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...

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...

in this article we implement an operational matrix of fractional integration for legendre polynomials. we proposed an algorithm to obtain an approximation solution for fractional differential equations, described in riemann-liouville sense, based on shifted legendre polynomials. this method was applied to solve linear multi-order fractional differential equation with initial conditions, and the...

In this paper, biochemical reaction problem is given in the form of a system of non-linear differential equations involving Caputo fractional derivative. The aim is to suggest an instrumental scheme to approximate the solution of this problem. To achieve this goal, the fractional derivation terms are expanded as the elements of shifted Legendre scaling functions. Then, applying operational matr...

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations of the first-kind is proposed. This problem is transformedto a nonlinear two-dimensional Volterra integral equation of the second-kind. The properties ofthe bivariate shifted Legendre functions are presented. The operational matrices of integrationtogether with the produ...

We consider the problem of recovering a hidden monic polynomial f(X) of degree d ≥ 1 over a finite field Fp of p elements given a black box which, for any x ∈ Fp, evaluates the quadratic character of f(x). We design a classical algorithm of complexity O(d2pd+ε) and also show that the quantum query complexity of this problem is O(d). Some of our results extend those of Wim van Dam, Sean Hallgren...

In this paper, interval Legendre wavelet method is investigated to approximated the solution of the interval Volterra-Fredholm-Hammerstein integral equation. The shifted interval Legendre polynomials are introduced and based on interval Legendre wavelet method is defined. The existence and uniqueness theorem for the interval Volterra-Fredholm-Hammerstein integral equations is proved. Some examp...

We consider a class of arithmetic equations over the complete lattice of integers (extended with−∞ and∞) and provide a polynomial time algorithm for computing least solutions. For systems of equations with addition and least upper bounds, this algorithm is a smooth generalization of the Bellman-Ford algorithm for computing the single source shortest path in presence of positive and negative edg...

We present explicit formulae and complexities of bit-parallel shifted polynomial basis (SPB) squarers in finite field GF (2)s generated by general irreducible trinomials x+x+1 (0 < k < n) and type-II irreducible pentanomials x + x + x + xk−1 + 1 (3 < k < (n − 3)/2). The complexities of the proposed squarers match or slightly outperform the previous best results. These formulae can also be used ...