Fourier transform on manifolds

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Thread starter redtree
Start date Mar 5, 2019
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Curvature Fourier Fourier transform Manifold Manifolds Transform

In summary, the Fourier transform on manifolds is a mathematical tool used to decompose functions defined on curved and multi-dimensional geometric objects, known as manifolds, into their frequency components. It differs from the traditional Fourier transform by taking into account the underlying geometry of the manifold in its calculations. Some real-world applications include signal processing, image and data analysis, and differential geometry. However, the Fourier transform on manifolds can be computationally expensive and may not be suitable for non-smooth functions. To learn more, resources such as books, research papers, and seminars are recommended, along with a strong understanding of traditional Fourier analysis and differential geometry.

Mar 5, 2019

redtree

Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?

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Mar 5, 2019

Orodruin

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The corresponding thing in a (Riemannian or pseudo-Riemannian) manifold would be to look for expansions in terms of eigenfunctions of the operator ##\nabla^2 = \nabla_a \nabla^a = g^{ab} \nabla_a \nabla_b##.

1. What is a Fourier transform on manifolds?

A Fourier transform on manifolds is a mathematical tool used to decompose a function defined on a manifold into its frequency components. It allows us to analyze the behavior of a function on a curved surface or space.

2. How is a Fourier transform on manifolds different from a traditional Fourier transform?

A traditional Fourier transform is defined on Euclidean spaces, while a Fourier transform on manifolds is defined on more general spaces with curvature. This makes it applicable to a wider range of problems and allows for a more accurate representation of functions on curved spaces.

3. What are some applications of Fourier transform on manifolds?

Fourier transform on manifolds has various applications in fields such as signal processing, image and data analysis, and differential geometry. It is also used in physics, particularly in quantum mechanics and general relativity.

4. How is a Fourier transform on manifolds calculated?

The calculation of a Fourier transform on manifolds involves first defining a suitable transform operator on the manifold, which depends on the geometry and structure of the manifold. The function is then decomposed into its frequency components using this operator.

5. What are some challenges in using Fourier transform on manifolds?

One of the main challenges in using Fourier transform on manifolds is the lack of a universal definition for the transform operator, as it depends on the specific manifold. Another challenge is the computational complexity, as the calculation of the transform can be more involved on curved spaces compared to Euclidean spaces.