The oscillation of differential transforms – entire Jacobi expansions

Let L = ( 1 − x 2 ) D 2 − ( ( β − α ) − ( α + β + 2 ) x ) D with α ⩾ − 1 2 , β ⩾ − 1 2 and D = d d x . Let f ∈ C ∞ [ − 1 , 1 ] , f ( x ) = ∑ n = 0 ∞ C n p n ( α , β ) ( x ) , with p n ( α , β ) ( x ) normalized Jacobi polynomials and the C n decrease sufficiently fast. Set L k = L ( L k − 1 ) , k ⩾ 2 . Let ρ > 1 . If the number of sign changes of ( L k f ) ( x ) in ( − 1 , 1 ) is O ( k 1 / ( ρ + 1 ) ) , then f extends to be an entire function of logarithmic order ⩽ ρ ρ − 1 . For Legendre expansions, the result holds with 1 ρ + 1 replaced with 1 ρ .