Estimates of the coefficients of the Jacobi expansion by measures of smoothness

For expansion by Jacobi polynomials we relate smoothness given by appropriate K-functionals in L p , 1 ⩽ p ⩽ 2 , to estimates on the coefficients in the ℓ q form. As a corollary for 1 < p ⩽ 2 , q = p p − 1 and a n the coefficients of the Legendre expansion of f ∈ L p [ − 1 , 1 ] , we obtain the estimate { ∑ n = m ∞ min ( 1 , t n ) q m n ( 1 q − 1 2 ) | a n | q } 1 / q ⩽ C ω φ m ( f , t ) p .