OPERATIONAL CALCULUS FOR THE CONTINUOUS LEGENDRE TRANSFORM WITH APPLICATIONS

This paper develops an operational calculus for the continuous Legendre transform introduced and studied by Butzer, Stens and Wehrens [1]. It is an extension of the work done by Churchill et al [2], [31 for the discrete case. In particular, a differentiation theorem and a convolution theorem are proved and the results are applied to the solution of some boundary value problems.

1 J Px(x)f(x)dx (1) (Tf)(A)where P(x)is the Legendre function and A >_ -].This transform has been introduced and studied by Butzer, Stens and Wehrens [1].The discrete analog of the transform ill (1) has been studied by Churchill  [2] and Churchill and Dolph [3].The object of this paper is to develop an operational calculus for the transforin which is useful in solving paxtial differential equations whose underlying differential forln is given by D=xx (1-x)  Ill section 2 we present the background material needed in the sequel.In section 3, ve derive the operational calculus for (1) including a convolution theorem and a table of transforms of some functions.In the last section we apply the results to solving some boundary value problems. 2. PRELIMINARIES.We recall basic properties of the transform (Tf)(A) (see [1]) and itnportant coutiguous relations that hold for the Legendre function.
the Legendre transform of some functions.
The first property in this direction involves the D as given in (2).

2+I
This will follow Dy applying the contiguous relation (3).
() u () L',(-)() i (TI)() xi, th, .oi oiion on f, the contiguous relation (5) and Theorem 3.2 yields (Tg The next operational property that we will derive involves the inverse of the differential op- erator D. We define the inverse of D, denoted by D-', by D-'(f(x)) g(x) if and only if D(g(x)) f(x).If (Tf)(A) is known, then we want to relate T((D-Xf))(A) to the transform of f.
We shall finally develop a convolution property for the Legendre transforin.In particular, we will show Theorem 3.4.If f(x) and g(x) are given functions for which (Tf)(A) and (Tg)(A) respectively exist, then their product (Tf)(A)(Tg)(A) is the transform of the function h(x) ](x),g(x) where h(x) is given by h(cos v)=f(cos)g(cosB)sinododO where cos/ cosacost + sinsintcos8 with 0 < c, t < r, a + r, < r and O is real.The variables a, t, astd may be interpreted as the sides of a spherical triangle on the unit hemisphere and is the angle between the sides a and u (see Figure 1).
Set x cos a and y cos B. Then he addition formula for gle Legendre function (6) will yield upon an ingegration wih respecg to from 0 to 1 P(cos .)dwhere cos cos cos + sin sin cos (s figure 1).In the spherical triangle PQR, we have cos cos c cos v + sin c sin v cos 8.
Using this relation along with the sine law and transfornation of co-ordinates, the double i,tegral can be written as: Hence, fo"fo" (o .)g(o 3)i,, (Tf)(A)(Tg)(A) -P,(cosv)sinv The expression in the bracket is a function of v and we then write 1 f(cosa)g(cosfl)sinoMadt? (5) This may be interpreted as a convolution product of f and g and (Th(cos v))(A) (Tf)(A)(Tg)(A).
Geometrically, the expression (15) is the mean value of f(cosa)g(cos) over the unit hemi- sphere x2+y2+z 1, z > O.To see this, we note that the element surface area is dS sinadadS.This is clear if we identify the coordinate transformation in Figure 1 by X COS sin t sin 0 sin c cos 0 Thus (15) reads l fsff(cosc,)g(cos3)dS.
We consider proble arising in heat conduction and in potentiM theory.
A. Heat Conduction Problem.Consider a non-homogeneous b with extremities at x 1 md is iulaed at these end points.Let u(x,t) be the temperature of the b at position z at time t.The one dimensiond heat equation with prescribed iNtid temperature is given by 0 (' oN(' where k, p d c e physical constt.srepresenting therm conductivity, density xd specific het rpective]y.We sume that the thermal conductivity k is given by k (1 z), being re comfit.he above equation reads 0 ( 0u(z,t)) pcOu The solution is given by -U(A,t) ---A(A-I-1)V(A,t) pc v(, o) G().