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Abstract Harmonic Analysis of Continuous Wavelet Transforms

Edgar Allan Poe

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Published in Abstract Harmonic Analysis Of Continuous Wavelet Transforms (Lecture Notes In Mathematics 1863)

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The continuous wavelet transform (CWT) has revolutionized the fields of signal processing, image processing, and time-frequency analysis. Its ability to decompose signals into a time-scale representation has made it an indispensable tool for analyzing complex data. However, to fully understand the inner workings of the CWT, a solid foundation in abstract harmonic analysis is essential.

In this comprehensive article, we delve into the abstract harmonic analysis of continuous wavelet transforms. We explore the key concepts, theorems, and applications that underpin this transformative tool, providing a thorough understanding for both seasoned researchers and aspiring students alike.

The CWT is rooted in the theory of abstract harmonic analysis, which provides a framework for studying the representation of functions in terms of frequency. At the heart of this theory lies the concept of the Fourier transform, which decomposes a function into its constituent sinusoids.

Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics 1863)

Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics Book 1863)

by Roger Joseph Boscovich

4.2 out of 5

Language	:	English
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The CWT extends the Fourier transform by introducing the concept of scale. By dilating and translating a single "mother" wavelet function, the CWT generates a family of wavelets that can capture both the frequency and scale content of a signal.

Several key theorems provide the mathematical underpinnings for the CWT. These theorems establish the existence, uniqueness, and properties of the CWT.

Existence and Uniqueness Theorem: Under certain conditions, the CWT of a square-integrable function exists and is unique.
Inversion Theorem: The CWT can be inverted, allowing for the reconstruction of the original signal from its wavelet representation.
Plancherel's Theorem: The CWT preserves the energy of the signal, meaning that the total energy in the signal is equal to the total energy in its wavelet representation.

The CWT has found widespread applications in various fields, including:

Signal Processing: Noise removal, feature extraction, signal compression
Image Processing: Image enhancement, texture analysis, object detection
Time-Frequency Analysis: Music analysis, speech recognition, biomedical signal processing

Abstract harmonic analysis provides the mathematical foundations for understanding the continuous wavelet transform. By exploring the key concepts, theorems, and applications, we gain a deeper appreciation for this transformative tool and its impact on various disciplines.

Whether you are a seasoned researcher seeking to advance your knowledge or an aspiring student eager to embark on a mathematical journey, this article offers a comprehensive guide to the abstract harmonic analysis of continuous wavelet transforms. Embrace the mathematical beauty and power of this transformative tool and unlock its potential for groundbreaking discoveries and innovations.

Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics Book 1863)

by Roger Joseph Boscovich

4.2 out of 5