Hilbert Book Model Project/Quaternionic Field Equations/Fourier Transform

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Fourier spaces[edit]

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.
Inner products[edit]

Remember that

Complex Fourier transform[edit]

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 .

Quaternionic Fourier transform[edit]

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.

Left oriented Fourier transform[edit]

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

Right oriented Fourier transform[edit]

Similarly for the right oriented Fourier transform

Conclusion[edit]

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 .