Abstract |
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Frames are nowadays a standard tool in many areas of mathematics, physics and engineering. However, there are situations where it is difficult, even impossible, to design an appropriate frame. Thus there is room for generalizations, obtained by relaxing the constraints. A first case is that of semi-frames, in which one frame bound only is satisfied. Accordingly, on has to distinguish between upper and lower semi-frames. We will summarize this construction. Even more, one may get rid of both bounds, but then one needs two basic functions and one is led to the notion of reproducing pair. It turns out that every reproducing pair generates two Hilbert spaces, conjugate dual of each other. We will discuss in detail their construction and provide a number of examples, both discrete and continuous. Next, we notice that, by their very definition, the natural environment of a reproducing pair is a partial inner product space (pip) with an $L^2$ central Hilbert space. A first possibility is to work in a rigged Hilbert space. Then, after describing the general construction, we will discuss two characteristic examples, namely, we take for the partial inner product space a Hilbert scale or a lattice of $L^p$ spaces. |