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In this paper localized fractals are studied by using harmonic wavelets. It will be shown that, harmonic wavelets are orthogonal to the Fourier basis. Starting from this, a method is defined for the decomposition of a suitable signal into the periodic and localized parts. For a given signal, the denoising will be done by simply performing a projection into the wavelet space of approximation. It is also shown that due to their self similarity property, a good approximation of fractals can be obtained by a very few instances of the wavelet series. Moreover, the reconstruction is independent on scale as it should be according to the scale invariance of fractals.