Oscilltory integrls Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto August, 0 Oscilltory integrls Suppose tht Φ C R d ), ψ DR d ), nd tht Φ is rel-vlued. I : 0, ) C by Iλ) = e Φx) ψx)dx, λ > 0. R d Define We cll Φ phse nd ψ n mplitude, nd Iλ) n oscilltory integrl. The following proof follows Stein nd Shkrchi. Theorem. If there is some c > 0 such tht Φ)x) c for ll x supp ψ, then for ech nonnegtive integer N there is some c N 0 such tht Iλ) c N λ N, λ > 0. Proof. There is some h DR d ), h 0, such tht hx) = for x supp ψ. Define : R d R d by = h Φ Φ, whose entries ech belong to DR d ), nd define L : C R d ) DR d ) by Lf = k k f = )f. L stisfies, doing integrtion by prts nd using the fct tht hs compct support, R d Lf)gdx = R d k k f)gdx = f k g)dx. R d Elis M. Stein nd Rmi Shkrchi, Functionl Anlysis, p. 35, Proposition.. Wlter Rudin, Functionl Anlysis, second ed., p. 6, Theorem 6.0.
Thus the trnspose of L is L t g = Furthermore, in supp ψ, k g) = g). Le Φ ) = e Φ k k Φ) = e Φ = e Φ. k Φ Φ kφ Thus for ny positive integer N nd for x supp ψ, Le Φ )x) = e Φx), hence Iλ) = L N e Φ )ψdx = R d e Φ L t ) N ψdx. R d But L t ) N ψ dx = λ N A N dx, R d R d where A = ψ) nd A n = A n ). With c N = A N dx <, R d we obtin Iλ) = e Φ L t ) N ψdx L R R t ) N ψ dx = c N λ N, d d completing the proof. The following is n estimte for one-dimensionl oscilltory integrl without n mplitude term. 3 Lemm. Let < b, nd suppose tht Φ C R) is rel-vlued, tht either Φ x) 0 for ll x [, b] or Φ x) 0 for ll x [, b], nd tht Φ x) for ll x [, b]. Then 3λ, λ > 0. 3 Elis M. Stein nd Rmi Shkrchi, Functionl Anlysis, p. 36, Proposition..
Proof. Write which stisfies Lf)gdx = L = d Φ dx, Φ f gdx = Φ fg b f g Φ ) dx. With f = e Φ nd g =, we hve Lf = e Φ nd hence e Φ dx = eφ b ) Φ e Φ Φ dx = eφ b Φ + e Φ Φ ) Φ dx. For λ > 0, using tht Φ x) for ll x [, b] the boundry terms hve bsolute vlue e Φb) Φ b) eφ) Φ ) λ Φ b) + λ Φ ) λ. Becuse Φ 0 or Φ 0 on [, b], e Φ Φ ) Φ dx λ Φ ) Φ dx λ = Φ ) Φ dx λ = λ Φ ) Φ b) λ ; the finl inequlity uses the fct tht the two terms inside the bsolute vlue re both, nd thus the bsolute vlue cn be bounded by the lrger of them. Putting together the two inequlities, e Φ dx λ + 3 λ = 3λ, λ > 0, proving the clim. Lemm 3. Let < b, nd suppose tht Φ C R) is rel-vlued, tht either Φ x) 0 for ll x [, b] or Φ x) 0 for ll x [, b], nd tht there is some µ > 0 such tht Φ x) µ for ll x [, b]. Then 3µ λ, λ > 0. 3
Proof. Φ is continuous on [, b], so, by the intermedite vlue theorem, either Φ x) µ for ll x [, b] or Φ x) µ for ll x [, b]. Let ɛ = in the first cse nd ɛ = in the second cse, nd define Φ 0 = ɛ Φ µ. Then pplying Lemm, for λ > 0 we hve, writing λ 0 = µλ, e 0Φ0x) dx 3λ 0, i.e. e iɛλφx) dx 3µλ). If ɛ = this is the clim. If ɛ =, then the bove integrl is the complex conjugte of the integrl in the clim, nd these hve the sme bsolute vlues. Theorem. Let < b, nd suppose tht Φ C R) is rel-vlued, tht either Φ x) 0 for ll x [, b] or Φ x) 0 for ll x [, b], nd there is some µ > 0 such tht Φ x) µ for ll x [, b]. Suppose lso tht ψ C R). Then with ) we hve c ψ = 3 ψb) + ψ x) dx e Φx) ψx)dx c ψµ λ., Proof. Define J : [, b] C by Jx) = x e Φu) du, which stisfies J x) = e Φx). Integrting by prts, e Φx) ψx)dx = J x)ψx)dx = Jx)ψx) nd s J) = 0 this is equl to Jb)ψb) Jx)ψ x)dx. Lemm 3 tells us tht Jx) 3µ λ for ll x [, b], so Jb)ψb) Jx)ψ x)dx 3µ λ ψb) + 3µ λ proving the clim. b Jx)ψ x)dx, ψ x) dx,
The following is vn der Corput s lemm. Lemm 5 vn der Corput s lemm). Let < b nd suppose tht Φ C R) is rel-vlued nd stisfies Φ x) for ll x [, b]. Then 8λ /, λ > 0. Proof. Becuse Φ is strictly incresing on [, b], Φ hs t most one zero in this intervl. If Φ x 0 ) = 0, then for x x 0 + λ / we hve Φ x) λ /, nd pplying Lemm 3 with µ = λ /, 3µ λ = 3λ /. [x 0+λ /,b] For x x 0 λ / we hve Φ x) λ /, nd pplying Lemm 3 with µ = λ /, [,x 0 λ / ] 3µ λ = 3λ /. But nd so [x 0 λ /,x 0+λ / ] [,b] = [,x 0 λ / ] [x 0 λ /,x 0+λ / ] [,b] + +, [x 0 λ /,x 0+λ / ] [,b] [x 0+λ /,b] 3λ / + λ / + 3λ / = 8λ /. dx λ /, If there is no x 0 [, b] such tht Φ x 0 ) = 0, then either Φ > 0 on [, b] or Φ < 0 on [, b]. In the first cse, becuse Φ is strictly incresing on [, b], Φ x) > λ / for x [ + λ /, b], nd pplying Lemm 3 with µ = λ / gives + [,+λ / ] [,b] λ / + 3µ λ = λ /. [+λ /,b] Elis M. Stein nd Rmi Shkrchi, Functionl Anlysis, p. 38, Proposition.3. 5
In the second cse, Φ x) < λ / for x [, b λ / ], nd pplying Lemm 3 with µ = λ / lso gives λ /. Therefore, if Φ does not hve zero on [, b] then λ / < 8λ /. Lemm 6. Let < b nd suppose tht Φ C R) is rel-vlued nd tht there is some µ > 0 such tht Φ x) µ for ll x [, b]. Then 8µ / λ /, λ > 0. Proof. Φ is continuous on [, b], so by the intermedite vlue theorem either Φ x) µ for ll x [, b] or Φ x) µ for ll x [, b]. Let ɛ = in the first cse nd ɛ = in the second cse, nd define Φ 0 = ɛ Φ µ. Then Φ 0x) for ll x [, b], nd pplying Lemm 5, e iµλφ0x) dx 8µλ) /, λ > 0, i.e. e iɛλφx) dx 8µλ) /, λ > 0. If ɛ = this is the inequlity in the clim. If ɛ =, then the bove integrl is the complex conjugte of the integrl in the clim, nd these hve the sme bsolute vlues. We use the bove to prove the following estimte which involves n mplitude. 5 Theorem 7. Let < b nd suppose tht Φ C R) is rel-vlued nd tht there is some µ > 0 such tht Φ x) µ for ll x [, b]. Suppose lso tht ψ C R). Then with ) c ψ = 8 ψb) + ψ x) dx 5 Elis M. Stein nd Rmi Shkrchi, Functionl Anlysis, p. 38, Corollry.., 6
we hve e Φx) ψx)dx c ψµ / λ /, λ > 0. Proof. Define J : [, b] C by Jx) = x e Φu) du, which stisfies J x) = e Φx). Integrting by prts, e Φx) ψx)dx = J x)ψx)dx = Jx)ψx) nd s J) = 0 this is equl to Jb)ψb) Jx)ψ x)dx. b Jx)ψ x)dx. But for ech x [, b] we hve by Lemm 6 tht Jx) 8µ / λ /, so Jb)ψb) Jx)ψ x)dx 8µ / λ / ψb) + 8µ / λ / ψ x) dx, completing the proof. Bessel functions For n Z, the nth Bessel function of the first kind J n : R R is Let I = J n λ) = π π 0 e sin x e inx dx, λ R. [ 0, π ] [ ] [ 3π, I =, π, I 3 = π, 5π ] [ ] 7π, I =, π, on which cos x, nd on which sin x I 5 = [ π, 3π ] [ 5π, I 6 =, 7π ],. Write Φx) = sin x nd ψx) = e inx. Φ x) = cosx) nd Φ x) = sinx), nd for I, I, I 3, I we pply Theorem with µ =. For ech of I, I, I 3, I we compute c ψ = 3 ) + πn, which gives us e Φx) ψx)dx c ψµ λ = 3 + πn ) λ. I k 7
For I 5 nd I 6, we pply Theorem 7 with µ =. For ech of I 5 nd I 6 we compute c ψ = 8 ) + πn, which gives us e Φx) ψx)dx c ψµ / λ / = 8 + πn ) / λ /. I k Therefore J n λ) π 3 + πn which shows tht for ech n Z, s λ. ) λ + J n λ) = O n λ / ) π 8 + πn ) / λ /, 8