Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 4 (2009), No. 1, pp. 75-96 MELLIN TRANSFORM FOR BOEHMIANS BY R. ROOPKUMAR Abstract A suitable Boehmian space is constructed to extend the distributional Mellin transform. Mellin transform of a Boehmian is defined as a quotient of analytic functions. We prove that the generalized Mellin transform has all its usual properties. We also discuss the relation between the Mellin transform and the Laplace transform in the context of Boehmians. 1. Introduction J. Mikusi´nski and P. Mikusi´nski [8] introduced Boehmians as a gener- alization of distributions by the motivation of regular operators [2]. An abstract construction of Boehmian space was given in [9] with two no- tions of convergence. Thereafter various Boehmian spaces have been de- fined and also various integral transforms have been extended on them. See [6, 10, 11, 13, 16, 18, 19, 21]. The main objective of introducing an integral transform to the context of Boehmians is to find a Boehmian space which is suitable for defining the integral transform and it is properly larger than the space of distributions where the particular integral transform has been al- ready discussed. If we construct Boehmian space by using distribution space as the top space (which contains the numerator sequences) of a Boehmian space, then obviously the Boehmian space contains the distribution space. In certain cases, such Boehmian spaces are even generalizing some other Boehmian spaces. For example, the Boehmian spaces introduced in [5, 6] Received January 17, 2005. AMS Subject Classification: Primary 44A12, 44A90; Secondary 40F12. Key words and phrases: Boehmians, convolution, distributions, Radon transforms. 75 76 R. ROOPKUMAR [March are properly larger than L p-Boehmians [4] and tempered Boehmians [12] respectively. In [1, 20], Mellin transform has been discussed on tempered Boehmians and Mellin transformable C∞-Boehmians respectively. In both papers, the definitions of Mellin transform are erroneous and the Mellin transform on M ′ a,b is not at all discussed. The details are given in the last section. M ′ Since the distributional Mellin transform is defined on the space a,b, B M ′ we construct a suitable Boehmian space M which properly contains a,b. To discuss the operational properties of the Mellin transform, we also de- fine various operations on the Boehmian space such as multiplication of a Boehmian by a function of the form (log x)k, multiplication of a Boehmian by a polynomial, derivative of a Boehmian, translation of a Boehmian, change of scale of a Boehmian and change of variable of a Boehmian by the function xρ. We define the Mellin transform of a Boehmian as a quotient of analytic functions, satisfying all the expected operational properties. We also provide the identification between BM and the Laplace transformable Boehmian space BL [13] and establish the relation between the Mellin transform and the Laplace transform in the context of Boehmian spaces. Finally, we show B M ′ that M is properly larger than a,b. 2. Preliminaries Let Ma,b be the space of all smooth functions on I = (0, ∞) satisfying k+1 (k) γk(φ) = sup{ζa,b(x)x |φ (x)| : x ∈ I} < ∞, k = 0, 1, 2,... (2.1) where x−a if 0 <x ≤ 1 ζa,b(x)= ( x−b if 1 <x< ∞ and a, b are fixed with 0 <a<b< ∞. The space Ma,b is a Fr´echet space with the sequence (γk)k∈N0 of semi-norms, where N0 is the set of all M ′ M non-negative integers. The dual space a,b of a,b is equipped with the ∗ M ′ weak topology. We say that fn → f as n → ∞ in a,b if hfn(x), φ(x)i− hf(x), φ(x)i→ 0 as n →∞ for each φ ∈ Ma,b. A locally integrable function 2009] MELLIN TRANSFORM FOR BOEHMIANS 77 f on I, with f(x)/(xζa,b(x)) is integrable on I, can be identified as a member M ′ of a,b by the map ∞ φ 7→ f(x)φ(x)dx (2.2) Z0 For instance, every Schwartz testing function f ∈ D(I) can be identified as M ′ M ′ a regular member of a,b. Mellin transform of f ∈ a,b is defined by s−1 (Mf)(s)= hf(x),x i, ∀s ∈ Ωa,b (2.3) where Ωa,b is the strip {s ∈ C : a < Re s < b}. It is well known that Mf is an analytic function with a polynomial growth. See [24, p.108]. A Mellin M ′ type convolution on a,b is defined by h(f ∨ g)(x), φ(x)i = hf(x), hg(y), φ(xy)ii, ∀φ ∈ Ma,b. (2.4) M ′ This convolution is a commutative binary operation on a,b. When f, g ∈ M ′ M ′ D a,b are regular functions or f ∈ a,b and g ∈ (I), f ∨ g is also regular [24, p.118]. In deed, ∞ x 1 0 f y g(y) y dy, ∀x ∈ I if f and g are regular. (f ∨ g)(x)= (2.5) R f(y), 1 g x , ∀x ∈ I if f ∈ M ′ , g ∈ D(I). y y a,b D E Theorem 2.1. (Inversion) Let F (s) = Mf for s ∈ Ωf . Then, in the sense of convergence in D ′(I), 1 σ+ir f(x) = lim F (s)x−sds, r→∞ 2πi Zσ−ir where a < σ < b. M ′ Definition 2.2. Let f ∈ a,b, k ∈ N0, α = u + iv ∈ C, r > 0 and ρ ∈ R \ {0}. We define k M ′ M ′ (1) L : a,b → a,b by k k k h(L f)(x), φ(x)i = h(log x) f(x), φ(x)i = hf(x), (log x) φ(x)i, ∀φ ∈ Ma,b. M ′ M ′ (2) Pα : a,b → a−u,b−u by α α h(Pαf)(x), ψ(x)i = hx f(x), ψ(x)i = hf(x),x ψ(x)i, ∀ψ ∈ Ma−u,b−u. 78 R. ROOPKUMAR [March k M ′ M ′ (3) D : a,b → a+k,b+k by h(Dkf)(x),χ(x)i = hf (k)(x),χ(x)i k k M = (−1) hf(x), (Dxχ)(x)i, ∀χ ∈ a+k,b+k. k M ′ M ′ (4) (DP1) : a,b → a,b by k k h((DP1) f)(x), φ(x)i = h(Dxx) f(x), φ(x)i k k = (−1) hf(x), ((xDx) φ)(x)i, ∀φ ∈ Ma,b. k M ′ M ′ (5) (P1D) : a,b → a,b by k k h((P1D) f)(x), φ(x)i = h(xDx) f(x), φ(x)i k k = (−1) hf(x), ((Dxx) φ)(x)i, ∀φ ∈ Ma,b. k k M ′ M ′ (6) (D P1 ) : a,b → a,b by k k k k h((D P1 )f)(x), φ(x)i = h(Dxx )f(x), φ(x)i k k k M = (−1) hf(x), ((x Dxφ)(x)i, ∀φ ∈ a,b. k k M ′ M ′ (7) (P1 D ) : a,b → a,b by k k k k h((P1 D )f)(x), φ(x)i = h(x Dx)f(x), φ(x)i k k k M = (−1) hf(x), ((Dxx )φ)(x)i, ∀φ ∈ a,b. M ′ M ′ (8) Mr : a,b → a,b by −1 −1 h(Mrf)(x), φ(x)i = hf(rx), φ(x)i = hf(x), r φ(r x)i, ∀φ ∈ Ma,b. M ′ M ′ (9) Eρ : a,b → c,d by ρ −1 (1−ρ)/ρ −ρ h(Eρf)(x), φ(x)i=hf(x ), φ(x)i=hf(x), |ρ| x φ(x )i, ∀φ∈Mc,d, where c = ρa, d = ρb if ρ> 0, c = ρb, d = ρa if ρ< 0. M ′ Theorem 2.3. Let f ∈ a,b, α = u + iv ∈ C, r > 0, k ∈ N0 and ρ ∈ R \ {0} and F (s) be the Mellin transform of f. Then k k (1) M(L f)= Ds F (s). (2) M(Pαf)= F (s + α). (3) M(Dkf) = (−1)k(s − k)(s − k + 1) · · · (s − 1)F (s − k). k k k (4) M((DP1) f) = (−1) (s − 1) F (s). k k k (5) M((P1D) f) = (−1) s F (s). k k k (6) M(D P1 f) = (−1) (s − k)(s − k + 1) · · · (s − 1)F (s). k k k (7) M(P1 D f) = (−1) s(s + 1) · · · (s + k − 1)F (s). −s (8) M(Mrf)= r F (s). −1 −1 (9) M(Eρf)= |ρ| F (ρ s). Remark 2.4. In the literature, the continuity of the distributional Mellin transform is not discussed. 2009] MELLIN TRANSFORM FOR BOEHMIANS 79 By introducing a topology on the range of Mellin transform as M ′ −1 M ′ U is open in M( a,b) if M (U) is open in a,b, we can get that the Mellin transform becomes a homeomorphism between M ′ M ′ a,b and M( a,b). A similar technique is followed in the context of Hilbert transform on the Schwartz testing function space D. See [14, p.114]. The Mellin transform is closely related with the Laplace transform as follows: Let La,b be the space of all smooth functions φ on (−∞, ∞) with k γk(φ)= sup |κa,b(t)D φ| < ∞, k = 0, 1, 2 . −∞<t<∞ where eat 0 ≤ t< ∞ κa,b = ( ebt −∞ <t< 0 L ′ and its dual is denoted by a,b. L ′ The distributional Laplace transform on a,b is defined by −st (Lf)(s)= hf(t), e i, s ∈ Ωa,b. L ′ The convolution on a,b is defined as follows. L ′ Definition 2.5.