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We present various inequalities for the sum Sn(t)=∑k=0nPk(t),where Pk denotes the Legendre polynomial of degree k. Among others we prove that the inequalities 25(1+t)≤Sn(t)and3−12(1−t2)≤Sn(t)hold for all n≥1 and t∈[−1,1]. The… Click to show full abstract
We present various inequalities for the sum Sn(t)=∑k=0nPk(t),where Pk denotes the Legendre polynomial of degree k. Among others we prove that the inequalities 25(1+t)≤Sn(t)and3−12(1−t2)≤Sn(t)hold for all n≥1 and t∈[−1,1]. The constant factors 2/5 and (3−1)/2 are sharp. This refines a classical result of FejA©r, who proved in 1908 that Sn(t) is nonnegative for all n≥1 and t∈[−1,1].
Keywords:
legendre polynomials;
inequalities legendre;
fej inequalities
Journal Title: Mathematische Nachrichten
Year Published: 2017
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Journal Title: Mathematische Nachrichten
Year Published: 2017
Link to full text (if available)
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recommendations!