Fourier - Bessel Transforms

263

The Bessel functions Jl,k> 0, can also be computed exphcitly using most of the techniques developed above. We content ourselves with stating only the final results.

Theorem 6. // ip^ is a character on Cf^ which is trivial on Cf ^ and non-trivial onÖ^-'\h^l,then

Jk { v ) = hvit)Wk{t)dt = 0 if N^g. If\v\ = q^'^^^>q, then

JkHv ) =<

0 , lSm<h/2,

X { v ) J x(2'n;xVl -т:^^')У^к{я>М)^х, h/2^m<h,

If \v\ = q^'^>q, then

4i^m<A±l

Xiv )

h + 1

j xi2Tvx^/l-TX^)\pk((p(x))dx, —Y-U^<h,

Jk4v ) = <

\x\=q -

[ \x\ = q Ux\=q

X { 2Tux^ ) xPk { ( p { T^x ) ) dx + l\ , m = h,v = x~^\ueU,

' } ■

X { 2xux^' ) dx + \\ ,m>h,v = T ^""u^ueU.

q'^^xi

Each of the integrals occurring in the above evaluation of J^ can be written as a finite sum of ^''-th roots of unity (the last integral is a sum of ^-th roots of unity as in Theorem 5.

As a final remark, we note that the subspaces Жо and J^q of J^o» defined in the remark after Theorem 1, are not invariant under the Fourier transform. For example, iffej^o, it follows from (3.7) and Theorem 4 that

f ( ] / ^R ) =if { Q ) \Q\jS { - RQ ) dQ= J f{Q)\Q\dQ,

к \Q\âq\R\~'

and this last integral is not zero, in general. A similar proof can be given show the non-invariance of Жо.

References

1 . BocHNER, S., and K. Chandrasekharan: Fourier transforms. Princeton, N.J.: Princeton

University Press 1949.

2 . BosECK, H. : Darstellungen von Matrixgruppen über topologischen Körpern I. Math. Nachr.

24 , 229—243 (1962).

3 . BouRBAKi, N.: Algèbre commutative, Ch. 5 and 6. Paris: Hermann 1964.

4 . Gelfand, I. M., and M. I. Graev: Representations of a group of the second order with

elements from a locally compact field. Uspehi Mat. Nauk. (Russian Math. Surveys) 18, 29—100 (1963).

18 Math Ann 174