Discrete FIR filter banks minimizing various measures of approximation error at the same time

We present the optimum discrete running approximation of time-limited signals by a FIR (finite impulse response) filter bank minimizing continuous long-range worst-case measures of error. Firstly, we provide a brief survey of the two conditions of the optimum approximation that uses a finite number of sample values. As an application, we obtain a running FIR approximation g(t) that is optimum in each of certain separate segments in the time axis. Secondly, we derive new continuous approximation satisfying a so-called “condition of discrete orthogonality” that uses interpolation functions with very long time-duration. This condition is one of the two conditions mentioned above. This new approximation has the same values at discrete t as the FIR approximation g(t). Thirdly, using the Rayley´s theorem and the mini-max theorem, we show that this new approximation satisfies the remaining condition of the two conditions. Hence, its discrete version g(t) is optimum in a given long interval but uses FIR interpolation functions. Finally, we present a favorable example of almost perfect reconstruction FIR filter bank having total bandwidth smaller than 2π.