Weak behaviour of Fourier-Jacobi series

Weighted Weak Behaviour of Fourier‐Jacobi Series

“…In the general case α > −1/2, α ≥ β (the case β ≥ α follows by symmetry), we prove in theorem 1 that the n-th partial sum operators are uniformly of restricted weak (p, p)-type when p is an endpoint of the interval of mean convergence, thus extending the above cited result (the question of the weak boundedness had already been answered in the negative in [2]). Now, however, uniform bounds are not available for Jacobi polynomials; therefore, a uniform weighted norm inequality is needed for operators of the form f −→ u n H(v n f ), where H is the Hilbert transform and (u n ), (v n ) are two sequences of weights involving Jacobi polynomials or their bounds.…”

“…If both α, β > −1/2, the authors proved (see [2]) that the n-th partial sum operators are uniformly of restricted weak (p, p)-type but not of of weak (p, p)-type when p is an endpoint of the interval of mean convergence. In theorems 2 and 3 we extend this result to weighted case f −→ uS n (u −1 f ), where u is also a Jacobi weight, that is,…”

“…The weak boundedness uS n f L p * (w) ≤ C uf L p (w) implies the following conditions (see [2], theorem 1, with the appropriate changes):…”

Strong and Weak Weighted Convergence of Jacobi Series

“…The result for this weight was extended to all Fourier Jacobi series by Guadalupe, Perez and Varona [5]; for example, when : ; &1Â2 and :> &1Â2, they obtained restricted weak convergence when p=4(:+1)Â(2:+3) or p=4(:+1)Â(2:+1), values of p for which they show weak convergence doesn't hold.…”

Mean and Weak Convergence of Fourier-Bessel Series

“…with a constant C independent of n, and p = p 0 or p 1 . This "end point problem" has been studied by different authors for another operators (see [2], [3], [6], [11]), but in the case of Fourier-Bessel series it seems to be new, even in the unweighted case. In theorem 3, we give necessary conditions on U, V for (0.3) to be true.…”