Math 248 Homework 1. Edward Burkard. Exercise 1. Prove the following Fourier Transforms where a > 0 and c R: f (x) = b. f(x c) = e.
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1 Math 48 Homework Ewar Burkar Exercise. Prove the following Fourier Transforms where a > an c : a. f(x) f(ξ) b. f(x c) e πicξ f(ξ) c. eπixc f(x) f(ξ c). f(ax) f(ξ) a e. f (x) πiξ f(ξ) f. xf(x) f(ξ) πi ξ g. (f g)(x) f(ξ)ĝ(ξ) a. b. c.. e. Suppose f C (). f(x) ef e πiξx f(x) x f(ξ) f(x c) e πiξx f(x c) x e πiξ(u+c) f(u) u e πicξ e πiξu f(u) u ux c ux e πixc f(x) e πicξ f(ξ) f(ax) f(ξ c) uax, a ux f (x) e πiξx e πixc f(x) x e πi(ξ c)x f(x) x e πiξax f(ax) x e πiξu f(u) x a a f(ξ) e πiξx f (x)x Let u e πiξx an v f (x)x, then u πiξe πiξx x an v f. So e πiξx f (x) x e πiξx f(x) + πiξe πiξx f(x) x e πiξx f(x) + πiξ f(ξ) πiξ f(ξ)
2 f(ξ)ĝ(ξ) f. g. (f g)(x) πi ξ f(ξ) e πiξx f(x) x πi ξ πi ξ e πiξx f(x) x ( πix)e πiξx f(x) x πi πi ( πi) e πiξx [xf(x)] x e πiξx [xf(x)] x F ubini s T heorem xf(x) e πiξx (f g)(x) x f(x y)g(y) y x e πiξ(x y+y) f(x y)g(y) x y e πiξy g(y) e πiξ(x y) f(x y) x y e πiξy g(y) f(ξ) y f(ξ) e πiξy g(y) y Exercise. Compute the Fourier Transforms of the following functions where a > : a. f(x) e πax b. f(x) x +a (Hint: Use contour integration an the esiue Theorem) c. f(x) e πa x a. Since f (x) aπxe πax, by Exercise.f above we have: ξ f(ξ) ( aπxe πax )(ξ) i (f a )(ξ) i a (πiξ) f(ξ) π a ξ f(ξ) So we are left with the ifferential equation: Solving for f(ξ), we see the general solution is: ξ f(ξ) π a ξ f(ξ) f(ξ) ce πξ a Consier the function f (ξ) e πξ a ˆf(ξ). So we have ξ f (ξ) π πξ πξ ξe a f(ξ) + e a a ξ f(ξ) π πξ π πξ ξe a f(ξ) ξe a f(ξ) a a so that f (ξ) is constant. To fin the constant, let ξ in orer to obtain f () f() e πax.
3 3 To compute this integral, consier the following ( ) ( f) (ξ) e πax e πa(x +y ) which follows from Fubini s theorem. By changing to polar coorinates this becomes ( f) (ξ) π Let u πar, then u πarr a a π re πar θr re πar r e u u or hence, f (ξ) e πξ a f(ξ) f() a f(ξ) πξ e a a b. Let s try to take the transform of f: f(ξ) To o this integral, use contour integration: e πiξx x + a x Cr e πiξz z + a z where C r (without loss of generality say r > a) is the semicircle in the upper complex plane (Im(z) ). Then we have that f(ξ) lim r ( First consier the secon integral. Since r > a we have so C r e πiξz ) z + a z e πiξz Cupper z + a z. z + a z a a z a r r a e πiξz Cupper z + a z Cupper Cupper πr (r a ) e πiξz z + a z r a z Since this tens to as r tens to infinity, by the esiue theorem: e f(ξ) lim r Cr πiξz z + a z πi lim To compute the resiue, consier the function r es(f; ia) then f (z) e πiξz z + ia
4 4 c. f(ξ) πi lim f (ia) eπaξ r 4πa f(x) e πa x e πiξx e πa x x e πiξx e πax x + (,] (,] ex( πiξ aπ) e π(ixξ ax) x + [, ) [, ) e πiξx e πax x e π(ixξ+ax) x πiξ + aπ + e x(πiξ aπ) πiξ aπ ( ) ( ) πiξ + aπ + πiξ aπ aπ πiξ + aπ + πiξ Exercise 3. Prove that e πx x. (Hint: e π(x +y ) ) Note that e πx > so e πx x >. Now consier ( e x) πx By switching to polar coorinates we have: let u πr, then So e πx x. π π π e π(x re πr θr e u θu +y ) xy e u u Exercise 4. Let f L (). Define F (x, ξ) : e πixξ f(x). Then F is continuous for each x. Show that G(ξ) : F (x, ξ)x is continuous. Let ξ be an arbitrary point. Now choose any sequence {ξ n } converging to ξ, an efine F n (x) F (x, ξ n ). Then since F is continuous we have: lim F n(x) lim F (x, ξ n ) x F (x, ξ) n n an F n F. Now since F L, by the ominate convergence theorem we have: lim G(ξ n) lim F n (x) x F (x, ξ) x G(ξ) n n So G is continuous. Exercise 5. If f(x), xf(x) L (), then f C () an f ξ πixf(ξ).
5 5 f(ξ + h) f(ξ) h ( e e πiξx πihx ) x h The integran in the above equation is boune by xf(x) which is an L function an tens to πixf(x)e πiξx pointwise. So by the Lebesgue Dominate Convergence Theorem ( e e πiξx πihx ) x πixf(ξ) h in the L norm. So letting h, we see that the first equation becomes: Moreover, f is continuous as it is ifferentiable. ξ f(ξ) f(ξ + h) f(ξ) lim h h ( e e πiξx πihx ) x h πixf(ξ) Exercise 6. x n f (m) S () n, m. Exercise 7. If x m f(x) L () m n, then f C n (). Moreover, m f ξ m ( πix)m f m n. Exercise 8. Fin an example of an f L x() such that f / L ξ (). Consier the function Clearly this function is in L x() as: f(x) rect(x) : rect(x) x The Fourier transform of rect(x) is: e πiξx rect(x) x { x > x rect(x) x e πiξx x sin πξ πξ : sinc(x) Claim: sinc / L ξ (). proof of claim: sinc(x) x Which iverges by the comparison test. Thus sinc / L ξ (). rect(x) is a function that satisfies the requirements. n n sinc(x) x sin(πx) πx sin(πn) πn πn x Exercise 9. Show that there is no constant < C,α < such that x α C,α x, x.
6 6 Exercise. Prove Leibniz formula for all. Exercise. () Show that S ( ) is a C(X) vector space. () Show that for every α, β, ρ α,β is a seminorm on S ( ). Exercise. () If f S ( m ) an g S ( n ), then the function of m+n variables f(x,..., x m )g(y,..., y n ) is in S ( m+n ). () If f S ( n ) an p(x) is a polynomial of n real variables, then p(x)f(x) is in S ( n ). (3) If f S ( n ), then for any multi-inex α, α f S ( n ). (4) f S ( n ) sup x α (x β f(x)) < α, β. Exercise 3. (x) Exercise 4. Show that such neighborhoos of are convex hence making them into a locally convex topological vector space. Exercise 5. Convolution is well-efine in L ( ) an Let f, g L ( ). f g f g. f g f g x f(x y)g(y)y x f(x y) g(y) yx T onelli s T hm f(x y) g(y) xy g(y) f(x y) xy g(y) f y f g(y) y f g < Thus since the L norm of f g is finite we get that f g is finite a.e. an thus f g is finite a.e. f g L ( ). Exercise 6. Show that x j (f g) f g x j.
7 7 Exercise 7. Give a complete proof of (πiξ) β f ( β f)(ξ) for f S ( ). Exercise 8. If f S ( n ) an g S ( m ), then f(x,..., x n )g(x n+,..., x n+m ) is in S ( n+m ). [f(x,..., x n )g(x n+,..., x n+m )] (ξ,..., ξ n+m ) f(ξ,..., ξ n )ĝ(ξ n+,..., ξ n+m ). Moreover, Exercise 9. Justify the use of Fubini s theorem in the proof of f g f ĝ in S ( ). Exercise. Prove f g f ĝ in L ( ). Exercise. Show that the Fourier Transform an the Inverse Fourier Transform F : S ( ) S ( ) (a) f f F : S ( ) S ( ) (b) f ˇf are continuous (as linear) maps with respect to the natural topology of the Schwartz space. Hint: (a) (b) is easy. To prove (a), use Prop 4 (eqns 6 & 7) combine with Leibnitz s rule. ρ Also note: f n f Pm (f f n ) as n, m n where P m (g) : γ + δ m ρ γδ(g). Exercise. Fin an prove the analogues of the properties of the Fourier Transform for the Inverse Fourier Transform. Exercise 3. Show ˇˆ f f. Exercise 4. Show ˇˆ f f f. Exercise 5. Calculate the Fourier Series of {x} : x [x] where [x] is the integer part of x. { Exercise 6. Calculate the Fourier Series of f(x) χ [n,n+) n even χ [n,n+) n o Exercise 7. Calculate the Fourier Series of f(x) { x + x < x + x
8 8 Exercise 8. Show f(ξ) e πiξa e πiξb as esire. πiξ is efine for ξ ( f() b a). e lim f(ξ) πiξa e πiξb lim ξ ξ πiξ L Hôpital πiae πiξa ( πib)e πiξb lim ξ πi πia + πib πi b a Exercise 9. Exercise 3. Exercise 3. Exercise 3. Exercise 33. Exercise 34.
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