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In an infinite dimensional Hilbert space, a Fourier transform accomplishes a complete transform of an old orthonormal base
to another orthonormal base
, such that none of the new base vectors can be written as a linear combination that does not include all the old base vectors.
The base vector
is eigenvector of a normal operator
with eigenvalues
. Base
is orthonormal.
Similarly, the base vector
is eigenvector of a normal operator
with eigenvalues
.
The inner product
is a function of both
and
coordinates.
Remember that function
can be represented with respect to an orthonormal base
and operator
as
These equations describe Fourier transform pairs
and the same continuum
. That continuum
is represented by
as well as by
and these functions correspond respectively to the operators
and
. So
and
describe the same thing, which is the continuum
.
The inner product
is a function that fulfills the following corollaries.
- Convolution of functions in the old base
representation becomes multiplication in the new base
representation.
- Similarly, convolution of functions in the new base
representation becomes multiplication in the old base
representation.
- Differentiation in the old base representation becomes multiplication by the new coordinate in the new base representation.
- Similarly, differentiation in the new base representation becomes multiplication by the old coordinate in the old base representation.
Remember that
Fourier transformation is well established for complex functions. We will apply that knowledge by establishing complex parameter spaces inside the quaternionic background parameter space.
If an
axis along the normalized vector
is drawn through the quaternionic background parameter space, then
Here
plays the role of parameter
along direction
and
plays the role of parameter
along direction
.
can be taken in an arbitrary direction and can start at an arbitrary location in the quaternionic background parameter space..
The inner product
relates to a two parametric function that along the direction
corresponds to
Here
and
are complex functions with complex imaginary base number
.
More generally the specification of the quaternionic Fourier must cope with the non-commuting multiplication of quaternionic functions.
We see in the formulas that this method merely achieves a rotation of parameter spaces and functions. In the complex number based Hilbert space, it would achieve no change at all.
The Fourier transform installs only a partial rotation. This results in left and right oriented Fourier transforms.
The left oriented Fourier transform
has an inverse
.
The left oriented Fourier transform is defined by:
For two members
and
of an orthonormal base
holds
For two members
and
of an orthonormal base
holds
The reverse transform is given by
Similarly for the right oriented Fourier transform
The extra value of the right oriented and left oriented Fourier transforms is low. The complex number based Fourier transform has much greater value for the spectral analysis of continuums. However that analysys then restricts to a single direction per case,
Important is the fact that Fourier transform pairs
describe the same continuum
.