THE WAVELET TRANSFORM

1. Front Matter

   Pages i-xiii

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2. An Overview

   • Ram Shankar Pathak

   Pages 1-20

3. TheWavelet Transform on Lp

   • Ram Shankar Pathak

   Pages 21-48

4. Composition ofWavelet Transforms

   • Ram Shankar Pathak

   Pages 49-65

5. TheWavelet Transform on Spaces of Type S

   • Ram Shankar Pathak

   Pages 67-82

6. The Wavelet Transform on Spaces of Type W

   • Ram Shankar Pathak

   Pages 83-91

7. The Wavelet Transform on a Generalized Sobolev Space

   • Ram Shankar Pathak

   Pages 93-107

8. A Class of Convolutions: Convolution for the Wavelet Transform

   • Ram Shankar Pathak

   Pages 109-128

9. The Wavelet Convolution Product

   • Ram Shankar Pathak

   Pages 129-136

10. Asymptotic Expansions of the Wavelet Transform when |b| is Large

    • Ram Shankar Pathak

    Pages 137-153

11. Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a

    • Ram Shankar Pathak

    Pages 155-170

12. Back Matter

    Pages 171-178

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The wavelet transform has emerged as one of the most promising function transforms with great potential in applications during the last four decades. The present monograph is an outcome of the recent researches by the author and his co-workers, most of which are not available in a book form. Nevertheless, it also contains the results of many other celebrated workers of the ?eld. The aim of the book is to enrich the theory of the wavelet transform and to provide new directions for further research in theory and applications of the wavelet transform. The book does not contain any sophisticated Mathematics. It is intended for graduate students of Mathematics, Physics and Engineering sciences, as well as interested researchers from other ?elds. The Fourier transform has wide applications in Pure and Applied Mathematics, Physics and Engineering sciences; but sometimes one has to make compromise with the results obtainedbytheFouriertransformwiththephysicalintuitions. ThereasonisthattheFourier transform does not re?ect the evolution over time of the (physical) spectrum and thus it contains no local information. The continuous wavelet transform (W f)(b,a), involving ? wavelet ?, translation parameterb and dilation parametera, overcomes these drawbacks of the Fourier transform by representing signals (time dependent functions) in the phase space (time/frequency) plane with a local frequency resolution. The Fourier transform is p n restricted to the domain L (R ) with 1 p 2, whereas the wavelet transform can be de?ned for 1 p

• Hilbert transformation
• Processing
• Sobolev space
• Sobolov space
• approximation theory
• convolution
• differential equation
• distribution
• equation
• form
• integral
• partial differential equation
• signal processing
• wavelet
• wavelet transform

• Department of Mathematics, Banaras Hindu University, Varanasi, India

  Ram Shankar Pathak

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