Shifted Jacobi Polynomials Research Articles

Shifted Jacobi series analysis of linear two-point boundary value problems

The shifted Jacobi polynomial functions approximation is extended to solving the linear ordinary differential equation of the two-point boundary value problem. The linear ordinary differential equation of boundary value problems is reduced to the linear functional differential equation of the initial value problem. A new time-domain approach to the derivation of a Jacobi transformation matrix together with the Jacobi integration matrix, the solution of the linear functional ordinary differential equation of the initial value problem can be obtained via shifted Jacobi series. Two examples are demonstrated and the satisfactory computational results are compared with those of the exact solution.

One-parameter Jacobi transformations of sequences

A Toeplitz transformation of a class of sequences is considered, in which the coefficients are multiples of shifted Jacobi polynomials, and depend upon a parameter. When the parameter is equal to +1, the transformation is not regular and is best applied to monotonic sequences [6]. When the parameter is equal to −1, the transformation is regular, and best applied to alternating sequences. In general, the transformation yields rational approximations. In the particular case of a logarithmic series, the rational approximations are precisely the [ n, n] Padé approximants; a particular case of a result of Luke [2]. Numerical examples are given, and errors estimated.

Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials

The Zernike aberration theory for constant amplitude circular apertures is extended to annular apertures having a Gaussian-like radial taper. Explicit expressions are obtained for the optimum balanced wave-front aberrations in terms of shifted Jacobi polynomials. Properties of the polynomials (e.g., Rodrigues formula, recurrence relations, derivatives, etc.) are investigated.

The use of the shifted jacobi polynomials in the synthesis of lowpass filters

AbstractIt is shown that the shifted Jacobi polynomials are particularly useful in solving the approximation problem of both the general and the monotonic polynomial lowpass filters. A new type of least squares monotonic filter is introduced.