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Saw

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## Summary:

- Fourier transform seems easily understandable as a change of basis operator. However, it is said that time and frequency domains are not bases but vector spaces. Why and what would be the consequences?

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts.

It makes sense, since it is the usual trick so often done in Physics: you have a problem that is not easily solved from the usual perspective; thus you decide to change perspective so as to simplify the operations; once that you have solved your problem, if needed, you shift back to the original perspective. A possible example, among many others: a convolution in the time basis is simply a multiplication in the frequency basis.

Certainly, the usual term is “time domain” and “frequency domain”. But I thought that this is because the analyzed vector is in this case at the same time a function (a signal), where the input (the domain) is the dimensions (time instants in the time basis; frequencies in the frequency basis) and the output is the magnitude of the signal in each dimension, although nothing should prevent the use of the term “basis” if one takes the vector instead of the function stance.

Furthermore, the meaning of the FT expression…

[tex]X(f) = \int_{ - \infty }^\infty {x(t){e^{ - i2\pi ft}}dt} [/tex]

seems to shine out under that change of basis approach:

- The object is a periodical signal (even if it is only so in the limit, at infinity), so it looks logical that the basis vector (here the basis function) is a rotating disk, represented for example by a complex exponential.

- In particular, a basis function in the frequency domain would be a disk rotating at a fixed frequency, and in order to obtain its coordinates in the time basis, one should project it over each time basis vector, i.e. each time instant.

- The next step would be carrying out a dot product (here called inner product) between the analyzing basis vector thus obtained and the analyzed signal, so as to obtain the amount or coordinate of the latter in the relevant frequency.

- There are some peculiarities, but they can be explained as logical adjustments: there is an integral in the inner product instead of a summation because we are adding a set of infinite continuous elements, the direct and indirect inverse transform have opposite signs in the exponent because that is the way to invert the operation (after all phases are added in the exponent)…

- It is true that a vector space with only two bases would look weird, but I have heard that there are fractional Fourier transforms and that the time and frequency bases are the “pure” examples (separated BTW by a right angle), but there are many other intermediate bases, which pick up part of time and part of frequency.

However, I am puzzled to hear that it is categorically said that FT is NOT a “change of basis” operator. See here for example in this sense: https://math.stackexchange.com/ques...basis-or-is-it-a-linear-transformation/123553.

I gather that this warning does not spoil the idea that the FT is a trick to carry out an operation in an easier way since it would still be a transformation from one vector space to a more convenient one.

But I do feel frustrated if this means that the easy understanding of the meaning of the equation is not valid.

So my questions are:

It makes sense, since it is the usual trick so often done in Physics: you have a problem that is not easily solved from the usual perspective; thus you decide to change perspective so as to simplify the operations; once that you have solved your problem, if needed, you shift back to the original perspective. A possible example, among many others: a convolution in the time basis is simply a multiplication in the frequency basis.

Certainly, the usual term is “time domain” and “frequency domain”. But I thought that this is because the analyzed vector is in this case at the same time a function (a signal), where the input (the domain) is the dimensions (time instants in the time basis; frequencies in the frequency basis) and the output is the magnitude of the signal in each dimension, although nothing should prevent the use of the term “basis” if one takes the vector instead of the function stance.

Furthermore, the meaning of the FT expression…

[tex]X(f) = \int_{ - \infty }^\infty {x(t){e^{ - i2\pi ft}}dt} [/tex]

seems to shine out under that change of basis approach:

- The object is a periodical signal (even if it is only so in the limit, at infinity), so it looks logical that the basis vector (here the basis function) is a rotating disk, represented for example by a complex exponential.

- In particular, a basis function in the frequency domain would be a disk rotating at a fixed frequency, and in order to obtain its coordinates in the time basis, one should project it over each time basis vector, i.e. each time instant.

- The next step would be carrying out a dot product (here called inner product) between the analyzing basis vector thus obtained and the analyzed signal, so as to obtain the amount or coordinate of the latter in the relevant frequency.

- There are some peculiarities, but they can be explained as logical adjustments: there is an integral in the inner product instead of a summation because we are adding a set of infinite continuous elements, the direct and indirect inverse transform have opposite signs in the exponent because that is the way to invert the operation (after all phases are added in the exponent)…

- It is true that a vector space with only two bases would look weird, but I have heard that there are fractional Fourier transforms and that the time and frequency bases are the “pure” examples (separated BTW by a right angle), but there are many other intermediate bases, which pick up part of time and part of frequency.

However, I am puzzled to hear that it is categorically said that FT is NOT a “change of basis” operator. See here for example in this sense: https://math.stackexchange.com/ques...basis-or-is-it-a-linear-transformation/123553.

I gather that this warning does not spoil the idea that the FT is a trick to carry out an operation in an easier way since it would still be a transformation from one vector space to a more convenient one.

But I do feel frustrated if this means that the easy understanding of the meaning of the equation is not valid.

So my questions are:

- Is the understanding that FT is not a change of basis but a transformation between vector spaces a peaceful idea or is there some debate about it?
- Even if the answer is that it’s a transformation between vector spaces, can we still use the idea that it is an inner product? With what then, if it is not with a “basis” function?
- The link above seems to identify the reason that there would be no canonical isomorphism between the two “spaces”. If so, why that? Really, I can hardly imagine things having more structural similarity than time and frequency, which are two sides of the same coin…

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