Let L be a big holomorphic line bundle on a compact complex manifold X. We show how to associate a convex function on the Okounkov body of L to any continuous metric e −ψ on L. We will call this the Chebyshev transform of ψ, denoted by c[ψ]. Our main theorem states that the integral of the difference of the Chebyshev transforms of two weights is equal to the relative energy of the weights, whic...

In a previous paper we have presented a new method of imposing boundary conditions in the pseudospectral Chebyshev approximation of a scalar hyperbolic equation. The novel idea of the new method is to collocate the equation at the boundary points as well as in the inner grid points, using the boundary conditions as penalty terms. In this paper we extend the above boundary treatment to the case ...

This study introduced new technique which is based on a combination of the least-squares (LST) with Chebyshev and Legendre polynomials for finding approximate solutions higher-order linear Fredholm-Volterra integro-differential equations (FVIDEs) subject to mixed conditions. Two examples second third-order FVIDEs are considered illustrate proposed method, numerical results comprised demonstrate...

We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.

We study linear transformations $$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$$ of the form $$T[x^n]=P_n(x)$$ where $$\{P_n(x)\}$$ is a real orthogonal polynomial system. With $$T=\sum \tfrac{Q_k(x)}{k!}D^k$$ , we seek to understand behavior transformation T by studying roots $$Q_k(x)$$ . prove four main things. First, show that only case are constant and an system when $$P_n(x)$$ shifted set ...

In this paper we describe a generic spectral Petrov–Galerkin method that is sparse and strictly banded for any linear ordinary differential equation with polynomial coefficients. The applies to all subdivisions of Jacobi polynomials (e.g., Chebyshev Legendre), utilizes well-known recurrence relations orthogonal polynomials, leads almost exactly the same discretized system equations as integrati...

The theory of orthogonal polynomials plays an important role in many branches of mathematics, such as approximation theory (best approximation, interpolation, quadrature), special functions, continued fractions, differential and integral equations. The notion of orthogonality originated from the theory of continued fractions, but later became an independent (and possibly more important) discipl...

the low earth orbiting (leo) satellites are widely used for geosciences applications. for most applications, precise orbital information of the satellites is required. a combination of the in suite observations and dynamic orbit yields the optimum solution. in order to obtain the combined optimal solution, one needs to analytically or numerically propagate the state vector epoch by an epoch bas...

Estimating the temperature field of a building envelope could be time-consuming task. The use reduced-order method is then proposed: Proper Generalized Decomposition method. solution transient heat equation re-written as function its parameters: boundary conditions, initial condition, etc. To avoid tremendous number parameters, condition parameterized. This usually done by using Orthogonal to p...

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...