Alexandre: MATLAB did not used to have the inverse cwt function in their wavelet toolbox as of 2008. Hence, at that time, it was filling a gap--that matlab hadn't yet implemented. I have not used the newer version of matlab wavelet toolbox, but my guess is that it is the same concept, but less full-fledged.
Zahra: CWT does not share the same notion of details and approximations--that is DWT only.
If by "instantaneous frequency" you mean the "equivalent Fourier frequency" (i.e. sine wave in infinite time domain at oscillating at a single frequency), then here's the answer. The CWT "pseudofrequency" depends on the mother wavelet you use for the transform. Each mother wavelet has a corresponding "center frequency", and the relation is given as:
f = centerfrq(mother)/(a*delta).
a = scale
delta = sampling period.
I strongly encourage you to read up on wavelet theory, in order to become a competent user of this code.
Thanks for this fantastic code!
A question regarding time series with gaps:
As I understand, the Haar wavelet will ignore gaps, as long as they are replaced by zeros and the time series is zero-meaned. If this is correct, could you include (either here or in the wave_bases.m file) code for the Haar wavelet?