Abstract :
Givenα, β>−1, letpn(x)=p(α, β)n(x),n=0, 1, 2,… be the sequence of Jacobi polynomials orthonormal on (−1, 1) with respect to the weightu(x)=(1−x)α (1+x)β. Denote by (SNf)(x) theNth partial sum of the Fourier–Jacobi series of the functionfon (−1, 1), so that (SNf)(x)=∑Nn=0 anpn(x), withan=∫1−1 f(x) pn(x) u(x) dx. For fixedp∈(1, ∞), we characterize the weightswsuch that limN→∞ ∫1−1 |[(SNf)(x)−f(x)] w(x)|p u(x) dx=0 whenever ∫1−1 |f(x) w(x)|p dx<∞, the weightswsuch that limN→∞ supλ>0 λ[∫ENλ w(x)p u(x) dx]1/p=0 whenever ∫1−1 |f(x) w(x)|p dx<∞, and the weightswsuch that limN→∞ supλ>0 λ[∫ENλ w(x)p u(x) dx]1/p=0 whenever ∫∞0 [∫Fλ w(x) dx]1/p dλ<∞; here,ENλ={x∈(−1, 1) ; |(SNf(x)−f(x)|>λ} andFλ={x∈(−1, 1): |f(x)|>λ}.
