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 transformsK.B. Wolf, "Integral Transforms in Science and Engineering", New York, Plenum Press, 1979. such as the Fourier transform, the fractional Fourier Transform,Almeida, Luís B. (1994). "The fractional Fourier transform and time–frequency representations". [https://ieeexplore.ieee.org/document/330368 IEEE Trans. Signal Process. 42 (11): 3084–3091]. the Laplace transform, and linear canonical transformations.J.J. Healy, M.A. Kutay, H.M. Ozaktas and J.T. Sheridan, "Linear Canonical Transforms: Theory and Applications", Springer, New York 2016. 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
References
Notes
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