The sampling theorem can be extended to cover all distributions in L. Schwartz's sense. As any band-limited distribution reduces to an entire function of exponential type, the sampling theorem for band-limited signals is nothing but an interpolation or extrapolation of such a function. Here, an explicit inter-extrapolation formula with equally-spaced sampling points on the negative time axis is presented. The formula can be applied to all band-limited signals appearing in practice because on additional requirement is imposed on signals such as being square-integrable, being bounded or being tempered. And moreover, since the sampling points are entirely contained in the negative time axis, a signal at any time can be reconstructed from the signal values up to the present without delay, and in principle a deterministic prediction is possible for arbitrarily far future. The formula is obtained combining the extension of Newton interpolation formula for polynomials with Mittag-Leffler summation for divergent series.

Markov renewal process (MRP) is widely used in communication theory, OR, physics and so on. This paper is concerned with the mathematically rigorous construction of the stationary MRP with finite states on the whole time axis. A joint (forward and backward) invariant probability exists uniquely for an irreducible MRP. A stationary MRP on the whole time axis is defined by setting the joint invariant probability at the point of origin and by forming the MRP in the positive direction and the reverse MRP in the negative direction. An elementary proof of the stationarity of the defined process is given by showing directly that any finite-dimensional distribution is invariant under the shift of time. The proof of the existence and uniqueness of the joint invariant probability is also presented. Once the stationarity of the defined MRP is established, various stochastic processes generated thereof can easily be shown to be stationary as well.

Sampling theorem for all bandlimited distributions which converges in the sense of distributions is presented. An inter-extrapolation formula with the most general sampling points on the negative time axis is provided and especially for equally-spaced sampling points an explicit formula is given, where it is proved that the Newton interpolating polynomial through the finite number of sample values converges with the Mittag-Leffler summation. These formulas are both proved to be convergent in the topology of distribution, hence a consistent theory is accomplished within the scope of distributions. As many linear operations on signals such as differentiation are continuous with respect to the distribution topology, the given formulas exhibit great facility when applied to signal analysis.

This is a sequel of my previous paper. There, the mathematical definition of the stationary Markov renewal process (MRP) was given with a complete proof that the defined process is really a stationary point process on the whole time axis. As a consequence, various stochastic processes appearing in practice which are derived from MRP can easily be shown to be stationary. The author has been interested in the calculation of the power spectrum of MRP generated signals in electrical or nervous systems. For this purpose, it was necessary to construct a stationary process generated by MRP and hence it was inevitable to introduce a stationary MRP on the whole time axis. In this paper, the ergodicity of the one-parameter group of measure-preserving transformations associated with the time shift of the stationary MRP is discussed. An irreducible MRP is either aperiodic or periodic with some period. Here, a proof is given that an aperiodic MRP is strongly mixing, from which ergodicity follows. On the contrary, as a periodic MRP is only ergodic without being mixing, it is directly proved ergodic. Once the ergodicity of MRP is established, various stationary processes derived thereof are easily shown to be ergodic as well. This means that the characteristics, among which the power spectrum of each sample process is our concern, are sample-wise invariant, which will be useful in many applications. Recently, as MRP and related stochatsic processes are widely used in many fields, e.g. physics, OR, communication theory, and bioscience, our results will find large utilities in both the theory and the application of stochastic processes.

A sampling theorem valid for all band-limited distributions with most generally arranged sampling points is presented. Any bandlimited signal, whose complex frequency band has a horizontal width less than 4πb, can be reconstructed from the values of the signal and its derivatives on the sampling points in the negative time axis, as long as the average density of the points exceeds the Nyquist rate 2b.