Looking for Material on Wavelets

In summary, the person is looking for a good beginner text on Wavelets at an undergraduate or early graduate level. They are missing background material in infinite dimensional vector spaces/function spaces and Fourier analysis. They received suggestions from others, including a website and book recommendations. They also mention a book that covers wavelets and various applications, though it has a nonconventional treatment of Lebesgue integration.
  • #1
stephenkeiths
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Hi, I'm looking for a good beginner text on Wavelets. Preferably an undergraduate or early graduate level. The background material that I'm missing most is infinite dimensional vector spaces/function spaces and Fourier analysis.

Any advice would really be appreciated!

Thanks!
 
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  • #2
stephenkeiths said:
Hi, I'm looking for a good beginner text on Wavelets. Preferably an undergraduate or early graduate level. The background material that I'm missing most is infinite dimensional vector spaces/function spaces and Fourier analysis.
Maybe this is not quite the right forum for wavelets, but there's lots of material on Gerry Kaiser's website: http://www.wavelets.com/

For the other stuff, maybe Folland's book on Fourier analysis that Micromass recommended recently. Or if you want something with more sophisticated functional analysis, maybe try Kreyszig.
 
  • #3
strangerep said:
Maybe this is not quite the right forum for wavelets, but there's lots of material on Gerry Kaiser's website: http://www.wavelets.com/

For the other stuff, maybe Folland's book on Fourier analysis that Micromass recommended recently. Or if you want something with more sophisticated functional analysis, maybe try Kreyszig.

Neither Folland or Kreyszig covers wavelets though :frown:
But https://www.amazon.com/dp/0122084381/?tag=pfamazon01-20 is a very good book which does covers the basics of wavelets and many other applications (older editions don't have wavelets, so be sure to get a new edition). Although I have to admit that his treatment of Lebesgue integration is rather nonconventional...

There are better functional analysis books out there, but if you're interested in seeing tons of applications then this is the best book.
 
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1. What are wavelets and how are they used in scientific research?

Wavelets are mathematical functions used to analyze data and signals in various fields of science. They are used in a wide range of applications, such as image and signal processing, data compression, and pattern recognition. Wavelets allow for a more efficient representation of data compared to traditional Fourier analysis, which makes them useful for analyzing signals with irregularities or discontinuities.

2. What are the advantages of using wavelets over other analysis techniques?

One of the main advantages of wavelets is their ability to capture localized features in data, making them suitable for analyzing signals with both high and low frequency components. They also have a multi-resolution property, meaning they can analyze data at different scales, providing a more detailed representation of the data compared to other techniques. Additionally, wavelets have a compact support, which means they only operate on a specific region of the data, making them computationally efficient.

3. Where can I find material on wavelets for beginners?

There are many resources available for beginners to learn about wavelets. Some popular books include "A Wavelet Tour of Signal Processing" by Stephane Mallat and "Wavelets and Filter Banks" by Gilbert Strang and Truong Nguyen. There are also online tutorials and courses, such as those offered by the IEEE Signal Processing Society and Coursera.

4. How are wavelets related to Fourier analysis?

Wavelets and Fourier analysis are both techniques used for analyzing signals and data. While Fourier analysis decomposes a signal into its frequency components, wavelets use a time-scale approach to analyze data at different scales. This allows wavelets to better capture localized features and provide a more detailed representation of the data.

5. What are some real-world applications of wavelets?

Wavelets have many practical applications in various fields, including image and signal processing, data compression, and biomedicine. They are used in medical imaging to enhance the quality of MRI and CT scans, in audio and video compression to reduce file sizes while maintaining quality, and in financial analysis to analyze stock market data. Wavelets also have applications in weather forecasting, speech recognition, and earthquake detection.

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