Standard Generals Questions - Real Analysis
Measure and integration
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Discuss the differences between Riemann and Lebesgue integration,
why Lebesgue integration is better. What are necessary and
sufficient conditions for a function to be Riemann integrable?
Give an example of a function that is Lebesgue integrable, but not
Riemann integrable. Give an example of an open dense subset of
[0,1] whose characteristic function is not Riemann integrable.
What are Borel sets? Are all Borel sets G-deltas or F-sigmas? Are
the rationals a G-delta? When is a set Lebesgue measurable? Why
are there more Lebesgue measurable sets than Borel sets? Give an
example of a set that is not Lebesgue measurable. Does every
measurable set in R have density points? Let E be a measurable set
in R. Must there exist points x and y in E such that (x+y)/2 is
also in E? Suppose E has positive measure. Show that the set of
differences E-E contains an interval.
Give an example of an open dense subset of [0,1] with measure 1/2.
Give an example of a residual set (countable intersection of open
dense subsets) with measure 0.
Let E_n be a sequence of measurable sets in [0,1] with m(E_n)
tending to 1. Does there exist a subsequence whose intersections
all have measure greater than 1/2?
Define absolute continuity of a measure. State the Radon-Nikodym
Thm. Give an example of a measure which is singular with respect
to Lebesgue measure. Give an example of a singular measure with
continuous distribution function.
State Fubini's Theorem.
Given f in L_1(X), prove that to each epsilon > 0 there is a delta
> 0 such that mu(A) < delta ==> \int_A |f| d(mu) < epsilon.
If \sum mu(A_i) is finite, show that the set of points lying in
infinitely many A_i has measure 0.
For what p is x^p integrable over [0,1]? What about x^p cos(1/x)?
Same questions in the sense of Riemann and Lebesgue integral.
For which real values of s is the integral over R of sin(x)/x^s
finite?
Suppose f is continuous and \int_1^{\infty} f(x)x^n dx = 0 for all
n >= 2. What can be said about f?
Fourier type questions
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What is the image of L^1 under the Fourier transform? State the
Fourier inversion formula. When does it hold? What about the
transform of an L^2 function? State and prove the Plancherel
identity. What is the Schwartz space? What is the dual of the
Schwartz space?
What is the Fourier transform of a compactly supported function?
State the Paley-Wiener Thm. When does a Fourier series converge to
its original function? What can you say about the Fourier series
of a smooth function? Does this condition imply the function
is smooth? State and prove the Poisson summation formula.
What is the transform of the characteristic function of an
interval? Compute the Fourier transform of 1/(1+x^2). What is the
Fourier transform of the Cantor function? Can you find functions
other than the Gaussian that are their own Fourier transform?
Let f be smooth of compact support, and F(s) = \int_0^{\infty}
f(x)exp(-sx) dx. (aka Laplace transform.) Is F always defined
for s real? Is it continuous? Differentiable? What is the
derivative? Is F analytic for complex s? How can we recover f
from F? What does this tell us about the general case (arbitrary
f)?
Suppose f is a function in L^1(R). Define g(x) to be the integral
over R of cos(xt)f(t). What can you say about g?
Similarly, define take f in L^1(R), and define g(x) = \int_R
exp(-ixt^2)f(t) dt. What can you say about it? Is it enough to
prove your statements for a dense subset of L^1?
How would you estimate the behaviour of \int_R x^2n exp(-x^4) dx?
What can you say about \int_R f(x) exp(ikg(x)) dx (as a function
of k)?
Convergence questions
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State and prove Fatou's Lemma. State and prove the monotone
convergence theorem. State and prove the dominated convergence
theorem.
Prove that a sequence of continuous functions on [0,1] that
converges monotonically to 0 converges uniformly (Dini's Thm).
If a sequence of functions converges uniformly on a finite measure
space, do their integrals converge to the integral of their limit?
On an infinite measure space?
Given a sequence of continuous functions from [0,1] -> R that
tends to 0 pointwise, do their integrals converge to 0?
Given a sequence of functions converging pointwise, when does the
limit of their integrals converge to the integral of their limit?
Same for derivatives?
Give a sequence of functions on [0,1] that converges nowhere, yet
with integrals converging to zero.
Given a sequence of integrable functions, when does the sequence
of their integrals converge?
Consider the series 1 - 1/2 + 1/3 - 1/4 + ... What does this
converge to? Can you prove that it converges? Can the series be
rearranged to converge to another value?
What is the limit as n tends to infinity of 1/n + 1/(n+1) + ... +
1/2n?
Differentiation
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What are necessary and sufficient conditions for a function for a
function to be the integral of its derivative?
State the Fundamental Thm of Calculus. Give exact conditions and
counterexamples...
Give an example of an (strictly?) increasing function which has
derivative a.e. 0.
Give an example of a function differentiable at every point of
[0,1], but with derivative not in L^1([0,1]).
Give an example of an everywhere continuous, nowhere
differentiable function on the unit interval.
Is the indefinite integral of a function in L^2(R) continuous?
Define the total variation of a real-valued function on a closed
interval. What can be said about functions of bounded variation?
Must a continuous function of bounded variation be absolutely
continuous? What can you say about monotonic functions? What is
Helly's Thm?
Suppose you have a sequence of functions f_n of total variation
that is bounded independent of n. What can you conclude?
What can you say about convex functions? Can a convex function
have countably many points of non-differentiability? Could these
points have an accumulation point? Could there be uncountably
many points of non-differentiability?
What does it mean for a function to satisfy a Lipschitz condition?
What does this imply about the distributional derivative? If f:R
-> R^n is Lipschitz, what can you say about the equation dx/dt =
f(x)? Do you how to prove this? Does the proof work if R^n is
replaced by a Banach space?
L^p spaces
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What is the dual space to L^p?
State and prove the Holder and Minkowski inequalities.
When is L^p contained in L^q? For what measure is this
containment reversed?
Let f be in L^p(R), g in L^q(R). What can you say about the
convolution f*g? What if p and q are conjugate exponents?
What is a Hilbert space? How would you prove that L^2(measure) is
a Hilbert space? Is L^{1/2} complete?
Define Hardy Littlewood Maximal function. State its properties.
Give a counterexample showing that L^11 does not map into L^1.
State the Arzela-Ascoli Thm. What is the analog for L^p spaces?
In other words, characterize the compact subsets of an L^p space.
Do you know what a weak derivative is? Let u be a compactly
supported function on R^2. Can you bound an L^p norm of u in
terms of the integral over R^2 of the length of the gradient of u?
Miscellany
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State the inverse function theorem.
What are Baire categories?
Let phi:B -> C (complex numbers) be a Banach algebra hom. What
can you say about its norm?
Is there a function on [0,1] which is discontinuous at every
rational? At every irrational?
What is the Lebesgue measure of the Cantor set? What is the
Hausdorff dimension of the Cantor set?
Describe the set of points x in R such that there are infinitely
many rational numbers p/q with |x-p/q| < exp(-q).
Given a smooth f:[0,1] -> R, describe how \int_0^1 exp(tf(x)) dx
behaves as t goes to infinity.
Prove the Riemann-Lebesgue Lemma.
Calculate \int_R (sin(x)/x)^2 dx.