The wavelet transform has a powerful time-frequency analysis and signal-coding tool suitable for use in the manipulation of complex nonstationary signals. This article provides an overview of the emerging role of wavelet-transform analysis in biomedical signal processing and analysis. It also provides a brief overview of the theory of the transform in its two distinct and very different forms: continuous and discrete. In conclusion, it has been shown that the wavelet transform is a flexible time-frequency decomposition tool that can form the basis of useful signal analysis, and coding schemes. It is envisaged that the future will see further application of the wavelet transform to biomedical signal analysis, as the emerging technologies based on them are honed for practical purposes.

This paper deals with Chebyshev wavelets. We analyze their properties computing their Fourier transform. Moreover, we discuss the differential properties of Chebyshev wavelets due to the connection coefficients. Uniform convergence of Chebyshev wavelets and their approximation error allow us to provide rigorous proofs. In particular, we expand the mother wavelet in Taylor series with an application both in fractional calculus and fractal geometry. Finally, we give two examples concerning the main properties proved.