Math 414-501 — Test 1 Review

General Information

Calculators. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra or storing programs or other material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure. There will be 4 to 6 questions, some with multiple parts. The problems will be similar to ones done for homework, and as examples in class and in the text. Here are some sample questions.

1. Define the term uniform convergence. Does the Fourier sine series for f(x) = x(π -x) converge uniformly to f on [0, π]? Give a reason for your answer.
2. Let f(x) = 2π - x, 0 ≤ x ≤ 2π. Find the Fourier series for f. Sketch three periods of the function to which f converges pointwise. Is the convergence uniform? Give a reason for your answer. Use the series you have found to evaluate the series
   1 − 1/3 + 1/5 − 1/7 + ...
   
3. State and prove the Riemann Lebesgue Lemma for a function that is continuously differentiable on the interval [a, b].

Topics Covered

1. Extensions of functions — periodic, even periodic, and odd periodic extensions. Be able to sketch extensions of functions.
2. Fourier series. Be able to compute Fourier series in either real or complex forms, and with prescribed period 2π on an intervals of the form [−π, π], [0, 2π], or [c − π, c + π]. Be able to know and use Lemma 1.3.
3. Fourier sine series (FS for odd, 2π-periodic extension) and Fourier cosine series (FS for even, 2π-periodic exrtension). Be able to compute FSS and FCS for functions defined on a half interval, [0,π].

Poitnwise convergence

1. Definition of pointwise convergence.
2. Definitions of piecewise continuous, jump discontinuity, and piecewise smooth.
3. Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that f is continuously differentiable. §1.3.1.
4. Fourier (Dirichlet) kernel, P. Know what P is and how to express partial sums in terms of P. §1.3.2
5. Be able to sketch a proof of Theorem 1.22, making use of the formula for P and the properties of P as well as the Riemann-Lesbegue Lemma. §1.3.2.
6. Know the Theorems 1.22 and 1.28. Be able to use them to decide what function an FS, FSS, or FCS converges to.

Uniform convergence

1. Definition of uniform convergence.
2. Conditions under which an FS, FSS, or FCS is uniformly convergent. Be able to apply these to determine whether or not an FS is uniformly convergent. These are all stated for periodic functions. To apply them on [−π,π], or [0,π], work with the appropriate periodic extension. §1.3.4.
3. Gibbs' phenomenon. Be able to briefly describe the Gibbs' phenomenon.

Mean convergence

1. Inner products — definition, L²[a,b] and its inner product (§0.3.1), orthogonal sets and orthonormal sets (§0.5.1).
2. Definition of mean (``energy'') convergence.
3. Best approximation property of partial sums. Be able to show Lemma 1.34, given that V_N has an orthogonal basis. §1.3.5.
4. Parseval's theorem. Know both the real and complex form. be able to use it to sum series similar to ones given in the homework.
5. Here are the main theorems on mean convergence. Be able to state them and briefly explain their significance.
   1. If f is in L², then the partial sums of the FS for f converge in the mean to f.
   2. Riesz-Fisher theorem (Synthesis problem). If Σ_k (|a_k|² + |b_k|²) < ∞, then there exists an f in L² such that the series formed from the a_k's and b_k's is the FS for f and converges in the mean to f.

Updated 2/19/2009.