Masaharu Arai

Aitken acceleration and Fibonacci numbers

The third author discovered numerically an interesting phenomenon that the Aitken acceleration to the ratiopn=fn−1/fn of the successive Fibonacci numbersfn−1 andpn in exactlyp2n. This fact has a natural extension to the case that Aitken acceleration is replaced by ε-algorithm and Fibonacci numbers can be replaced by solutions of second order difference equations. As a byproduct we will show ann-plication formula of cotz.

Restoration of lost samples by oversampling near the Nyquist rate

A formula in matrix form for restoration is given and the matrix with the sampling rate near the Nyquist rate is investigated. Elements of this matrix can be expanded in Laurent series of the sampling rate parameter, which is defined by the quotient of the Nyquist rate and the sampling rate. The Nyquist rate corresponds to a pole. First terms of these Laurent series near the Nyquist rate are given.