Finite Fourier Research Articles

On the spectral decomposition of stationary time series using walsh functions. I

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

On the spectral decomposition of stationary time series using walsh functions. I

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.

On uniform asymptotic expansions of finite Laplace and Fourier integrals

SynopsisA uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.

A new technique for the determination of coronal magnetic fields: A fixed mesh solution to Laplace's equation using line-of-sight boundary conditions

A new method for computing potential magnetic field configurations in the solar atmosphere is described. A discrete approximation to Laplace's equation is solved in the domain R ⊙ ≤ r ≤R 1, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π (R 1being an arbitrary radial distance from the solar center). The method utilizes the measured line-of-sight magnetic fields directly as the boundary condition at the solar surface and constrains the field to become radial at the outer boundary, R 1. First the differential equation and boundary conditions are reduced to a set of two-dimensional equations in r, θ by Fourier transforming out the periodic φ dependence. Next each transformed boundary condition is converted to a Dirichlet surface condition. Then each two-dimensional equation with standard Dirichlet-Dirichlet boundary conditions is solved for the Fourier coefficient it determines. Finally, the solution of the original three dimensional equation is obtained through inverse Fourier transformation. The primary numerical tools in this technique are the use of a finite fast Fourier transform technique and also a generalized cyclic reduction algorithm developed at NCAR. Any extraneous monopole component present in the data can be removed if so desired. The code was developed for the HAO solar-interplanetary modeling effort in response to the following specific requirements: The spatial resolution of the present code is such that measured surface line-of-sight magnetic fields to a resolution of 2.8° in both latitude and longitude can be adequately treated. Comparisons with the Altschuler-Newkirk code shows that the two methods are more-or-less equivalent for computing large-scale fields at ≈2R ⊙but that the fixed mesh method is capable of much greater accuracy close to the Sun.

Series solution of the biharmonic equation in the rectangular domain with some applications to mechanics

SummaryA double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.

Sequential least-squares Fourier estimation with fading memory

Necessary and sufficient conditions are developed which insure finite fading memory least-square Fourier coefficient estimates from time and frequency domain data are equal. The implementation of such filters are also presented.

On the eigenproblem for convolution integral equations

This paper concerns the eigenproblem for convolution integral equations whose kernels can be expressed as finite or infinite Fourier transforms of integrable functions. A procedure which closely parallels previous work on displacement integral equations is derived and the problem of existence is treated. Approximations are obtained for both the eigenvalues and the eigenfunctions.

The Plane-Stress Problem for a Doubly Connected Region in Which the Medium Undergoes Elastic and Creep Deformation

A theoretical method of analysis is revealed for handling the plane-stress problem of a doubly connected region in which the medium undergoes elastic and creep deformation. The solution is developed in small increments of time as a sequence of solutions to elastostatic problems of two types. The first elastostatic problem involves surface forces only and is solved by the alternating method of Schwarz in conjunction with the method of Muskhelishvili. The second elastostatic problem is of the “initial-stress” type and is solved by the use of finite differences and double Fourier series. A numerical example, evaluated with the aid of an IBM 7090 electronic data-processing machine, is presented for the case of a uniformly stretched plate in the shape of a circular annulus.