Is the Nyquist Rate Enough?

The Nyquist rate was derived using infinite time interval. But all our applications are based on finite time intervals. In digital communications we repeat our processes usually over the symbol time interval. The Nyquist rate will provide very few samples on this small interval. It will be very difficult to recover the symbol function from so few samples. In this paper we provide a mathematical proof that faster sample rate is meaningful, necessary, and it provides additional information about the function. We extend the sampling theorem for signals defined over finite time measurement interval. Our proofs use the very well known mathematical concept of infinite dimensionality of function space.