TY - JOUR
AU - Offord, A. C.
AB - O N HANKEL TRANSFORMS. 49 By A. C. OFFORD. 12 December, 1933.—Read 14 December, 1933.] [Received Let us write S (x) = x±J (x), v v where J {x) is Bessel's function and $\(v) > — \. Then two functions f(x) and F(x) are said to be Hankel transforms f of one another when they are connected by the formulae F{x)=\ S (xu)f{u)du, Jo f(x) = f S u) F(u) du, Jo where the integrals are usually interpreted in some generalized sense. When v= — \, S (xu) becomes •y/(2J7r) cosxu, and these formulae reduce to the formulae for Fourier cosine transforms. In a recent paper $ on Fourier transforms the following classes of functions were considered. DEFINITION A. A function f(x) belongs to the class H if it belongs to L(0, X)§for all finite X and is such that I ("u / U \ \\ (1 )cosxuf(u)du I Jo v / for all co^O and x^O, M being a number \\. | Cf. Bochner, 1, 180; Plancherel, 7; Titchmarsh, 8 and 9; Watson, 10. I Offord, 6. A knowledge of this paper is not assumed. § By IF (a, b) we mean the class of functions whose p-th 
TI - On Hankel Transforms
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s2-39.1.49
DA - 1935-01-01
UR - https://www.deepdyve.com/lp/wiley/on-hankel-transforms-bjPXxpPJms
SP - 49
EP - 67
VL - s2-39
IS - 1
DP - DeepDyve
ER -