In this note we prove mean convergence of Lagrange interpolation at the zeros para-orthogonal polynomials for measures on unit circle which do not belong to Szegő’s class. When measure is in class, proved functions disk algebra.

Lagrange multipliers offer a way to find the extrema of an objective function f(x1, x2, . . . , xn) subject to a constraint function g(x1, x2, . . . , xn) = 0; for example, we may wish to maximize a+ b+ c subject to a2 + b2 + c2 = 1. However, there are lots of tiny details that need to be checked in order to completely solve a problem with Lagrange multipliers. On an olympiad the use of Lagrang...

Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger t...

Let f be a real differentiable function in an open interval I with one-to-one derivative. We observe that if the Lagrange mean L [ f ] of a generator f is conditionally positively homogeneous, then f must be of the class C∞ and the function g(x) := x f ′ (x)− f (x) , x ∈ I, is also a generator of L [ f ] i.e. that L [g] = L [ f ].We show that this fact and a result on equality of two Lagrange m...

It is well-known that the equations for a simple fluid can be cast into what is called their Lagrange formulation. We introduce a notion of a generalized Lagrange formulation, which is applicable to a wide variety of systems of partial differential equations. These include numerous systems of physical interest, in particular, those for various material media in general relativity. There is prov...

In this paper, the system of Fredholm integral equations of the second kind is investigated by using a modified degenerate kernel  method (MDKM). To construct a MDKM the source function is approximated by the same way of producing degenerate kernel. The interpolation is used to make the needed approximations. Lagrange polynomials are adopted for the interpolation. The equivalency of  proposed m...

This paper examines the optimum design of FIR precoders or equalizers for multiple-input multiple-output (MIMO) frequencyselective wireless channels. For the case of a left-coprime FIR channel, which arises generically when the number nT of transmit antennas is larger than the number nR of receive antennas, the Bezout matrix identity can be employed to design an FIR MIMO precoder that equalizes...

A Lagrange Sliding-Colliding Interface enables the free interaction of several meshes during the Lagrange Finite difference time step. It is based on dynamically identifying the boundary edges of the other meshes which may collide with a certain boundary vertex and identifying a "penetration point" for each such edge. From those non-additive possible collisions, the minimal sufficient set of in...