On the Approximation of Continuous Functions by Fourier-Legendre Sums

ISSN： 0021-9045

Source： Journal of Approximation Theory, Vol.86, Iss.2, 1996-08, pp. : 197-215

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Abstract

Let { X n } [infinity] 0 be the orthonormal system of Legendre polynomials on [-1, 1]. For f∈ C [-1, 1] let S n ( f ) ( n +1∈ N ) be the n th partial sum of the Fourier-Legendre series of the function f . Some refinements of the classical inequality parallelf-S_n(f)parallel_{C[-1,1]}leqAcdot(n+1)^{1/2}E_n(f)_C,qquadA=mbox{ const.}qquad Agt 0,eqno (1) involving best approximation in L p -norms are discussed. For a class of examples we obtain better order estimates than those that can be derived from (1). Furthermore, we show that the results are best possible in a certain sense. It turns out that only in two particular cases ( p = 4 3 and p =4) there is no proof of optimality of the results. In conclusion, we give without proof a generalization of the main theorem to the ultraspherical case.