In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.

By convention, the translation and scale invariant functions of Legendre moments are achieved by using a combination of the corresponding invariants of geometric moments. They can also be accomplished by normalizing the translated and/or scaled images using complex or geometric moments. However, the derivation of these functions is not based on Legendre polynomials. This is mainly due to the fa...

Abstract. It was shown by P. J. Davis that the Newton-Cotes quadrature formula is convergent if the integrand is an analytic function that is regular in a sufficiently large region of the complex plane containing the interval of integration. In the present paper, a bound on the error of the Newton-Cotes quadrature formula for analytic functions is derived. Also the bounds on the Legendre polyno...

We state and prove an apparently hitherto unrecorded evaluation of certain Legendre symbols: if p is prime, p 6= 2, and ab = p − 1, then the Legendre symbol ( b p ) is given by ( b p ) = (−1).

We give via variational approach and multiresolution expansion in the base of compactly supported wavelets the explicit time description of four following problems: dynamics and optimal dynamics for some important electromechanical system, Galerkin approximation for beam equation, computations of Melnikov functions for perturbed Hamiltonian systems. We also consider wavelet parametrization in F...