I have found some applications of the **Frame Theory** in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection.


As we know, any frame in a Hilbert space is itself a Bessel sequence.  

There is a spectrum of weakening sequences between the frames and Bessel sequences, like frame sequences, K-frames, K-frame sequences and Upper semi-frames.



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> Q1. Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames? Do such of these weakening sequences only determine a generalization or an extension of the notion of the frames or does there exist any necessity (I mean application(s) in engineering sciences) based on each of these weakening sequences have been formulated and considered?
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> Q2. Any other weakening sequence(s)?
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Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?
To answer the first question, frame sequences are just countable frames. A somewhat more standard name for upper semi-frames are *Bessel sequences*, that is countable sequences $\{f\_n\}$ in a Hilbert space $\mathcal{H}$ which satisfy 


$ \sum\limits\_{n=1}^\infty |\langle f, f\_{n} \rangle |^2 \leq B \|f\|^2 \hspace{2 mm} \text{for some positive}\ B>0 \text{ and all } f\in \mathcal{H}.$


This bound is often easier to establish for a system than the lower bound which additionally holds for frames. The lower frame bound is important for invertibility related questions.


$K$-frames are a relatively recent development, motivated by function systems that arise in sampling problems. They are mentioned in a paper by Feichtinger and Werther, motivated by the fact that some interesting classes of functions (which are not frames) generate proper subspaces that they themselves are not contained in. These systems still have frame-like properties when operator $K$ is bounded.


The second question has different answers depending on who you speak to. One can strengthen properties in the definition of a frame to get things that are still not orthonormal bases but closer to them like *Riesz bases*. The other direction seems to be what you are interested in.


One direction to generalize frame theory is to Banach spaces. Once there one can find systems of functions which frames are proper subsets of. For instance, there are unconditional quasi-bases which frames are examples of but not the other way around (the unconditional here refers to the fact that rearrangements of expansions converge *unconditionally* while a quasi-basis allows for one to re-write any function as an expansion in terms of a sequence of functions and an associated dual system much like one can for frames).


A good introduction to frame theory in its connections to sampling theory is *Foundations of Time-Frequency Analysis* by K. Gröchenig.


