This paper presents a study on Fourier series , based on the following Sobolev-type inner product: with M_k,i > 0, where is the set of orthornomal polynomials with respect to the inner product. It is proved that and for under the (standard) assumption that [f(x) - f(t)]/(x - t) ∈ L² (&mgr;), where &mgr; is the Jacobi measure d&mgr;(x) = (1 - x)^&agr; (1 + x)^&bgr; dx, &agr;, &bgr; > -1.

The uniform convergence is, furthermore, obtained over [-1 + &egr;,1 - &egr;] for every &egr; ∈ (0,1) if f satisfies the Lipschitz condition |f(x) - f(t)| ≤M |h|^&eegr; for |h|<&dgr; uniformly in [-1,1]. A weaker convergence result is also derived. More specifically, the Fourier series converges almost everywhere in [-1,1] if the modulus of continuity of f satisfies the following estimate:

sup{| f (x₁) - f(x₂)|: x₁, x₂ ∈[-1,1], |x₁ - x₂| ≤ &dgr;} = O(log^-(1+&egr;)) 1/&dgr;) for &egr;> 0.

The estimates and behavior of the Fourier series when the mass points a_k ∈[-1,1] remain open.