On the existence of the band-limited interpolation of non-band-limited signals

Abstract

The distribution theory serves as an important theoretical foundation for some approaches arose from the engineering intuition. Particular examples are approaches based on the delta-“function”. In this work, we show that the usual construction of a band-limited interpolation (BLI) of signals “vanishing” at infinity (e.g., in [1], [2]), using the delta-“function”, is erroneous, both in the distributional sense and in the tempered distributional sense. The latter sense is in particular important for analyzing the frequency behaviour of that method — the aliasing error and the truncation error. Furthermore, we show that it is possible to construct a BLI without using the delta-“function”. This can in particular be done easily for the space of signals having integrable frequencies. If one consider another notion of band-limited functions, a BLI can even be given for the space of continuous signals “vanishing” at infinity. For the space of continuous signals, we answer the question whether there exists a BLI negatively.