This work completely characterizes the behavior of Cesaro means of any order of the Jacobi polynomials. In particular, pointwise estimates are derived for the Cesaro mean kernel. Complete answers are given for the convergence almost everywhere of partial sums of Cesaro means of functions belonging to the critical $L^p$ spaces. This characterization is deduced from weak type estimates for the maximal partial sum operator. The methods used are fairly general and should apply to other series of special functions.
Facts and definitions An absolute value estimate for $3(1-y)\leq 2(1-x)$ A basic estimate for $3(1-y)\leq 2(1-x)$ A kernel estimate for $3(1-y)\leq 2(1-x)$ and $-1\leq \theta \leq 0$ A reduction lemma A kernel estimate for $3(1-y)\leq 2(1-x)$ and $\theta \geq-1$ A Cesaro kernel estimate for $t\leq s/2$ A basic estimate for separated arguments A reduction lemma for separated arguments A kernel estimate for separated arguments Cesaro kernel estimate for $t\leq s-b$ Cesaro kernel estimate for $s$ near $t$ Kernel estimates A weak type lemma Lemmas for the upper critical value Proofs of theorems (1.1)-(1.3) Norm estimates for $p$ not between the critical values A polynomial norm inequality A lower bound for a norm of the kernel Some limitations of the basic result Growth of Cesaro means.