DUAL INTEGRAL EQUATIONS INVOLVING LEGENDRE ...

58 P. K. BANERJI, DESHNA LOONKER
∫ ∞
h(α) = τ tanh(π τ)f(τ)P 1 (cosh α)dτ = h −
2(α); α > α 1 . (2.2)
0
2 +iτ
where α 1 is an arbitrary positive number. By virtue of the validity of embedding
from the linear spaces into their duals [2, pp. 121-23], for the distribution spaces,
considered in this paper, the imbeddings are D → M α β
(I) → S → E and their duals
E ′ → S ′ → M ′ β α → D ′ for the generalized Mehler-Fock transform of order zero.
The equation (1.31) is obtained due to the property
∫ b
∫ d
f(h(t))ϕ(t)dt = f(x) ϕ(h−1 (x))
a
c h ′ (h −1 dx, (2.3)
(x))
where f is locally integrable on [a, b). If x 0 ∈ [c, d), then
T h {δ(x − x 0 ); t} = δ(h(t) − x 0 ) = δ(t − h−1 (x 0 ))
h ′ (h −1 (x 0 )) . (2.4)
The operator T h −1 : D 41[a, ′ b) → D 41[c, ′ d)is defined by
(T h ) −1 = T h −1. (2.5)
The distributional interpretation of (1.28), in terms of T h and T h −1, is, see [3, p.
188],
[ ( g
)]
f = T h x −α
+ ∗ T h −1
h ′ , (2.6)
and the solution for (1.28) is
sin απ d
g(t) =
π dt T (
h x
−α
+ ∗ T h −1(f); t ) , (2.7)
where f is the usual locally integrable function.
We notice [7, pp. 407-408] that the involvement of functions F 1 (x) and F 2 (x)in
terms of the prescribed functions g 1 (α) and h 2 (α) are given by
and
F 1 (x) = √ 1 ∫
d x
g 1 (α) sinh α dα
√ (2.8)
π dx 0 (cosh x − cosh α)
F 2 (x) = √ 1 ∫ ∞
h 2 (α) sinh α dα
√ , x ≥ α 1 (2.9)
π (cosh α − cosh x)
x
The function F 1 (x), that is given by (2.8), is similar to the distributional solution
obtained in (1.30). Further, by considering cosh α as a strictly increasing function,
the function F 1 (x) in (2.8) is similar to the distributional solution given by (2.7).
By virtue of relations (1.28) – (1.31) and (2.3) – (2.7), we justify to state that g 1 (α)
is a locally integrable function and since the properties of the operators cosh α and
cosh x has the correspondence D ′ 41[0, ∞)→ D ′ 41(x, ∞], the function F 1 (x)turns out
to be locally integrable and hence it is, indeed, true that (2.8) is the required distributional
solution. This allow us, simultaneously, to confirm that F 1 (x)is defined
on spaces D ′ , M ′ β α ,E ′ , S ′ , respectively, by considering distributional generalized
Mehler-Fock transform of order zero, i.e. ,