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On Fourier Transforms of Radial Functions and Distributions

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On Fourier Transforms of Radial Functions and Distributions ON FOURIER TRANSFORMS OF RADIAL FUNCTIONS AND DISTRIBUTIONS LOUKAS GRAFAKOS AND GERALD TESCHL Abstract. We find a formula that relates the Fourier transform of a radial function on Rn with the Fourier transform of the same function defined on Rn+2. This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t 7! f(jtj) and the two-dimensional function (x1; x2) 7! f(j(x1; x2)j). We prove analogous results for radial tempered distributions. 1. Introduction The Fourier transform of a function Φ in L1(Rn) is defined by the con- vergent integral Z −2πix·ξ Fn(Φ)(ξ) = Φ(x)e dx : Rn If the function Φ is radial, i.e., Φ(x) = '(jxj) for some function ' on the line, then its Fourier transform is also radial and we use the notation Fn(Φ)(ξ) = Fn(')(r) ; where r = jξj. In this article, we will show that there is a relationship be- tween Fn(')(r) and Fn+2(')(r) as functions of the positive real variable r. We have the following result. Theorem 1.1. Let n ≥ 1. Suppose that f is a function on the real line such that the functions f(j · j) are in L1(Rn+2) and also in L1(Rn). Then we have 1 1 d (1) F (f)(r) = − F (f)(r) r > 0 : n+2 2π r dr n Moreover, the following formula is valid for all even Schwartz functions ' on the real line: 1 1 d (2) F (')(r) = F s−n+1 ('(s)sn) (r); r > 0 : n+2 2π r2 n ds 1991 Mathematics Subject Classification. Primary 42B10, 42A10; Secondary 42B37. Key words and phrases. Radial Fourier transform, Hankel transform. Grafakos' research was supported by the NSF (USA) under grant DMS 0900946. Teschl's work was supported by the Austrian Science Fund (FWF) under Grant No. Y330. 1 2 L. GRAFAKOS AND G. TESCHL Using the fact that the Fourier transform is a unitary operator on L2(Rn) we may extend (1) to the case where the functions f(j·j) are in L2(Rn+2) and in L2(Rn). Moreover, in Section 4 we extend (1) to tempered distributions. Applications are given in the last section. Corollary 1.2. Let f(r) be a function on [0; 1) and k some positive integer such the functions x ! f(jxj) are absolutely integrable over Rn for all n with 1 ≤ n ≤ 2k + 2. Then we have k ` 1 X (−1)`(2k − ` − 1)! 1 d F (f)(ρ) = F (f)(ρ) 2k+1 (2π)k 2k−`(k − `)! (` − 1)! ρ2k−` dρ 1 `=1 and k ` 1 X (−1)`(2k − ` − 1)! 1 d F (f)(ρ) = F (f)(ρ): 2k+2 (2π)k 2k−`(k − `)! (` − 1)! ρ2k−` dρ 2 `=1 The Corollary can be obtained using (1) by induction on k. The simple details are omitted. Again, absolute integrability can be replaced by square integrability. 2. The proof The Fourier transform of an integrable radial function f(jxj) on Rn is given by Z 1 n s 2 −1 Fn(f)(jξj) =2π f(s) J n (2πsjξj) s ds 2 −1 0 jξj Z 1 n n−1 =(2π) 2 f(s)J n (2πsjξj) s ds ; e2 −1 0 −ν where Jeν(x) = x Jν(x), and Jν is the classical Bessel function of order ν. This formula can be found in many textbooks, and we refer to, e.g., [3, Sect. B.5] or [10, Sect. IV.1] for a proof. Moreover, this formula makes sense for all integers n ≥ 1, even n = 1, in which case r 2 cos t J (t) = p : −1=2 π t Let us set Z 1 n n−1 H n (f)(r) = (2π) 2 f(s)J n (2πsr) s ds : 2 −1 e2 −1 0 Then we make use of B.2.(1) in [3], i.e., the identity d (3) Jeν(r) = −rJeν+1(r); dr which is also valid when ν = −1=2, since r 2 sin t J (t) = p : 1=2 π t ON RADIAL FOURIER TRANSFORMS 3 In view of (3), it is straightforward to verify that 1 d n n − H −1(f)(r) = 2πH (f)(r) = 2πH n+2 −1(f)(r) ; r dr 2 2 2 provided f is such that interchanging differentiation with the integral defin- ing H n is permissible. For this to happen, we need to have that 2 −1 Z 1 d n−1 jf(s)j J n −1(rs) s ds < 1 e2 0 dr and thus it will be sufficient to have Z 1 Z 1 2 2 n−1 rs n−1 (4) jf(s)j rs jJ n (rs)js ds ≤ c jf(s)j s ds < 1 e2 n+1 0 0 (1 + rs) 2 −n=2−1=2 1 n+2 since jJ n (s)j ≤ c(1 + s) . But since f(j · j) is in L (R ) we have e2 Z 1=r Z 1 n+1 n+1 (5) jf(s)js ds + jf(s)js 2 ds < 1 0 1=r and this certainly implies (4) for all r > 0. We conclude (1) whenever (5) holds. We note that the appearance of condition (5) is natural as indicated in [8] (Lemma 25.1). To prove (2) we argue as follows. We have Z 1 −n+1 d n n d n H n r ('(r)r ) (r) = (2π) 2 '(s)s J n (2πsr) ds 2 −1 e2 −1 dr 0 ds and integrating by parts the preceding expression becomes 1 n Z 2 +2 n 2 (2π) '(s)s sr Jen+2 −1(2πsr) ds 0 2 2 which is equal to 2πr H n+2 (')(r). This proves (2). 2 −1 Remark 2.1. Note that we have 2π H (f)(r) = H (f(s)sν)(2πr); ν rν ν where Z 1 1 Hν(f)(r) = f(s)Jν(rs)s ds; ν ≥ − ; 0 2 is the Hankel transform. This of course ties in with the fact that the Hankel transform also arises naturally as the spectral transformation associated with the radial part of the Laplacian −∆; we refer to [4, Sect. 5] and the refer- ences therein for further information. Moreover, note that [6] contains the associated recursion from Theorem 1.1 for the Hankel transform, but only for even Schwartz functions. This recursion was rediscovered in connection with the radial Fourier transform in [9] for the case of Schwartz functions. See also [5] for related results. A transference theorem for radial multipliers which exploits the connection between the Fourier transform of radial functions on Rn and Rn+2 was obtained in [1]. This multiplier theorem is based on an identity dual to (3). 4 L. GRAFAKOS AND G. TESCHL 3. Radial distributions We denote by S(Rn) the space of Schwartz functions on Rn and by S0(Rn) the space of tempered distributions on Rn. A Schwartz function is called radial if for all orthogonal transformations A 2 O(n) (that is, for all rotations on Rn) we have ' = ' ◦ A: n We denote the set of all radial Schwartz functions by Srad(R ). For further background on radial distributions we refer to Treves [13, Lect. 5]. Observe that in the one-dimensional case the radial Schwartz functions are precisely the even Schwartz functions, that is: Srad(R) = Seven(R) = f' 2 S(R): '(x) = '(−x)g: Similarly, a distribution u 2 S0(Rn) is called radial if for all orthogonal transformations A 2 O(n) we have u = u ◦ A: This means that h u; ' i = h u; ' ◦ A i n 0 n for all Schwartz functions ' on R . We denote by Srad(R ) the space of all radial tempered distributions on Rn. We also denote by Sn−1 the (n − 1)- n dimensional unit sphere on R and by !n−1 its surface area. Given a general, non necessarily radial, Schwartz function there is a nat- ural homomorphism Z n o 1 S(R ) !Srad(R);'(x) 7! ' (r) = '(rθ) dθ !n−1 Sn−1 o 1 with the understanding that when n = 1, then ' (x) = 2 ('(x) + '(−x)). Conversely, given an even Schwartz function on R we can define a corre- sponding radial Schwartz function via n O Srad(R) !Srad(R );'(r) 7! ' (x) = '(jxj): The map ' 7! 'O is a homomorphism; the proof of this fact is omitted since a stronger statement is proved at the end of this section. Both facts require the following lemma: Lemma 3.1. Suppose that f is a smooth even function on R. Then there is a smooth function g on the real line such that f(x) = g(x2) for all x 2 R. Moreover, one has for t ≥ 0 (6) jg(k)(t)j ≤ C(k) sup jf (2k)(s)j: p 0≤s≤ t ON RADIAL FOURIER TRANSFORMS 5 Proof. By Whitney's theorem [14], there is a smooth function g on the real line such that f(t) = g(t2) for all real t. To see the last assertion we use the following representation of the re- mainder in Taylor's theorem: g(k)(t2) 2k Z t f (2k)(s) = (2t)−2k+1k (t2 − s2)k−1 ds k! k 0 (2k)! 2k Z 1 f (2k)(st) = 2−2kk (1 − s2)k−1 ds k 0 (2k)! from which one easily derives (6). This yields in particular that g(k)(0) f (2k)(0) = k! (2k)! since 2k Z 1 2kΓ(k)Γ(1=2) 2−2kk (1 − s2)k−1ds = 2−2kk = 1: k 0 k Γ(k + 1=2) The composition ' 7! ('o)O = 'rad gives rise to a homomorphism from n n S(R ) !Srad(R ) which reduces to the identity map on radial Schwarz functions.
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