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Geometric algebra integral transforms and their applications
1Department of Natural Science, College of Liberal Arts, International Christian University , Mitaka, Tokyo, Japan.
Summary
Geometric algebra unifies integral transforms like Fourier and wavelets into higher-dimensional Clifford algebra. This review explores these generalized transforms and their applications in math, physics, and engineering.
Area of Science:
- Mathematics
- Physics
- Engineering
Background:
- Integral transforms like Fourier and wavelets are fundamental in various scientific fields.
- Geometric algebra provides a unified framework for mathematical concepts.
- Generalizing transforms within geometric algebra offers new analytical possibilities.
Purpose of the Study:
- To introduce the foundations of generalized integral transforms in geometric algebra.
- To provide examples of higher-dimensional quaternionic, octonionic, and Clifford GA transforms.
- To highlight applications of these generalized transforms in mathematics, physics, and engineering.
Main Methods:
- Reviewing existing literature on integral transforms and geometric algebra.
- Developing theoretical frameworks for generalizing transforms within Clifford algebra.
- Presenting representative examples of novel transforms.
Main Results:
- Demonstrated the generalization of real and complex integral transforms to higher dimensions using geometric algebra.
- Introduced quaternionic, octonionic, and Clifford GA versions of transforms.
- Showcased the applicability of these generalized transforms.
Conclusions:
- Geometric algebra offers a powerful framework for unifying and extending integral transforms.
- These generalized transforms have significant potential in advanced mathematics, physics, and engineering.
- This work lays the groundwork for future research in higher-dimensional transform analysis.
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