We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the approximation of continuous functions by Chebyshev interpolation and Chebyshev series and how to efficiently compute such approximations. We conclude the pap...

In this paper, we introduce a generalized Chebyshev collocation method (GCCM) based on the generalized Chebyshev polynomials for solving stiff systems. For employing a technique of the embedded Runge-Kutta method used in explicit schemes, the property of the generalized Chebyshev polynomials is used, in which the nodes for the higher degree polynomial are overlapped with those for the lower deg...

In this paper, a generalized rational Chebyshev approximation problem is considered. The problem is this: To minimize the maximum absolute value of the "criterion function" of the error. By imposing a rather natural restriction on the criterion function, the problem is solved completely; the existence, the uniqueness and the characterization of the best approximation are clarified and interesti...

We study the kernels Kn,s(z) in the remainder terms Rn,s(f) of the Gauss-Turán quadrature formulae for analytic functions on elliptical contours with foci at ±1, when the weight ω is a generalized Chebyshev weight function. For the generalized Chebyshev weight of the first (third) kind, it is shown that the modulus of the kernel |Kn,s(z)| attains its maximum on the real axis (positive real semi...

In this paper‎, two inverse problems of determining an unknown source term in a parabolic‎ equation are considered‎. ‎First‎, ‎the unknown source term is ‎estimated in the form of a combination of Chebyshev functions‎. ‎Then‎, ‎a numerical algorithm based on Chebyshev polynomials is presented for obtaining the solution of the problem‎. ‎For solving the problem‎, ‎the operational matrices of int...

Odd degree Chebyshev polynomials over a ring of modulo 2 have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observe...