A Legendre Polynomial Integral

Further Extensions of a Legendre Function Integral

Divergent Legendre-sobolev Polynomial Series

Let be introduced the Sobolev-type inner product (f, g) = 1 2 Z 1 −1 f(x)g(x)dx + M [f ′(1)g′(1) + f ′(−1)g′(−1)], where M ≥ 0. In this paper we will prove that for 1 ≤ p ≤ 4 3 there are functions f ∈ L([−1, 1]) whose Fourier expansion in terms of the orthonormal polynomials with respect to the above Sobolev inner product are divergent almost everywhere on [−1, 1]. We also show that, for some v...

Legendre-type integrands and convex integral functions

In this paper, we study the properties of integral functionals induced on LE(S, μ) by closed convex functions on a Euclidean space E. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and...

Linear complexity of Legendre-polynomial quotients

We continue to investigate binary sequence (fu) over {0, 1} defined by (−1)fu = ( (u−u)/p p ) for integers u ≥ 0, where ( · p ) is the Legendre symbol and we restrict ( 0 p ) = 1. In an earlier work, the linear complexity of (fu) was determined for w = p − 1 under the assumption of 2p−1 6≡ 1 (mod p2). In this work, we give possible values on the linear complexity of (fu) for all 1 ≤ w < p− 1 un...

On a Zonal Polynomial Integral