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 and coverage. There will be 4 to 6 questions, some with multiple parts. The test will cover sections 0.1-0.5, 1.2, and 1.3.1-1.3.3 in the text. The problems will be similar to ones done for homework, and as examples in class and in the text. A short table of integrals will be provided.

Topics Covered

Inner Product Spaces

• Inner products
  1. Definitions of real and complex inner products. Schwarz and triangle inequalities. Be able to compute the angle between two vectors. §0.2, §0.4.
  2. Examples of inner product spaces. ℝⁿ, ℂⁿ, L² and ℓ². §0.2, §0.3.
• Types of convergence
  1. Pointwise convergence. f_n(x) → f(x) for each fixed x. §0.3.1
  2. Convergence in L² (mean convergence). ||f− f_n||_L² → 0.  §0.3.1.
  3. Uniform convergence. max_x|f_n(x) − f(x)| → 0.  §0.3.2.
  4. Know the definitions for these types of convergence. Be able to give examples illustrating the differences among them.
• Orthogonality
  1. Orthogonal and orthonormal sets of vectors, orthonormal bases, and orthogonal complements, V^⊥. Know the definitions for these terms. Know how to write a vector in terms of an orthonormal basis, and how to calculate the coefficients. Be able to do problems similar to ones assigned in homework. §0.5.1.
  2. Orthogonal projections and "least-squares" minimization problems. Be able to find orthogonal projections and to solve least-squares minimization problems. §0.5.2.
  3. Gram-Schmidt process. Be able to use this to find a an orthonormal basis for a space. §0.5.3.

Fourier Series

• Calculating Fourier Series
  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 extension). Be able to compute FSS and FCS for functions defined on a half interval, [0,π].
• Pointwise convergence
  1. Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that f is continuously differentiable. §1.3.1.
  2. Fourier (Dirichlet) kernel, P. Know what P is and how to express partial sums in terms of P. §1.3.2
  3. 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-Lebesgue Lemma. §1.3.2.
  4. Know the Theorems 1.22 and 1.28. Be able to use them to decide what function an FS, FSS, or FCS converges to.

Updated 2/20/2010.