Transform theory

In mathematics, transform theory is the study of transforms, which relate a function in one domain to another function in a second domain. The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or diagonalized as in spectral theory.

Main examples of transforms that are both well known and widely applicable include integral transforms^[1] such as the Fourier transform, the fractional Fourier Transform,^[2] the Laplace transform, and linear canonical transformations.^[3] These transformations are used in signal processing, optics, and quantum mechanics.

Spectral theory

In spectral theory, the spectral theorem says that if A is an n×n self-adjoint matrix, there is an orthonormal basis of eigenvectors of A. This implies that A is diagonalizable.

Furthermore, each eigenvalue is real.

Transforms

• Laplace transform
• Fourier transform
• Fractional Fourier Transform
• Linear canonical transformation
• Wavelet transform
• Hankel transform
• Joukowsky transform
• Mellin transform
• Z-transform

References

• Keener, James P. 2000. Principles of Applied Mathematics: Transformation and Approximation. Cambridge: Westview Press. ISBN 0-7382-0129-4