4 expanded tags

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).

The basic algorithm is to find dual sets of eigenvectors/eigenfunctions parametrized by a continuous (e.g., $\omega$ below) or discrete index (e.g., $n$ below), that satisfy completeness and orthogonality relations encapsulated in Dirac delta function resolutions such as that for the FT

$$\delta(x-y)= \int_{-\infty}^{\infty}\exp(i2\pi \omega x)\exp(-i2\pi \omega y)d\omega$$

giving

$$\int_{-\infty}^{\infty}f(y)\delta(x-y)dy=f(x)=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\int_{-\infty}^{\infty}f(y)\exp(-i2\pi \omega y) dy d\omega$$

$$=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\hat{f}(\omega) d\omega,$$

or that for the eigenvectors of Sturm-Liouville differential operators over finite domains

$$\delta(x-y)=\sum_{n=0}^{\infty }\Psi_n(x)\Psi_n^*(y)$$

giving

$$f(x)=\sum_{n=0}^{\infty }\Psi_n(x)\int_{a}^{b}f(y)\Psi_n^*(y) dy,$$

or Kronecker delta resolutions such as that for the associated Laguerre functions

$$\frac{(n+\alpha)!}{n!}\delta_{mn}=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$$

giving

$$f(x)=\sum_{n=0}^{\infty }\frac{n!L_{n}^{\alpha}(x)}{(n+\alpha)!}\hat{f}_n$$

with

$$\hat{f}_n=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$

The basic "physical" operation (BPO) at work here can be regarded as destructive/constructive interference; the product at a point of the value of the function (to be resolved) with the corresponding value of an eigenfunction has a negative or positive value (or phase factor) that may sum constructively or destructively with products at other points (seen as a matched filtering or correlation by replacing $y$ with $x-z$ above). Alternatively, the BPO may be viewed as projection of vectors onto a set of orthonormal axes. In addition, if the function and operations are discretized and/or the domains restricted (in one space or its dual or both, as for the DFT) aliasing (which seems analogous to the introduction of equivalence classes) is introduced and periodicity imposed.

Can you explain the machinery behind the Mukai-Fourier transform in terms of these BPOs or close analogies?

3 changed infinite to finite range of integration as originally intended

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).

The basic algorithm is to find dual sets of eigenvectors/eigenfunctions parametrized by a continuous (e.g., $\omega$ below) or discrete index (e.g., $n$ below), that satisfy completeness and orthogonality relations encapsulated in Dirac delta function resolutions such as that for the FT

$$\delta(x-y)= \int_{-\infty}^{\infty}\exp(i2\pi \omega x)\exp(-i2\pi \omega y)d\omega$$

giving

$$\int_{-\infty}^{\infty}f(y)\delta(x-y)dy=f(x)=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\int_{-\infty}^{\infty}f(y)\exp(-i2\pi \omega y) dy d\omega$$

$$=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\hat{f}(\omega) d\omega,$$

or that for the eigenvectors of Sturm-Liouville differential operators over finite domains

$$\delta(x-y)=\sum_{n=0}^{\infty }\Psi_n(x)\Psi_n^*(y)$$

giving

$$f(x)=\sum_{n=0}^{\infty }\Psi_n(x)\int_{-\infty}^{\infty}f(y)\Psi_n^*(y) \Psi_n(x)\int_{a}^{b}f(y)\Psi_n^*(y) dy,$$

or Kronecker delta resolutions such as that for the associated Laguerre functions

$$\frac{(n+\alpha)!}{n!}\delta_{mn}=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$$

giving

$$f(x)=\sum_{n=0}^{\infty }\frac{n!L_{n}^{\alpha}(x)}{(n+\alpha)!}\hat{f}_n$$

with

$$\hat{f}_n=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$

The basic "physical" operation (BPO) at work here can be regarded as destructive/constructive interference; the product at a point of the value of the function (to be resolved) with the corresponding value of an eigenfunction has a negative or positive value that may sum constructively or destructively with products at other points (seen as a matched filtering or correlation by replacing $y$ with $x-z$ above). Alternatively, the BPO may be viewed as projection of vectors onto a set of orthonormal axes. In addition, if the function and operations are discretized and/or the domains restricted (in one space or its dual or both, as for the DFT) aliasing (which seems analogous to the introduction of equivalence classes) is introduced and periodicity imposed.

Can you explain the machinery behind the Mukai-Fourier transform in terms of these BPOs or close analogies?

2 added 2 characters in body

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).

The basic algorithm is to find dual sets of eigenvectors/eigenfunctions parametrized by a continuous (e.g., $\omega$ below) or discrete index (e.g., $n$ below), that satisfy completeness and orthogonality relations encapsulated in Dirac delta function resolutions such as that for the FT

$$\delta(x-y)= \int_{-\infty}^{\infty}\exp(i2\pi \omega x)\exp(-i2\pi \omega y)d\omega$$

giving

$$\int_{-\infty}^{\infty}f(y)\delta(x-y)dy=f(x)=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\int_{-\infty}^{\infty}f(y)\exp(-i2\pi \omega y) dy d\omega$$

$$=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\hat{f}(\omega) d\omega,$$

or that for the eigenvectors of Sturm-Liouville differential operators over finite domains

$$\delta(x-y)=\sum_{n=0}^{\infty }\Psi_n(x)\Psi_n^*(y)$$

giving

$$f(x)=\sum_{n=0}^{\infty }\Psi_n(x)\int_{-\infty}^{\infty}f(y)\Psi_n^*(y) dy,$$

or Kronecker delta resolutions such as that for the associated Laguerre functions

$$\frac{(n+\alpha)!}{n!}\delta_{mn}=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$$

giving

$$f(x)=\sum_{n=0}^{\infty }\frac{n!L_{n}^{\alpha}(x)}{(n+\alpha)!}\hat{f}_n$$

with

$$\hat{f}n=\int{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$$\hat{f}_n=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$ The basic "physical" operation (BPO) at work here can be regarded as destructive/constructive interference; the product at a point of the value of the function (to be resolved) with the corresponding value of an eigenfunction has a negative or positive value that may sum constructively or destructively with products at other points (seen as a matched filtering or correlation by replacing$y$with$x-z\$ above). Alternatively, the BPO may be viewed as projection of vectors onto a set of orthonormal axes. In addition, if the function and operations are discretized and/or the domains restricted (in one space or its dual or both, as for the DFT) aliasing (which seems analogous to the introduction of equivalence classes) is introduced and periodicity imposed.

Can you explain the machinery behind the Mukai-Fourier transform in terms of these BPOs or close analogies?

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