An Overview
Ram Shankar Pathak · Atlantis studies in mathematics for engineering and science · 2009
In this chapter we present some elements of theory of classical and distributional Fourier and Hilbert transforms that are to be exploited in the development of the theory of the wavelet transform in L p -spaces ( $$1 \leqslant p<\infty$$ ) and in certain distribution spaces. For this purpose an elementary theory of distributions is given that is essential for proper understanding of various developments in the theory of wavelet transform presented in this book. The relationship between the wavelet transform and Fourier transform is well known. We shall find relations of the wavelet transform with Hilbert transform and Riesz fractional integral operator. The application of Fourier transform restricts itself to L p -spaces with $$1 \leqslant p \leqslant 2$$ but Hilbert transform approach gives results valid in L p -spaces with ( $$1 \leqslant p<\infty$$ ). Hilbert transform has applications in signal processing, aerofoil problems, dispersion relations, high-energy physics, potential theory problems and others [53]. Therefore, using aforesaid relation the wavelet transform can also be applied to tackle all such problems. A brief idea about convolution operator, and asymptotic expansion of a general integral transform is also given, that will form a basis for development of corresponding results for the wavelet transform.