A comprehensive approach to the construction of polynomial window functions is proposed. In particular the B-Splines with knots coincident at the borders are, for given degree, the polynomials having the largest variance in the time domain. Linear combination of B-Splines can be used to compute polynomial windows up to any high degree. The advantage of this approach lies in the fact that B-Splines can be computed with good accuracy even for high orders with a simple iterative algorithm. A comparison with the class of Bessel- Weighting-Patterns which includes the Kaiser- Window, is presented.

This letter contrasts two interpolating pulses with four sampling interval durations which interpolate cubic polynomials between successive samples. One yields an everywhere differentiable interpolant and the other an interpolant which is more band-limited despite not being everywhere differentiable. The continual need to interpolate intermediate values between data samples which arises in image processing makes it important to consider how interpolants differ from one another. This makes it of interest to compare two similar interpolating pulses which have the somewhat atypical behavior that the one with a continuous derivative is not the one with the more band-limited spectrum.

The Newton-Gregory interpolation formula is shown to lead to a sampling theorem for signals having Laplace transforms with pole frequencies less than a submultiple of the sampling rate. Plots of spectra computed by Fourier transforming polynomials fitted to short sample sequences are exhibited which accord with this sampling theorem.

When the process of deconvolution or inverse filtering is implemented digitally, a high sampling rate can accentuate the sensitivity to small data errors which besets deconvolution. Conversely, a low sampling rate can cause high analog frequencies to be misrepresented as lower alias frequencies within the digital frequency baseband. This paper considers using alias-free digital signals generated by Poisson transformation to avert the trade-off between these two complications.

Sequences with properties which make them attractive discrete-time signals for simulating analog signals are generated by the Poisson transform. It is shown that a finite number of Poisson transform sequence elements gives a signal representation equivalent to a best mean square representation with either Laguerre polynomials or functions.

It is shown that the Poisson transform used in the statistical theory of photodetection corresponds to a simple sign measurement and has mathematical properties which suggest its use in other areas of signal theory.