Why should a Fourier transform not be a change of basis?

In summary, the Fourier transform can be seen as a change of basis operator or a transformation between vector spaces, depending on mathematical semantics. However, it is commonly used and understood as a change of basis in physics and engineering contexts. There may be debates about its true nature, but for practical purposes, it can still be thought of as an inner product and used to simplify operations in different domains. The reason for the lack of a canonical isomorphism between the time and frequency domains is not clear, but both domains are closely related and can be thought of as different perspectives of the same underlying signal.
  • #36
I agree that is true, but it looks to me like the locally compact group considered in the OP and throughout the throughout the thread is ##(\mathbb{R},+),## so I'm not sure why the Fourier transform on a circle is directly relevant.

Working over ##\mathbb{R}##, you can definitely view the Fourier transform as map from a space to itself, e.g. use the space of Schwartz functions. I still agree it doesn't look like a change of basis.
 
Physics news on Phys.org
  • #37
The other heuristic that suggests looking at the Fourier transform as something like a change of basis is the fact that it diagonalizes differentiation. Wikipedia's article on Spectral Theory makes the interesting comment:

"There have been three main ways to formulate spectral theory, each of which find use in different domains. After Hilbert's initial formulation, the later development of abstract Hilbert spaces and the spectral theory of single normal operators on them were well suited to the requirements of physics, exemplified by the work of von Neumann.[5] The further theory built on this to address Banach algebras in general. This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative harmonic analysis.

The difference can be seen in making the connection with Fourier analysis. The Fourier transform on the real line is in one sense the spectral theory of differentiation qua differential operator. But for that to cover the phenomena one has already to deal with generalized eigenfunctions (for example, by means of a rigged Hilbert space). On the other hand it is simple to construct a group algebra, the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of Pontryagin duality."
 
  • #38
Infrared said:
I agree that is true, but it looks to me like the locally compact group considered in the OP and throughout the throughout the thread is ##(\mathbb{R},+),## so I'm not sure why the Fourier transform on a circle is directly relevant.

Working over ##\mathbb{R}##, you can definitely view the Fourier transform as map from a space to itself, e.g. use the space of Schwartz functions. I still agree it doesn't look like a change of basis.
Well, I was thinking about Fourier in general, and the circle case makes the point obvious. I also think that it doesn't look like a change of bases. Even more, I am not sure how standard the terminology about position and momentum basis is, but it is somewhat misleading. The spaces in question have countable bases, while these position and momentum "bases" are uncountable.
 
  • #39
In some cases, the energy basis is a true basis that is countable, while position and momentum "bases" are generalized "bases" that are uncountable. See the rigged Hilbert spaces references above. It is interesting that even for vector spaces, whether the basis is countable or not depends on the definition, and may differ between a Fourier basis and a Hamel basis.
 
  • #40
atyy said:
In some cases, the energy basis is a true basis that is countable, while position and momentum "bases" are generalized "bases" that are uncountable. See the rigged Hilbert spaces references above. It is interesting that even for vector spaces, whether the basis is countable or not depends on the definition, and may differ between a Fourier basis and a Hamel basis.
The rigged Hilbert spaces do not change anything about the fact that the position "basis" is uncountable. I didn't see where in the references it is called a basis.
 
  • Like
Likes FactChecker
  • #43
martinbn said:
I didn't see where in the references it is called a basis.

Take a look at post $34.
 
  • #44
atyy said:
Take a look at post $34.
I did, it desn't change anything. Refferences to rigged Hilbert spaces will not change the fact that the exponentals are uncountable and the spaces are seperable.
 
<h2>1. Why can't a Fourier transform be considered a simple change of basis?</h2><p>A Fourier transform is a mathematical operation that converts a signal from its original domain (such as time or space) to a representation in the frequency domain. This involves decomposing the signal into its constituent frequencies, and therefore cannot be considered a simple change of basis. Unlike a traditional change of basis, a Fourier transform involves a complex mathematical process that fundamentally alters the representation of the signal.</p><h2>2. What is the difference between a Fourier transform and a change of basis?</h2><p>A change of basis involves converting a vector or signal from one set of basis vectors to another, while a Fourier transform involves decomposing a signal into its frequency components. While both processes involve a change in representation, a Fourier transform is a more complex and specialized operation that is specific to signals and their frequencies.</p><h2>3. Can a Fourier transform be reversed like a change of basis?</h2><p>A Fourier transform can be reversed using an inverse Fourier transform, but this is not the same as a traditional change of basis. The inverse Fourier transform converts a signal from the frequency domain back to its original domain, but it does not simply switch between two sets of basis vectors. The inverse Fourier transform is a complex mathematical operation that involves integrating over the entire frequency spectrum.</p><h2>4. Why is a Fourier transform important in signal processing?</h2><p>A Fourier transform is important in signal processing because it allows us to analyze signals in the frequency domain. This can provide valuable insights into the underlying components and patterns of a signal, which can be useful for tasks such as filtering, compression, and noise reduction. In addition, many physical phenomena can be described and understood in terms of their frequency components, making the Fourier transform a powerful tool in scientific research.</p><h2>5. Are there any situations where a Fourier transform can be considered a change of basis?</h2><p>In some cases, a Fourier transform can be considered a change of basis. For example, if the original signal is already represented in terms of its frequency components (such as a sinusoidal signal), then a Fourier transform can be seen as simply converting between different sets of basis vectors. However, this is a special case and does not apply to most signals, which require a more complex transformation to be represented in the frequency domain.</p>

1. Why can't a Fourier transform be considered a simple change of basis?

A Fourier transform is a mathematical operation that converts a signal from its original domain (such as time or space) to a representation in the frequency domain. This involves decomposing the signal into its constituent frequencies, and therefore cannot be considered a simple change of basis. Unlike a traditional change of basis, a Fourier transform involves a complex mathematical process that fundamentally alters the representation of the signal.

2. What is the difference between a Fourier transform and a change of basis?

A change of basis involves converting a vector or signal from one set of basis vectors to another, while a Fourier transform involves decomposing a signal into its frequency components. While both processes involve a change in representation, a Fourier transform is a more complex and specialized operation that is specific to signals and their frequencies.

3. Can a Fourier transform be reversed like a change of basis?

A Fourier transform can be reversed using an inverse Fourier transform, but this is not the same as a traditional change of basis. The inverse Fourier transform converts a signal from the frequency domain back to its original domain, but it does not simply switch between two sets of basis vectors. The inverse Fourier transform is a complex mathematical operation that involves integrating over the entire frequency spectrum.

4. Why is a Fourier transform important in signal processing?

A Fourier transform is important in signal processing because it allows us to analyze signals in the frequency domain. This can provide valuable insights into the underlying components and patterns of a signal, which can be useful for tasks such as filtering, compression, and noise reduction. In addition, many physical phenomena can be described and understood in terms of their frequency components, making the Fourier transform a powerful tool in scientific research.

5. Are there any situations where a Fourier transform can be considered a change of basis?

In some cases, a Fourier transform can be considered a change of basis. For example, if the original signal is already represented in terms of its frequency components (such as a sinusoidal signal), then a Fourier transform can be seen as simply converting between different sets of basis vectors. However, this is a special case and does not apply to most signals, which require a more complex transformation to be represented in the frequency domain.

Similar threads

Replies
12
Views
3K
  • Classical Physics
2
Replies
47
Views
1K
  • Quantum Physics
Replies
4
Views
752
  • Linear and Abstract Algebra
Replies
8
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
9
Views
2K
  • Linear and Abstract Algebra
Replies
16
Views
4K
Replies
4
Views
2K
  • Atomic and Condensed Matter
Replies
1
Views
854
  • Calculus and Beyond Homework Help
Replies
1
Views
733
Back
Top