In the band-limited signal space, the mutual relation between continuous time signal and discrete time signal is expressed by the sampling theorem of Whittaker-Someya-Shannon. This theorem consists of an orthonormal expansion formula using sinc functions. In that formula, the expansion coefficients are identical to the sample values of signals. In general, the bandlimited signal space is not always suited to model the signals in nature. The authors have proposed a new model for signal processing based on finite times continuously differentiable functions. This paper aims to complete the sampling theorem for the spline signal spaces, which corresponds to the sampling theorem of Whittaker-Someya-Shannon in the band-limited signal space. Since the obtained sampling theorem gives the simplest representation of signals, it is considered to be the most fundamental characterization of spline functions used for signal processing. The biorthonormal basis derived in this paper is considered to be a system of delta functions at sampling points with some continuous differentiability.

General exercise approaches are not convenient for some people in undertaking appropriate exercise due to the limited variety of present programs at existing exercise machines. Moreover, continuous support by one sports doctor is only available for a limited number of users. In this paper, therefore, we propose an Internet-based technical framework, which is designed on multi-tiered client/server architecture, for integrating and easily upgrading exercise programs. By applying the technical framework, a cycle ergometer health promotion system was developed for providing personally fitted. We also presented some facilities to assist sports doctors in quickly designing and remotely improving individual exercise protocols against cycle ergometer exercise based on a history database. Then we evaluated the Internet-based cycle ergometer system during two months of feasibility experiments for six elderly persons in terms of usability. As a result, the Internet-based cycle ergometer system was effective for continuously supporting the personal fitting procedure.