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<ref>K.B. Wolf, "''Integral Transforms in Science and Engineering''", New York, Plenum Press, 1979.</ref> such as the Fourier transform, the fractional Fourier Transform,<ref>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].</ref> the Laplace transform, and linear canonical transformations.<ref>J.J. Healy, M.A. Kutay, H.M. Ozaktas and J.T. Sheridan, "''Linear Canonical Transforms: Theory and Applications''", Springer, New York 2016.</ref> 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}}
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== Notes == <references /> {{linear-algebra-stub}}