require a function to be continuous, or differentiable), and you may be expected to at least mention this when applying such results.

Homework 1 (Due Jan 22)

From textbook: Chapter 1, Exercises 1, 2, 3, 4(bchi), 5; Chapter 2, Exercises 2, 4, 5, 7, 8.

Homework 2 (Due Jan 29)

Available in PDF format here.

Homework 3 (Due Feb 5)

Available in PDF format here.
Errata: In Q4(b), “mean zero sense” should read “mean square sense”. In Q5b, “assume that” should read “ensure that”. All references to “Plancherel’s theorem” should read “Parseval’s identity”. In Q7, “uniformly to h” should read “uniformly to f”. In Q8(c), “for 0 < alpha < ½” should read “for ½ < alpha < 1”.

Homework 4 (Due Feb 26) - note postponement of two weeks

Available in PDF format here.

Homework 5 (Due Mar 4)

Available in PDF format here.

Homework 6 (Due Mar 11)

Available in PDF format here.
Errata: In Q6 (a), all the phases in the formulae for psi and its Fourier transform have the wrong sign (those with a + sign should be -, and vice versa). Similarly, in Q7(de), both occurrences of the + sign should instead be a - sign.

Homework 7 (Due Mar 18)

Available in PDF format here.
Errata: In Q2: there should be an "f" after the "a_0". In Q7(a), 1/N_2 should be 1/N_1.