On the Fourier transform of rotationally invariant distributions

We present an extension of the Poisson–Bochner formula for the Fourier transform of rotationally invariant distributions by analytic continuation “with respect to the dimension”. As application of this extension, a new derivation of the fundamental solution of the Euler– Poisson–Darboux operator is given.


Introduction and notation
If f ∈ L 1 (R n ) is rotationally invariant, i.e., if f (x) = g(|x|) with g(ρ)ρ n−1 ∈ L 1 ((0, ∞)), then the classical Poisson-Bochner formula expresses the Fourier transform F f ∈ C 0 (R n ) of f by the integral A generalization for functions in weighted L 1 -spaces, i.e., for g ∈ L 1 loc ((0, ∞)) fulfilling ∞ 0 |g(ρ)|ρ n−1 (1 + ρ) ( for φ ∈ D(R n \{0}).) A further generalization by means of partial integration can be found in [24,25], see also [31, Ex. 1.6.13 (a), p. 102]. A limit representation of the Fourier transform of radial temperate distributions is given in [26, (2.109), p. 140]. Let us mention that, e.g., the forward fundamental solution E of the wave operator ∂ 2 t − n is given by see Examples 2.2, 2.4 below. In this case, for t > 0, g(ρ) = (2π) −n ρ −1 sin(tρ) and f (x) = g(|x|) is not integrable nor does g satisfy condition (1.2). In order to calculate this important Fourier transform, different approximation methods were used, compare [13, pp. 177-183], [38, p. 51 The main purpose of this paper consists in generalizing formula (1.1) so as to yield a representation of the Fourier transform F S for arbitrary radially symmetric temperate distributions S. This is done by analytic continuation with respect to the index λ = n 2 − 1 of the Bessel function in (1.1), see Theorems 2.1, 2.3. So in a way, we use "analytic continuation with respect to the dimension n" of the underlying space R n . Heuristically, this procedure goes back, at least, to A. Weinstein, comp. [39, p. 44]: "The viewpoint of spaces of 'fractional dimensions' due to Weinstein is very fruitful and led to fundamental solutions in the large of the iterated EPD-equation." In [11, p. 8], the Bochner transform T n is defined by for suitable functions ϕ and n ∈ N. Whereas in [16] the connection between T n and T n+2 is rederived, see [33, (25.14 ), Lemma 25.1 , p. 359; Engl. transl. p. 486] and [32, p. 270], and in [12,27], the general connection between T n and T n+q , q ∈ N, is investigated, the present study is concerned with the analytic continuation of the function λ → T λ for complex λ.
In order to illustrate our method, we first apply it to the wave equation (Examples 2.2, 2.4) and then, in Sect. 3, to the Euler-Poisson-Darboux equation. In Propositions 3.2, 3.3, we derive in this way the fundamental solution E of the EPD-operator. (For the concept of fundamental solutions of linear partial differential operators with non-constant coefficients, see [36, pp. 138-142], [23, p. 29], [9, pp. 11-14].) This more complicated fundamental solution was given in [10] and verified therein by series expansion, see [3,4] for a recapitulation. Our deduction of E based on the analytic continuation of the Poisson-Bochner formula is different from that in [3,4,10] and seems to be new, comp. [2, p. 478]: "We do not know how to obtain an explicit formula (or formulas) for the inverse Fourier transform ofF(ξ, y; b) when b = 0, a problem that merits to be investigated." Let us introduce some notation. We employ the standard notation for the distribution spaces D , S , E , the dual spaces of the spaces D, S, E of "test functions", of "rapidly decreasing functions" and of C ∞ functions, respectively, see [18,20,36]. In order to display the active variable in a distribution, say x ∈ R n , we use notation as T (x) or T ∈ D (R n x ). Furthermore, we use the spaces D L p , For the evaluation of a distribution T ∈ E on a test function φ ∈ E, we use angle brackets, i.e., φ, T or, more precisely E φ, T E . More generally, if φ ∈ E⊗ F and T ∈ E for distribution spaces E, F, then E⊗ F φ, T E symbolizes the vector-valued scalar product (E⊗ F) × E → F, see [34,35] for more information on vector-valued distributions. (In all tensor products of this study, both factors are complete and at least one of the factors is nuclear and hence E⊗ π F = E⊗ F and we simply write E⊗ F.) The Heaviside function is denoted by Y , see [36, p. 36], and we set The function μ → χ μ can be analytically continued in S (R 1 ) and thus yields an entire function see [18, (3.2.17), p. 73]. We write δ τ (t) ∈ D (R 1 t ), τ ∈ R, for the delta distribution with support in τ, which is the derivative of Y (t − τ ), i.e., φ, δ τ = φ(τ ) for φ ∈ D(R 1 ). In contrast, δ ∈ D (R n ) without any subscript stands for the delta distribution at the origin.
The pull-back We use the Fourier transform F in the form this being extended to S by continuity. We write |S n−1 | for the hypersurface area 2π n/2 / ( n 2 ) of the unit sphere S n−1 in R n . For j ∈ N and w ∈ C, we use Pochhammer's symbol (w) 0 = 1, (w) j = w · (w + 1) · · · · · (w + j − 1). J λ and N λ denote, as usual, the Bessel functions of the first and of the second kind.

Analytic continuation of the Poisson-Bochner formula
Let us first rewrite (1.1) in a more symmetrical fashion by the following n-dimensional integral, still under the assumptions that f ∈ L 1 (R n ) and f is radially symmetric: We note incidentally that formula (2.1) allows a generalization (which follows, e.g., by density) for S ∈ D L 1 (R n ) ∩ S r (R n ), i.e., for radially symmetric integrable distributions S. Then F S is a continuous function given by belongs to the completed tensor product S(R n x )⊗ S (R n ξ ), and therefore the Fourier transform of S ∈ S r (R n ) can be written in the form (by applying [35,Prop. 4,p. 41]). Note that formula (2.3) allows to represent F x (|x| −1 sin(t|x|)) by the S -valued scalar product .
However, formula (2.4) cannot be evaluated for fixed ξ. In the following two theorems, we shall therefore imbed the kernel K (x, ξ) into an analytic family of kernels K λ (x, ξ) such that F S, S ∈ S r (R n ), can be obtained by analytic continuation with respect to λ. Let us mention that depends holomorphically on λ ∈ C (see below), but that these kernels do not belong to . This is the reason for the more complicated choices of K λ below.
Proof (a) Let us first show that the mapping is entire. From [15, 8.411.8] and using analytic continuation, we obtain the representation [34,Prop. 28,p. 98], and since is entire and E ⊂ O M , we conclude that also the mapping in (2.6) is entire, see [35,Prop. 4,p. 41].
(b) The distribution-valued function can analytically be continued to C\(−N) and has simple poles for λ = −k, k ∈ N, with the residues vanishes of order 2k at ξ = 0 and hence its product with Res λ=−k F λ (ξ ) vanishes.
is also well-defined and depends holomorphically on λ in S (R n ξ ). By analytic continuation, we conclude that F ξ K λ is represented by the continuous function R n x → S (R n ξ ) which, for x = 0, is given by the equation (2.7) If n = 2 or n = 1, then the same conclusion can be reached by proving (2.7) for Re λ < − 3 2 with the help of formula (2.2). (Note that K λ (x, ξ) ∈ D L 1 (R n ξ ) for Re λ < − 3 2 and fixed x ∈ R n \{0}.) Hence (2.7) is valid for n ∈ N, x = 0 and each λ ∈ C.
Because the support of the distribution |η| 2−n χ n/2−λ−2 (|η| 2 − 1) does not contain the origin η = 0, we conclude that ψ is, with all its derivatives, rapidly decreasing for |x| → ∞ and hence ψ ∈ S(R n ). This means that F ξ K λ and thus also K λ belong to S(R n and represent E t by partial Fourier transform, i.e., (a) The distribution-valued function U λ in (2.5) corresponding to S = sin(t|x|)/|x|, t > 0 fixed, is given by If Re λ > n − 1, then is a continuous function with values in L 1 (R n x ). Under this assumption on λ, we therefore obtain that U λ ∈ C(R n ξ ) is given by The distributions χ −1/2+λ (|ξ | 2 − t 2 ) ∈ S (R n ξ ) depend C ∞ on t > 0 and hence the last formula holds by analytic continuation for each λ ∈ C and t > 0. This implies in accordance with [31, Lemma 3.3.5, p. 218 (for k = 1)].
(c) For even n and Re λ > n − 1 we obtain the following from (2.8): The integral in (2.9) is absolutely convergent for Re λ > 1 and yields a continuous function of t and ξ depending analytically on λ. However, this integral is more complicated than the one in the case of odd n (see [ sin(t|x|)/|x| and assuming n even with n ≥ 6 we can insert λ = n 2 − 1 into (2.9), and we obtain by analytic continuation (Note that J −k (s) = (−1) k J k (s) for k ∈ N and s ∈ R. Let us also mention that the last formula can be deduced as well for n = 2 or n = 4 upon using a further differentiation with respect to t.) In order to evaluate the last integral, let us assume ξ = 0 and consider the analytic distribution-valued function By means of the series expansion of the Bessel function, we infer that T can be analytically continued to C\(− 1 2 − N 0 ) having simple poles in − 1 2 − N 0 (see also [22, 2.4, p. 193]). Furthermore, is also well-defined and analytic. A classical formula (see [29, II, 13.9, p. 164]) furnishes By analytic continuation we deduce from (2.10) and this yields exactly as in the case of odd n. Let us mention that a unified deduction of (2.11) independent of the parity of n is given in [31, Lemma 3.3.5, p. 218].

Theorem 2.3
Let K λ be defined as in Theorem 2.1 and assume λ ∈ C\(−N). Then the kernel and it depends therein holomorphically on λ ∈ C\(−N). If S ∈ S r (R n ) and S is C ∞ in a neighborhood of 0, then S(x) ·K λ (x, ξ) belongs to D L 1 (R n x )⊗ S (R n ξ ) and depends therein holomorphically on λ ∈ C\(−N). Finally, is holomorphic.
Therefore,Ũ λ (ξ ) ∈ L 1 loc (R n ξ ) for −1 < Re λ < − 1 2 , andŨ λ (ξ ) is given, for ξ = 0, by the absolutely convergent integral that we have encountered already in (2.10). By analytic continuation, we thus obtaiñ Hence we deduce from Theorem 2.3 the following expression for the forward fundamental solution E of ∂ 2 t − n : (2.13) (As said above, we interpret E as a continuous function of t with values in S (R n x ) and vanishing for t ≤ 0. Furthermore, for t > 0, the composition h * T of T = χ ( (1.4).) The representation of E in (2.13) was given already in [

Remark 2.5
Let us eventually observe that we could also employ the kernel for the analytic continuation of the Poisson-Bochner formula, yet only for a restricted class of distributions S. In fact, by partial Fourier transformation, it follows that Hence, for S ∈ D L 1 ,−n (R n x ) = (1 + |x| 2 ) n/2 · D L 1 (R n x ), the function (2.14) is entire and F S = 2 n/2−1 ( n 2 ) · U 0 n/2−1 if S is rotationally invariant. If S = sin(t|x|)/|x|, t > 0 fixed, then the assumption S ∈ D L 1 ,−n (R n ) is satisfied and U 0 λ in (2.14) would yield the same representation of F S as in Example 2.2. If, in contrast, S = 1, then S / ∈ D L 1 ,−n (R n ) and the entire distribution-valued function U 0 in (2.14) does not exist. Note, however, that U andŨ in (2.5) and in (2.12), respectively, remain meaningful and yield and F 1 = 2 n/2−1 ( n 2 )U n/2−1 = 2 n/2−1 ( n 2 )Ũ n/2−1 = (2π) n δ as expected.

The fundamental solution of the Euler-Poisson-Darboux operator
Let us turn now to the Euler-Poisson-Darboux operator acting on the space of distributions defined in the right half-space is strictly hyperbolic with respect to t, it has a unique fundamental solution Moreover, the strict hyperbolicity of P α (t, ∂ t , ∂ x ) implies that E α,τ depends C ∞ on t for t ≥ τ and that the support of E α,τ is contained in the propagation cone {(t, x) ∈ R n+1 ; t ≥ τ + |x|}. In particular, E α,τ ∈ C ∞ ([τ, ∞))⊗ E (R n ) and the partial Fourier transform S α,τ of E α,τ with respect to x fulfills i.e., S α,τ is an infinitely differentiable mapping from [τ, ∞) into O M (R n ). By constructing the Green function of the ordinary differential operator ∂ 2 t + (2α + 1)t −1 ∂ t + |x| 2 , we next derive an explicit representation of S α,τ . Proposition 3.1 For τ > 0 and α ∈ C, we have [Here N α , α ∈ C, denote the Bessel functions of the second kind.] Proof Upon Fourier transform with respect to x, (3.2) yields This ordinary differential equation arises by substitution from Bessel's equation, and the vector space of its homogeneous solutions is generated by t −α J α (t|x|) and t −α N α (t|x|), see [ then the constants C 1 , C 2 are determined by the following system of linear equations: The Wronskian determinant [21, A, 17.1, p. 72], and the power series of J α and N α yield C = 2 π . Hence and this furnishes formula (3.3).
In order to evaluate the Fourier transform Our deduction of E α,τ is different and seems to be new.
Let us mention that an earlier appearance of this fundamental solution in the form of a "Riemann function" can be found in [40, p. 361, last line]. In fact, (Note that H n (α+2) is defined erroneously in [40, p. 357 , the unique solution of (3.2), is given by (3.5) [As said above, we interpret E α,τ as a continuous function of t with values in E (R n x ) and vanishing for t ≤ τ. Furthermore, for t > τ, the composition h * T of T = χ (1− (1.4), and so is the multiplication with the C ∞ function given by 2 Proof (a) In order to apply Theorem 2.3, let us first check that S α,τ (t, x) in (3.5) is a C ∞ function of x in a neighborhood of 0. In fact, for α ∈ C\Z, we have then we see, analogously as in Example 2.4, which is the special case of α = − 1 2 , thatŨ λ (ξ ) ∈ L 1 loc (R n ) and that the evaluation in (3.6) furnishes an absolutely convergent integral for ξ = 0. Hence we obtain the following for −1 < Re λ < − 1 2 , 0 < τ < t and ξ = 0 fixed: According to [28, 10.