I believe that the approximation with be a filtered version of your original signal that will have frequency components depending on the order of filtering used.

In de-noising (decomposition-thresholding-reconstruction) problem, you can say a sampled signal is sum of approximation and detail coefficients, because both of them are output of high-pass and low-pass filter.

Yes the last level approximation and the remaining details can be added together to create the original signal. It can be said that the original signal has been decomposed into the details and the last level approximation that you have. Both of what the people before me have said is also true.

Wavelet transform if seen in the frequency domain is the decomposition of the time frequency plain and the complete coverage of the plane is guaranteed. 

hi

If I understood correctly it seems that what you are looking for is the  ability to localize and spot peculiarities in the time domain signal while eliminating noise.Seems quite a task as I expect an ecg signal of a critically ill patient to be non stationary.

My 2c views.:

1. We need a criteria to distinguish noise (if random). Deterministic noise e,g, hum could be filtered.

2. EMD method seems more appropriate compared to wavelet transform for this application as we can identify modes which we can classify.

3 It may be difficult to identify and classify  "peculiarities " from wavelet transform analysis without further use of Bayesian methods etc.

4. A hunch  SVD (PCA) method could be tried if you are looking only for speed/data compression

Cheers

This may be a more mathematical answer than you are looking for, but nonetheless, here it is.  Suppose momentarily that you just have a wavelet expansion of a noiseless signal.  Then since the wavelet forms a basis, you need all of the coefficients to represent the signal.  In that sense, they are exact.

However, one of the great features of wavelets is that they allow you to essentially prioritize different scales.  For example, if you have a signal that is very flat or doesn't have much variation, then you may care more about the coefficients of the wavelets that have large support, whereas if you have a spike in the signal, you can "zoom in" as it were by using the smaller scale wavelet elements.  This is what your question made me think of, in that, the wavelet elements with large support allow you to see coarse features in the signal, whereas the small support elements allow you to see fine detail.  However, this says nothing about the coefficients. 

Dear Rastogi, 

from theoretical point of view,  the main merits of wavelet transform is the ability to capture the signal in time-frequency components. By other words,  if you have a signal that is very flat or doesn't have much variation, then you may care more about the coefficients of the wavelets that have large support..

In addition, choice of wavelet basis function affects “compactness” of wavelet decomposition. so the approximation with  a filtered version of your original signal that will have frequency components is mainly based on the filtering order.