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Tools from Harmonic Analysis TOOLS FROM HARMONIC ANALYSIS BRADLY STADIE Abstract. The Fourier transform can be thought of as a map that decom- poses a function into oscillatory functions. In this paper, we will apply this decomposition to help us gain valuable insights into the behavior of our original function. Some particular properties of a function that the Fourier transform will help us examine include smoothness, localization, and its L2 norm. We will conclude with a section on the uncertainty principle, which says though these transformations are useful there is a limit to the amount of information they can convey. Contents 1. Basic Properties of the Fourier Transformation 1 2. Plancherel's Theorem and the Fourier Inversion Formula 5 3. The Hausdorff-Young and Young Inequalities 9 4. The Uncertainty Principle 10 Acknowledgments 14 References 14 1. Basic Properties of the Fourier Transformation Let f 2 L1(Rn). Then we define its Fourier transform f^ : Rn ! C by Z f^(ξ) = e−2πix·ξf(x)dx We can generalize this definition in the following way. Let M(Rn) be the space of finite complex-valued measures on Rn with the norm kµk = jµj(Rn) where jµj is the total variation. We then define Z µ^(ξ) = e−2πix·ξdµ(x) There are several basic formulas for Fourier transform that we will find useful throughout this paper. In particular: 2 Let Γ(x) = e−πjxj . Then 2 (1.1) Γ(^ ξ) = e−πjξj Date: AUGUST 26, 2011. 1 2 BRADLY STADIE n Let τ 2 R . Define fτ (x) = f(x − τ). Then ^ −2πiτ·ξ ^ (1.2) fτ (ξ) = e f(ξ) 2πix·τ Let eτ (x) = e Then ^ (1.3) edτ f(ξ) = f(ξ − τ) Let T be an invertible linear map from Rn to Rn. Further, let T −t be the inverse transpose of T . Then (1.4) f[◦ T = j det(T )j−1f^◦ T −t Let T be a dilation (i.e. T x = rx for some r > 0). Then as a special case of (1.4) we have (1.5) f\(rx) = r−nf^(r−1ξ) Finally, Let f~(x) = f(−x). Then ^ ¯ (1.6) f~ = f^ Two important properties we will use to relate f to f^ are those of smoothness and localization. We say our function f is smooth if it is infinitely differentiable. In order to define localization, we first let D(x; r) = fy 2 Rn : jy − xj < rg. With this in mind, we now say that f is localized in space if for some R ≥ 0 we have suppf ⊂ D(0;R). These concepts of smoothness and localization will relate f and f^ in the following way. Suppose our function f is localized in space, then f^ should be smooth. Con- versely if f is smooth than f^ should be localized. We will shortly prove rigorous version of both of these principals, starting first by showing that if f is localized in space then f is smooth. To aid us in this persuit, we will make use of multiindex notation. Specifically, a we define a multiindex to be a vector α 2 Rn whose components are nonnegative integers. If α is a multiindex then by definition α1 αn α @ @ (1.7) D = α1 ··· αn @x1 @xn α n αj (1.8) x = Πj=1xj The length of α, denoted by jαj, is defined as X (1.9) jαj = αj j TOOLS FROM HARMONIC ANALYSIS 3 Theorem 1.10. Let µ 2 M(Rn) and supp (µ) be compact. Then µ^ is in C1 and α α (1.11) D µ^ = ((−2πix) µ))b Further, if supp µ ⊂ D(0;R) then α jαj (1.12) kD µ^k1 ≤ (2πR) kµk Proof. Note that for any α the measure (2πix)αµ is again a finite measure with compact support. Hence, if we can prove thatµ ^ is C1 and that (1.8) holds with jαj = 1, then the fact thatµ ^ 2 C1 and (1.8) will follow by induction. n Fix j 2 f1; :::; ng and let ej be the jth standard basis vector. Fix ξ 2 R and consider the difference quotient µ^(ξ + he ) − µ^(ξ) ∆(h) = j h Z e−2πihxj − 1 = e−2πiξ·xdµ(x) h e−2πihxj −1 e−2πihxj −1 As h ! 0 the quantity h converges pointwise to −2πixj. Further, j h j ≤ e−2πihxj −1 −2πiξ·x 2πjxjj for each h. Hence, the integrand h e is dominated by j2πxjj, which is a bounded function on the support of µ. Thus, by applying the dominated convergence theorem we see that Z e−2πihxj − 1 lim ∆(h) = lim e−2πiξ·xdµ(x) h!0 h!0 h Z −2πiξ·x = −2πixje dµ(x) Hence, we see that (1.8) holds when jαj = 1. And so by induction (1.8) holds in general. To obtain (1.9) we first note if µ 2 M(Rn) then the following holds n kµ^k1 ≤ kµkM(R ) For given any ξ we have Z jµ^(ξ)j = j e−2πix·ξdµ(x)j Z ≤ je−2πix·ξjdjµj(x) = kµk Using this fact and (1.8) we see that j(2πix)αµj ≤ (2πR)jαjkµk. And so we obtain (1.9). Having seen that µ being localized impliesµ ^ is smooth, we now wish to show that µ being smooth impliesµ ^ is localized. More specifically, we wish to prove the following result. 4 BRADLY STADIE Theorem 1.13. Suppose that f is CN and that Dαf 2 L1 for all α with 0 ≤ jαj ≤ N. Then when jaj ≤ N; we have (1.14) Dcαf(ξ) = (2πiξ)αf^(ξ) And furthermore (1.15) jf^(ξ)j ≤ C(1 + jξj)−N For a suitable constant C In order to prove this result, we will first need a preliminary lemma. We begin by letting φ : Rn ! R be a C1 function with the following properties (1) φ(x) = 1 if jxj ≤ 1 (2) φ(x) = 0 if jxj ≥ 2 (3) 0 ≤ φ ≤ (1) (4) φ is radial x Now define φk = φ( k ); in other words, rescale the function φ to live on a scale k instead of a scale 1. If α is multiindex, then there is a constant Cα such that α Cα α jD φkj ≤ kjαj uniformly in α. Furthermore, if α 6= 0 then the support of D φ is contained in the region k ≤ jxj ≤ 2k With this appropriately rescaled function in mind we have the following lemma. N α 1 Lemma 1.16. If f is C ;D f 2 L , for all α with jαj ≤ N and if we let fk = φkf α α then limk!1 kD fk − D fk1 = 0 for all α with jαj ≤ N Proof. It is immediately seen that α α lim kφkD f − D fk1 = 0 k!1 and so in this case it is sufficient to show that α α (1.17) lim kD (φkf) − φkD fk1 = 0 k!1 By an application of the Leibniz rule we have α α X α−β β D (φkf) − φkD f = cβD fD φk 0<β≤α Where the various cβ's are constants. Hence we have α α X β α−β kD (φkf) − φkD fk1 ≤ C kD φkk1kD fkL1(fx:jx|≥kg) 0<β≤α −1 X α−β ≤ Ck kD kL1(fx:jx|≥kg) 0<β≤α The last line goes to zero since the L1 norms are taken only over the region jxj ≥ k TOOLS FROM HARMONIC ANALYSIS 5 We are now in a position to give a proof of Theorem 1.10 Proof. If f is C1 with compact support, then by integration by parts we have Z Z @f −2πix·ξ −2πix·ξ (x)e dx = 2πiξj e f(x)dx @xj And so (1.11) holds when jαj = 1. We then use induction to prove (1.11) for all α. To finish the proof, we need now remove the compact support assumption. Let fk be as in Lemma 1.12. Then (1.11) holds for fk. We now let k ! 1 so that we may α pass from our series of functions fk to our desired function f. On one hand, D\fk α ^ converges uniformly to D f as k ! 1 by Lemma 1.13. On the other hand, fk ^ α ^ α ^ converges uniformly to f and so (2πiξ) fk converges to (2πiξ) f pointwise. This proves (1.11) To prove (1.12) observe that (1.11) together with the boundedness of the fourier transform implies that ξαf^ 2 L1 if jαj ≤ N. By the appropriate estimation utilizing the fact that −1 N X α N CN (1 + jξj) ≤ jξ ≤ CN (1 + jξj) α≤N and so (1.12) follows. 2. Plancherel's Theorem and the Fourier Inversion Formula The aim of this section will be to develop both the Plancherel theorem and the Fourier inversion formula. The Plancherel theorem will ultimately allow us to relate the L2 norm of a function to the L2 norm of its Fourier Transform. This ability to compare norms will become essential when discussing the uncertainty principle. Before we get to the main results of this section we first take care of a couple of essential definitions. α β Definition 2.1. Let kfk = sup n jx D f(x)j where α and β are multi- α,β x2R indices. We define the Schwartz space S(Rn) to be the space of C1 functions that decay faster than any polynomial. To be precise, n 1 n (2.2) S(R ) = ff 2 C (R )j kfkα,β ≤ 1 8α; βg As we will soon see, the Schwartz space has many useful properties.
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