- For f ∈ L
^{1}, show that if F^{−1}f ∈ L^{1}or Ff ∈ L^{1}, then f equals a continuous bounded function a.e. - Suppose f,g ∈ L
^{1}(ℝ) and Ff = Fg. Show f = g a.e. - Show that if f ∈ L
^{1}, then lim_{|x| →∞}Ff= 0 . - ↑ Suppose f ∗ f = f or f ∗ f = 0 and f ∈ L
^{1}(ℝ). Show f = 0. - For this problem define ∫
_{a}^{∞}fdt ≡ lim_{r→∞}∫_{a}^{r}fdt. Note this coincides with the Lebesgue integral when f ∈ L^{1}. Show- ∫
_{0}^{∞}du = - lim
_{r→∞}∫_{δ}^{∞}du = 0 whenever δ > 0. - If f ∈ L
^{1}, then lim_{r→∞}∫_{ℝ}sinfdu = 0.

Hint: For the first two, use

= ∫_{0}^{∞}e^{−ut}dt and apply Fubini’s theorem to ∫_{0}^{R}sinu∫_{ℝ}e^{−ut}dtdu. For the last part, first establish it for f ∈ C_{c}^{∞}and then use the density of this set in L^{1}to obtain the result. This is sometimes called the Riemann Lebesgue lemma. - ∫
- ↑Suppose that g ∈ L
^{1}and that at some x > 0, g is locally Holder continuous from the right and from the left. This meansexists,

exists and there exist constants K,δ > 0 and r ∈ (0,1] such that for

< δ,for y > x and

for y < x. Show that under these conditions,

- Let g ∈ L
^{1}and suppose g is locally Holder continuous from the right and from the left at x. Show that thenThis is very interesting. If g ∈ L

^{2}, this shows F^{−1}=, the midpoint of the jump in g at the point, x. In particular, if g ∈G, F^{−1}= g. Hint: Show the left side of the above equation reduces toand then use Problem 6 to obtain the result.

- ↑ A measurable function g defined on has exponential growth if≤ Ce
^{ηt}for some η. For Re> η, define the Laplace Transform byAssume that g has exponential growth as above and is Holder continuous from the right and from the left at t. Pick γ > η. Show that

This formula is sometimes written in the form

and is called the complex inversion integral for Laplace transforms. It can be used to find inverse Laplace transforms. Hint: Plug in the formula for the Laplace transform and then massage to get it in the form of the preceding problem.

- Suppose f ∈G. Show F(f
_{xj})(t) = it_{j}Ff(t). - Let f ∈G and let k be a positive integer.
One could also define

Show both ||||

_{k,2}and ||||||_{k,2}are norms on G and that they are equivalent. These are Sobolev space norms. For which values of k does the second norm make sense? How about the first norm? - ↑Define H
^{k}(ℝ^{n}),k ≥ 0 by f ∈ L^{2}(ℝ^{n}) such thatShow H

^{k}(ℝ^{n}) is a Banach space, and that if k is a positive integer, H^{k}(ℝ^{n}) ={ f ∈ L^{2}(ℝ^{n}) : there exists {u_{j}}⊆G with ||u_{j}− f||_{2}→ 0 and {u_{j}} is a Cauchy sequence in ||||_{k,2}of Problem 10}. This is one way to define Sobolev Spaces. Hint: One way to do the second part of this is to define a new measure μ byThen show μ is a Borel measure which is inner and outer regular and show there exists

such that g_{m}∈G and g_{m}→ Ff in L^{2}(μ). Thus g_{m}= Ff_{m},f_{m}∈G because F maps G onto G. Then by Problem 10,is Cauchy in the norm ||||_{k,2}. - ↑ If 2k > n, show that if f ∈ H
^{k}(ℝ^{n}), then f equals a bounded continuous function a.e. Hint: Show that for k this large, Ff ∈ L^{1}(ℝ^{n}), and then use Problem 1. To do this, writeSo

Use the Cauchy Schwarz inequality. This is an example of a Sobolev imbedding Theorem.

- Let u ∈G. Then Fu ∈G and so, in particular, it makes sense to form the
integral,
where

= x ∈ ℝ^{n}. For u ∈G, define γu≡ u. Find a constant such that Fequals this constant times the above integral. Hint: By the dominated convergence theoremNow use the definition of the Fourier transform and Fubini’s theorem as required in order to obtain the desired relationship.

- Let h=
^{2}+. Show that h^{′}= 0 and h= π∕4. Then let x →∞ to conclude that ∫_{0}^{∞}e^{−t2 }dt =∕2. Show that ∫_{ −∞}^{∞}e^{−t2 }dt =and that ∫_{−∞}^{∞}e^{−ct2 }dt =. - Recall that for f a function, f
_{y}= f. Find a relationship between Ff_{y}and Ffgiven that f ∈ L^{1}. - For f ∈ L
^{1}, simplify Ff. - For f ∈ L
^{1}and c a nonzero real number, show Ff= Fgwhere g= f. - Suppose that f ∈ L
^{1}and that ∫dx < ∞. Find a way to use the Fourier transform of f to compute ∫ xfdx. - Let be a probability space and let X : Ω → ℝ
^{n}be a random variable. This means X^{−1}∈ℱ. Define a measure λ_{X}on the Borel sets of ℝ^{n}as follows. For E a Borel set,Explain why this is well defined. Next explain why λ

_{X}can be considered a Radon probability measure by completion. Explain why λ_{X}∈G^{∗}ifwhere G is the collection of functions used to define the Fourier transform.

- Using the above problem, the characteristic function of this measure (random variable)
is
Show this always exists for any such random variable and is continuous. Next show that for two random variables X,Y,λ

_{X}= λ_{Y }if and only if ϕ_{X}= ϕ_{Y }for all y. In other words, show the distribution measures are the same if and only if the characteristic functions are the same. A lot more can be concluded by looking at characteristic functions of this sort. The important thing about these characteristic functions is that they always exist, unlike moment generating functions.

Review Of Some Linear Algebra

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