- 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. - 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 6}. 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 6,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.

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