What is the intuition behind this and what does it signify? I do not have a strong mathematical background but have worked on wavelets as a signal processing tool. I would like to learn the mathematics behind it.
Sami is right. The problem arises when the wavelet in hand is a frame and not a basis. In this case the question is whether the coefficients of the frame (i.e. the projections of the signal on the elements of the frame/wavelet) can determine uniquely the signal. A tight frame means that there is a unique correspondance between a signal and its frame coefficients just like a basis. Having this property enables signal processing by means of wavelet transform + coefficient manipulation + going back from the manipulated coefficient back to an ameliorated signal.
To be honest I'm not an expert on this field either, but when one considers a frame as an overcomplete basis you would expect that the mapping for a point in the frame can be represented in many ways. I.e. single vector can be presented in multiple ways. This leads to situation where inner product of the frame (basis) and a coefficient vector may have different values for the same "point", thus the frame have different "lower" and "upper" bounds. With a special case, like with wavelets here, the bounds in fact are the same and the frame is considered to be tight and in "energy-wise" the presentation is unique for a point.
I'm not 100% sure if this is the reason for wavelets having the property to describe signal "frequency" behavior locally, instead of global analysis like with Fourier basis. I'm also interested in the correct answer as I may have misunderstood something...
Sami is right. The problem arises when the wavelet in hand is a frame and not a basis. In this case the question is whether the coefficients of the frame (i.e. the projections of the signal on the elements of the frame/wavelet) can determine uniquely the signal. A tight frame means that there is a unique correspondance between a signal and its frame coefficients just like a basis. Having this property enables signal processing by means of wavelet transform + coefficient manipulation + going back from the manipulated coefficient back to an ameliorated signal.
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