equiconvergent

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1: 30.4 Functions of the First Kind

… ►It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for $-1\le x\le 1$.

2: 18.2 General Orthogonal Polynomials

… ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with $p_n(x)=T_n\left(x\right)$, a Chebyshev polynomial of the first kind, see Table 18.3.1. … ►For OP’s with orthogonality measure in $𝒮$ Nevai (1979, pp. 148–150) generalized Szegő’s equiconvergence theorem. …