On the rate of growth of Bessel–Fourier coefficients for integrable functions

It is known that the Bessel–Fourier coefficients f ν , m of a function f , such that x s + 1 / 2 f ( x ) is integrable over [ 0 , 1 ] with 0 ⩽ s ⩽ ν + 1 / 2 , satisfy j ν , m − s − 1 / 2 f ν , m → 0 . We show a partial converse, namely that for 0 ⩽ α < 1 / 2 and any non-negative a m → 0 , there is a function f such that x α + 1 + ν f ( x ) is integrable and its Bessel–Fourier coefficients f ν , m satisfy j ν , m − α f ν , m ⩾ a m and j ν , m − α f ν , m → 0 . For ν = 0 , we conjecture that the same should be true when α = 1 2 , and discuss some consequences of this conjecture for the divergence Bessel–Fourier expansions of radial functions on the disc.