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.
References
- Keener, James P. 2000. Principles of Applied Mathematics: Transformation and Approximation. Cambridge: Westview Press. ISBN 0-7382-0129-4
Notes
- ^ K.B. Wolf, "Integral Transforms in Science and Engineering", New York, Plenum Press, 1979.
- ^ Almeida, Luís B. (1994). "The fractional Fourier transform and time–frequency representations". IEEE Trans. Signal Process. 42 (11): 3084–3091.
- ^ J.J. Healy, M.A. Kutay, H.M. Ozaktas and J.T. Sheridan, "Linear Canonical Transforms: Theory and Applications", Springer, New York 2016.
