Applied/Numerical Analysis Qualifying Exam
August 13, 2010
Policy on misprints: The qualifying exam committee tries to proofread
exams as carefully as possible. Nevertheless, the exam may contain a few
misprints. If you are convinced a problem has been stated incorrectly, in-
dicate your interpretation in writing your answer. In such cases, do not
interpret the problem so that it becomes trivial.
Part 1: Applied Analysis
Instructions: Do any 3 of the 4 problems in this part of the exam. Show
all of your work clearly. Please indicate which of the 4 problems you are
skipping.
1. Let Hbe a complex (separable) Hilbert space, with h·,·i and k ·k being
the inner product and norm.
(a) Define the term compact linear operator on H.
(b) Let K:H → H be compact. Show: If λ6= 0 is an eigenvalue of
K, then it has finite multiplicity.
2. Let hf, g i=R1
−1f(x)g(x)w(x)dx, where w∈C[−1,1], w(x)>0, and
w(−x) = w(x). Let {φn(x)}∞
n=0 be the orthogonal polynomials gener-
ated by using the Gram-Schmidt process on {1, x, x2...}. Assume that
φn(x) = xn+ lower powers.
(a) Show that φn(−x) = (−1)nφn(x).
(b) Show that φnis orthogonal to all polynomials of degree ≤n−1.
(c) Show that φn(x) satisfies this recurrence relation:
φn+1(x) = xφn(x)−cnφn−1(x), n≥1, where cn=hφn, xni
kφn−1k2.
3. Define D[φ] = R1
0(φ′2+qφ2)dx and H[φ] = R1
0φ2dx. Throughout, we
require that φ∈C(1)[0,1] and that φ(0) = 0.
(a) Let σ≥0. Minimize D[φ] + σφ2(1) subject to the constraint
H[φ] = 1. Find the resulting Sturm-Liouville eigenvalue problem,
including boundary conditions at x= 1.
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