Geodesic
Expansion of Continuous Differentiable Functions in Fourier Legendre Series
Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 823-827

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Let 1.1 denote the nth partial sum of the Fourier Legendre series of a function ƒ(x). The references available to us, except (5), prove only that Sn(ƒ, x) converges uniformly to ƒ(x) in [— 1, 1] if ƒ(x) has a continuous second derivative on [—1, 1]. Very recently Suetin (5) has shown by employing a theorem of A. F. Timan (7) (which is a stronger form of Jackson's theorem) that Sn(ƒ, x) converges uniformly to ƒ(x) ƒ(x) belongs to a Lipschitz class of order greater than 1/2 in [—1, 1]. More generally he has proved the following theorem. 
Saxena, R. B. Expansion of Continuous Differentiable Functions in Fourier Legendre Series. Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 823-827. doi: 10.4153/CJM-1967-076-1
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