MEHLER-FOCK TRANSFORMS OF GENERALIZED FUNCTIONS VIA ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3243–3253 S 0002-9939(97)03957-9

MEHLER-FOCK TRANSFORMS OF GENERALIZED FUNCTIONS VIA THE METHOD OF ADJOINTS ´ BENITO J. GONZALEZ AND EMILIO R. NEGRIN (Communicated by Palle E. T. Jorgensen)

Abstract. In this paper we analyse the Mehler-Fock transform of generalized functions via the method of adjoints. For a distribution of compact support, we prove that its Mehler-Fock transform agrees with its transform via the kernel method. A Paley-Wiener type theorem is established.

1. Introduction The Mehler-Fock transform of a distribution f of compact support on I (from now on I = (0, ∞)) was defined by the kernel method in [10] as: (1.1)

F (τ ) = hf (x), P− 12 +iτ (cosh x)i,

τ > 0,

where, as it is usual, P− 12 +iτ is the Legendre function of first kind and order zero. Related studies on Mehler-Fock transforms of generalized functions have been carried out in [3], [4], [5], [8], amongst others. In this paper, we study the Mehler-Fock transform by using the method of adjoints. For this purpose, it will be required to introduce two Fr´echet spaces V and W satisfying the condition that the linear operator L : W −→ V given by Z ∞ (1.2) P− 12 +iτ (cosh x)ψ(τ )dτ, x > 0, (Lψ) (x) = 0

be continuous. Specifically, we denote by V the vector space consisting of all complex-valued functions φ which possess derivatives of all orders on I and such that, for all k ∈ N ∪ {0}, γk (φ) = sup (sinh x)Akx φ(x) < ∞, x∈I

where Ax is the differential operator Ax ≡ Dx2 + (coth x)Dx ≡ (sinh x)−1 Dx (sinh x)Dx , and Dx denotes derivative with respect to the x-variable. The topology of V is that generated by the family of seminorms {γk }k∈N∪{0} . Received by the editors January 2, 1996 and, in revised form, April 24, 1996. 1991 Mathematics Subject Classification. Primary 44A15, 46F12. Key words and phrases. Mehler-Fock transform, Legendre function, generalized functions, distributions of compact support, Paley-Wiener type theorem. c

1997 American Mathematical Society

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License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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´ BENITO J. GONZALEZ AND EMILIO R. NEGRIN

Moreover, W is the vector space of all complex-valued even functions ψ on R such that r 1 + τ 2 cosh(πτ )ψ(τ ) ∈ L1 (R) τ −1 4 and r 3 x −1 1 2 2 +τ cosh(πτ )ψ(τ ) (x) < ∞, ρr (ψ) = sup x e F τ 4 x∈I for all r ∈ N ∪ {0}, and F denotes Fourier transform with respect to the τ -variable. The space W is equipped with the topology generated by the family of seminorms {ρr }r∈N∪{0} . As usual V 0 and W 0 denote the dual spaces of V and W, respectively. The Mehler-Fock transform of the generalized function f ∈ V 0 is given by L0 f, where L0 is the adjoint of L. Here, we establish the link between these two methods. It will be proved that both definitions agree for distributions of compact support on I. We also establish a Paley-Wiener type theorem. It is proved that every smooth function F such that |F (τ )| ≤ C · (1 + τ 2r ), for all τ > 0, some nonnegative integer r and some constant C, is the Mehler-Fock transform of a generalized function f ∈ V 0. In this article we extend this analysis to the so-called Mehler-Fock transform of order n ∈ N. The corresponding results for the Kontorovich-Lebedev transform were carried out in [7]. 2. The operator L and its adjoint The next result is the key to analysing the Mehler-Fock transform of generalized functions via the method of adjoints. Theorem 2.1. The operator L, given by (1.2), is a continuous linear mapping from W in V . Proof. First observe that for all τ > 0, Ax P− 12 +iτ (cosh x) = −

1 2 + τ P− 12 +iτ (cosh x). 4

So, for any ψ ∈ W and all k ∈ N ∪ {0}, we have k Z ∞ 1 + τ 2 P− 12 +iτ (cosh x)ψ(τ )dτ. Akx (Lψ) (x) = (−1)k (2.1) 4 0 Using the integral representation (cf. [1, (11), p. 156]), which is valid for all x > 0 and all τ > 0, (2.2) √ Z ∞ 2 −1 P− 12 +iτ (cosh x) = cosh(πτ ) (cosh x + cosh ξ) 2 cos(τ ξ)dξ π 0 and integrating (2.2) by parts, one obtains Z cosh(πτ ) ∞ −3 P− 12 +iτ (cosh x) = √ (cosh x + cosh ξ) 2 (sin(τ ξ))(sinh ξ)dξ. 2πτ 0


MEHLER-FOCK TRANSFORMS OF GENERALIZED FUNCTIONS

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Substituting this expression in (2.1) and reversing the order of integration, it follows that Z 3 (−1)k ∞ (2.3) Akx (Lψ)(x) = √ (cosh x + cosh ξ)− 2 (sinh ξ) 2π 0 ! k Z ∞ 1 −1 2 +τ τ cosh(πτ ) ψ(τ ) sin(τ ξ)dτ dξ. × 4 0 Note that the integral into the brackets is the Fourier sine transform of an odd function of the τ -variable, and consequently it is the Fourier transform of this function. Therefore, Z (−1)k ∞ −3 k (2.4) Ax (Lψ)(x) = √ (cosh x + cosh ξ) 2 (sinh ξ) 2π 0 #! " k 1 −1 2 +τ ψ(τ ) (ξ) dξ. × F τ cosh(πτ ) 4 Thus, for all x > 0 we have − 32 Z ∞ cosh ξ (sinh x)Ak (Lψ)(x) = √ sinh x 3 1+ (sinh ξ) x 2π(cosh x) 2 0 cosh x #! " k 1 −1 2 +τ ψ(τ ) (ξ) dξ × F τ cosh(πτ ) 4 which is less than or equal to Z ∞ 1 3 3 √ e−ξ ξ − 2 (sinh ξ) eξ ξ 2 2π 0

"

1 + τ2 F τ −1 cosh(πτ ) 4

k

#! ψ(τ )

(ξ) dξ

≤ M ρk (ψ), where M is a suitable real constant. Thus, for all k ∈ N ∪ {0}, γk (Lψ) ≤ M ρk (ψ). This concludes the continuity of L.

Now, the adjoint of the operator L, which is given by hL0 f, ψi = hf, Lψi, f ∈ V 0 , ψ ∈ W, is a continuous linear mapping from V 0 into W 0 when we provide V 0 and W 0 with their weak∗ -topologies. The Mehler-Fock transform of the generalized function f is the member of W 0 given by L0 f . 3. The link between both methods The next two lemmas allow us to establish the link between the two definitions of Mehler-Fock transforms of generalized functions. For it, set Z ∞ ψφ (τ ) = τ tanh(πτ ) (3.1) (sinh x)P− 12 +iτ (cosh x)φ(x)dx, 0

for any φ ∈ V and all τ > 0.



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Lemma 3.1. For all φ ∈ V the function ψφ given by (3.1) satisfies i) ψφ (τ ) = O(τ 2 ), as τ → 0; ii) f or all p ∈ N, ψφ (τ ) = O(τ −p ), as τ → ∞. Proof. i) Taking into account that tanh(πτ ) = O(τ ) for τ → 0, it follows that for τ small enough, Z ∞ 2 |ψφ (τ )| ≤ C1 τ (sinh x)P− 12 +iτ (cosh x)φ(x) dx ≤ C2 τ 2 , 0

for some suitable real constants C1 and C2 . ii) If A0x denotes the adjoint of Ax , namely, A0x = Dx (sinh x)Dx (sinh x)−1 , one has h i 1 0 2 + τ (sinh x)P− 12 +iτ (cosh x), Ax (sinh x)P− 12 +iτ (cosh x) = − 4 and so, for all k ∈ N ∪ {0}, (3.2) (−1)k

1 + τ2 4

k

Z ψφ (τ ) = τ tanh(πτ )

∞ 0

k

(A0x )

h i (sinh x)P− 12 +iτ (cosh x) φ(x)dx.

Note that the integral in the right-hand side of (3.2) is equal to Z ∞ (sinh x)P− 12 +iτ (cosh x)Akx φ(x)dx. 0

In fact, integrating by parts, Z ∞ h i A0x (sinh x)P− 12 +iτ (cosh x) φ(x)dx 0 ∞ = (sinh x) Dx P− 12 +iτ (cosh x) φ(x) 0 Z ∞ − (sinh x) Dx P− 12 +iτ (cosh x) (Dx φ(x))dx 0 ∞ = (sinh x) Dx P− 12 +iτ (cosh x) φ(x) 0∞ − (sinh x)P− 12 +iτ (cosh x)(Dx φ(x)) 0 Z ∞ + (sinh x)P− 12 +iτ (cosh x)Ax φ(x)dx. 0

Observe that the remainder terms of the above integration by parts tend to zero as x → 0 and x → ∞. Indeed, since φ ∈ V, it follows that (sinh x)φ(x) is bounded on I, and Dx P− 12 +iτ (cosh x) −→ 0, as x → 0 and x → ∞. Hence, (sinh x)(Dx P− 12 +iτ (cosh x))φ(x) −→ 0, as x → 0 and x → ∞. Also, since for all φ ∈ V, the function (sinh x)Dx φ(x) is bounded on I, and taking x into account that P− 12 +iτ (cosh x) = O(e− 2 ) as x → ∞, it follows that (sinh x)P− 12 +iτ (cosh x)Dx φ(x) −→ 0, as x → ∞.



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On the other hand, for any φ ∈ V one has (sinh x)Dx φ(x) = O(x), as x → 0. Thus, (sinh x)P− 12 +iτ (cosh x)Dx φ(x) −→ 0, as x → 0. This proves that Z ∞ 1 2 + τ ψφ (τ ) = τ tanh(πτ ) (sinh x)P− 12 +iτ (cosh x)Ax φ(x)dx. − 4 0 Now, for all k ∈ N, k > 1, one obtains Z ∞ h i k (A0x ) (sinh x)P− 12 +iτ (cosh x) φ(x)dx 0

= (−1)k−1

= (−1)

k−1

1 + τ2 4

k−1 Z

1 + τ2 4

∞

0

k−1 Z

h i A0x (sinh x)P− 12 +iτ (cosh x) φ(x)dx

∞ 0

(sinh x)P− 12 +iτ (cosh x)Ax φ(x)dx.

Iterating this process one arrives at k Z ∞ k 1 + τ 2 ψφ (τ ) = τ tanh(πτ ) (−1) (sinh x)P− 12 +iτ (cosh x)Akx φ(x)dx. 4 0 Taking into account that φ ∈ V and the asymptotic behavior of P− 12 +iτ (cosh x) as x → 0 and x → ∞, one concludes that k 1 + τ 2 |ψφ (τ )| ≤ M τ, ∀τ > 0, 4 for some suitable nonnegative constant M. Thus, the lemma holds.

Lemma 3.2. For all ψ ∈ W, there exists φ ∈ V such that ψ = ψφ , where ψφ is given by (3.1). Proof. Let ψ ∈ W and set φ = Lψ. Note that, for x > 0, one has Z ∞ (Lψφ )(x) = P− 12 +iτ (cosh x)ψφ (τ )dτ 0

Z

∞

= 0

P− 12 +iτ (cosh x)τ tanh(πτ )

Z

∞ 0

(sinh t)P− 12 +iτ (cosh t)φ(t)dt dτ,

and from the inversion formula of the Mehler-Fock transform, the above expression x is equal to φ(x), which holds whenever φ(x)e 2 ∈ L1 (0, ∞) (cf. [2]). Since φ ∈ V, this condition is satisfied. Thus, Lψφ = φ = Lψ. Now, we will establish that L is one to one. For it, assume that Lψ = 0. Taking k = 0 in (2.4) one has for all x > 0, Z ∞ 1 −3 (Lψ)(x) = √ (cosh x + cosh ξ) 2 (sinh ξ) F τ −1 cosh(πτ )ψ(τ ) (ξ) dξ = 0. 2π 0



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The change of variables t + 1 = cosh ξ yields Z ∞ 1 −3 0 = (Lψ)(x) = √ (cosh x + 1 + t) 2 Ψ(t)dt, ∀x > 0, 2π 0 where p Ψ(t) = F τ −1 cosh(πτ )ψ(τ ) (log (t + 1 + t2 + 2t)). Finally, since

Z

∞

− 32

(cosh x + 1 + t)

Ψ(t)dt

0

is the generalized Stieltjes transform of the function Ψ, from its inversion formula (cf. [9, p. 180]) it follows that Ψ ≡ 0. Therefore, p F τ −1 cosh(πτ )ψ(τ ) (log (t + 1 + t2 + 2t)) = 0, ∀t > 0, whence ψ ≡ 0.

Theorem 3.1. Let f ∈ E 0 (I). Then, for any ψ ∈ W, one has Z ∞ 0 (3.3) F (τ )ψ(τ )dτ, hL f, ψi = 0

where the function F (τ ) is given by (1.1). Proof. First, observe that D(I) ⊂ V ⊂ E(I) and the topology of V is stronger than the one induced on it by E(I). Thus, we may regard E 0 (I) as a vector subspace of V 0. By [6, Proposition 2, p. 97], the function F (τ ) satisfies i) For τ → 0, (3.4)

F (τ ) = O(1),

ii) there exists r ∈ N ∪ {0} such that for τ → ∞, F (τ ) = O τ 2r . (3.5) Using Lemma 3.1 and Lemma 3.2 it follows that for any ψ ∈ W one has i) ψ(τ ) = O(τ 2 ), as τ → 0; ii) for all p ∈ N, ψ(τ ) = O(τ −p ), as τ → ∞. Thus, the integral of the right-hand side of (3.3) makes sense. Clearly, if f ∈ D(I), the result of this theorem holds. In fact, assume that f has its support contained in the closed interval [a, b] ⊂ I; then, using Fubini’s theorem, one has hL0 f, ψi = hf, Lψi Z

Z

b

f (x)(Lψ)(x)dx =

= a

Z

Z

0

P− 12 +iτ (cosh x)ψ(τ )dτ dx Z

b

ψ(τ ) 0

∞

f (x) a

∞

=

Z

b

a

f (x)P− 12 +iτ (cosh x)dx dτ =

∞

F (τ )ψ(τ )dτ. 0

So, the result holds for f ∈ D(I). Now, for f ∈ E 0 (I) and using [11, Theorem 28.2(i), p. 301], there exists a sequence of functions {fm }m∈N belonging to D(I) which converges to f in E 0 (I),



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and hence, in V 0 . Furthermore, from Theorem 2.1 it follows that L0 is a continuous mapping from V 0 into W 0 . Thus {L0 fm }m∈N converges to L0 f in W 0 as m → ∞. Since, for all ψ ∈ W, Z ∞ 0 hL fm , ψi = Fm (τ )ψ(τ )dτ 0

where Fm (τ ) = hfm (x), P− 12 +iτ (cosh x)i, it follows that

Z

0

hL f, ψi = lim

(3.6)

m→∞

0

∞

Fm (τ )ψ(τ )dτ.

Clearly, from the asymptotic growth of Fm and ψ, the limit in (3.6) goes into the integral. Also, noting that lim Fm (τ ) = lim hfm (x), P− 12 +iτ (cosh x)i

m→∞

m→∞

= hf (x), P− 12 +iτ (cosh x)i = F (τ ),

the result holds. 4. A Paley-Wiener type theorem

The following theorem proves that any smooth function which satisfies conditions of type (3.4) and (3.5) is the Mehler-Fock transformation of an element of V 0 . Theorem 4.1. Let F (τ ) be a smooth function on I such that there exist a nonnegative integer r and a constant C with |F (τ )| ≤ C · 1 + τ 2r for all τ > 0. Then there exists a f ∈ V 0 such that Z +∞ 0 F (τ )ψ(τ )dτ, ∀ψ ∈ W. hL f, ψi = 0

Moreover, for all φ ∈ V, Z hf, φi = lim h T →∞

0

T

τ tanh(πτ )(sinh x)P− 12 +iτ (cosh τ )F (τ )dτ, φ(x)i.

Proof. Let G(τ ) be a smooth function on (0, ∞) such that (4.1) |G(τ )| ≤ M1 · 1 + τ −α , where α > 2 and M1 is a real constant. Let us define, for any x > 0, Z ∞ g(x) = (4.2) τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x)G(τ )dτ. 0

Note that for τ → 0, τ tanh(πτ ) = O(τ 2 ), and for τ → ∞, τ tanh(πτ ) = O(τ ). On the other hand, P− 12 +iτ (cosh x) ≤ P− 12 (cosh x) , ∀x > 0, ∀τ > 0.



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So, (4.2) exists and it is a locally integrable function on I. Moreover, Z ∞ Z ∞ Z ∞ (sinh x)−1 g(x) dx = τ tanh(πτ )P− 12 +iτ (cosh x)G(τ )dτ dx 0

0

0

Z ∞ τ | tanh(πτ )||G(τ )|dτ dx ≤ P− 12 (cosh x) 0 0 Z ∞ ≤ M2 · P− 12 (cosh x) dx < ∞, Z

∞

0

where M2 is a real constant. Therefore, the function g yields to a regular member in V 0 , denoted by Tg , which is given by Z ∞ hTg , φi = g(x)φ(x)dx, ∀φ ∈ V. 0

Also for any ψ ∈ W denoting φ = Lψ, and using Lemma 3.2, one obtains Z ∞ 0 hL Tg , ψi = hTg , Lψi = g(x)φ(x)dx Z

0

Z

∞

∞

φ(x)

= 0

0

τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x)G(τ )dτ dx,

which, by Fubini’s theorem, is equal to Z ∞ Z ∞ Z G(τ ) τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x)φ(x)dx dτ = 0

0

Thus, for all ψ ∈ W, hL0 Tg , ψi =

(4.3)

Z

∞

G(τ )ψ(τ )dτ. 0

∞

G(τ )ψ(τ )dτ. 0

Now, for all τ > 0, set G(τ ) =

1 4

F (τ ) r+p . + τ2

Clearly, for p > α/2, this function G satisfies condition (4.1). So, there exists a function g given by (4.2) such that the regular member Tg in V 0 satisfies (4.3). Set f = (−A0x )r+p Tg . Note that, for all φ ∈ V , Z ∞ r+p 0 r+p g(x)(−Ax ) φ(x)dx. hf, φi = h(−Ax ) Tg , φi = 0

Thus, for all ψ ∈ W, one obtains hL0 f, ψi = hTg , (−Ax )r+p (Lψ)i = Z

∞

= 0

* Tg , L

"

1 + τ2 4

#+

r+p

ψ(τ )

r+p Z ∞ 1 2 +τ G(τ ) ψ(τ )dτ = F (τ )ψ(τ )dτ. 4 0

On the other hand, for any φ ∈ V, Z ∞ hf, φi = h τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x)G(τ )dτ, (−Ax )r+p φ(x)i 0



Z = lim h T →∞

Z = lim h T →∞

Z = lim h T →∞

T 0

T 0

0

T

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τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x)G(τ )dτ, (−Ax )r+p φ(x)i

h i τ tanh(πτ )(−A0x )r+p (sinh x)P− 12 +iτ (cosh x) G(τ )dτ, φ(x)i r+p 1 + τ2 τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x) G(τ )dτ, φ(x)i 4

Z = lim h T →∞

0

T

τ tanh(πτ )(sinh x)P− 12 +iτ (cosh x)F (τ )dτ, φ(x)i.

5. Mehler-Fock transforms of order n Starting from the integral representation [1, (11), p. 156], which is valid for all n ∈ N ∪ {0}, all τ > 0 and all x > 0, P−n 1 +iτ (cosh x) 2 r Z ∞ 1 Γ(n + 12 )(sinh x)n 2 (cosh x + cosh ξ)−n− 2 cos(τ ξ)dξ, = π Γ(n + 12 + iτ )Γ(n + 12 − iτ ) 0 the results of the previous sections are readily extended to the so-called MehlerFock transform of order n ∈ N, the kernel of which is given by P−n 1 +iτ (cosh x), the 2 associated Legendre function of first kind and order n. Specifically, one introduces the corresponding spaces V and W ; namely, V is the vector space consisting of all complex-valued functions φ which possess derivatives of all orders on I and such that, for all k ∈ N ∪ {0}, γk (φ) = sup (sinh x)Akx φ(x) < ∞, x∈I

where Ax is the differential operator Ax ≡ (sinh x)−n−1 Dx (sinh x)2n+1 Dx (sinh x)−n , and W is the vector space of all complex-valued even functions ψ on R such that r 2 n + 12 + τ 2 ψ(τ ) ∈ L1 (R) τ Γ n + 12 + iτ Γ n + 12 − iτ and

r    1 2 2 + τ ψ(τ ) n + 3 x 2     2 ρr (ψ) = sup x e F (x) < ∞, 1 1 τ Γ n + 2 + iτ Γ n + 2 − iτ x∈I

for all r ∈ N ∪ {0}, and F denotes Fourier transform with respect to the τ -variable. The operator L : W → V, given by Z ∞ (Lψ)(x) = P−n 1 +iτ (cosh x)ψ(τ )dτ, 0

2

becomes a continuous linear mapping from W in V .


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Also, for any φ ∈ V and all τ > 0 set 1 1 1 ψφ (τ ) = τ sinh(πτ )Γ n + + iτ Γ n + − iτ π 2 2 Z ∞ (sinh x)P−n 1 +iτ (cosh x)φ(x)dx. × 2

0

It is obtained that i) ψφ (τ ) = O(τ 2n+2 ), as τ → 0; ii) for all p ∈ N, ψφ (τ ) = O(τ −p ), as τ → ∞. Also, for any ψ ∈ W there exists φ ∈ V such that ψ = ψφ . Now, for any f ∈ E 0 (I), the function F (τ ) = hf (x), P−n 1 +iτ (cosh x)i, τ > 0, 2

satisfies i) F (τ ) = O(1), as τ → 0; ii) there exists r ∈ N ∪ {0} such that F (τ ) = O(τ 2r ), as τ → ∞. Thus the corresponding result of Theorem 3.1 holds in this setting; namely, for all f ∈ E 0 (I), Z ∞ F (τ )ψ(τ )dτ. hL0 f, ψi = 0

Also the corresponding Paley-Wiener type theorem establishes that Theorem 5.1. Let F (τ ) be a smooth function on I such that there exist a nonnegative integer r and a constant C with |F (τ )| ≤ C · 1 + τ 2r for all τ > 0. Then there exists an f ∈ V 0 such that Z +∞ F (τ )ψ(τ )dτ, ∀ψ ∈ W. hL0 f, ψi = 0

Moreover, for all φ ∈ V,

Z hf, φi = lim h T →∞

where S(τ ) =

T

S(τ )G(τ, x)F (τ )dτ, φ(x)i,

0

1 1 1 τ sinh(πτ )Γ n + + iτ Γ n + − iτ π 2 2

and G(τ, x) = (sinh x) · P−−n (cosh x). 1 +iτ 2

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6. J. Horv´ ath, Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, Reading, MA, 1966. MR 34:4863 7. B. Lisena, On the Generalized Kontorovich-Lebedev Transform, Rend. Mat. Appl. 9 (VII) (1989), 87–101. MR 91c:44005 8. R.S. Pathak and R.K. Pandey, The generalized Mehler-Fock transformation of distributions, Arabian J. Sci. Engrg. 10 (1) (1985), 39–57. MR 86b:44005 9. D.B. Sumner, An inversion formula for the generalized Stieltjes transform, Bull. Amer. Math. Soc. 55 (1949), 174–183. MR 10:370f 10. U.N. Tiwari and J.N. Pandey, The Mehler-Fock transform of distributions, Rocky Mountain J. Math. 10 (1980), 401–408. MR 81m:46058 11. F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967. MR 37:726 ´ lisis Matema ´ tico, Universidad de La Laguna, 38271 Canary IsDepartamento de Ana lands, Spain E-mail address: [email protected] E-mail address: [email protected]
