Transform theory

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In mathematics, transform theory is the study of transforms. 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.

Contents 1 Spectral theory 2 Transforms 2.1 Laplace transform 2.2 Fourier transform 2.3 Mellin tranform 2.4 Hankel transform 2.5 Z transforms 3 See Also 4 References

[edit] Spectral theory

In Spectral theory the spectral decomposition theorem gives the conditions under which a matrix may be diagonalized. It says that if A is an n x n self-adjoint matrix, there is an orthogonal basis x_i for which (1) A is diagonalized by x (2) the elements of x are orthogonal (3) the inverse of the transpose of x is x (4) xAx^T is a diagonal matrix.

[edit] Transforms

[edit] Laplace transform

Laplace transform

[edit] Fourier transform

Fourier transform

[edit] Mellin tranform

Mellin transform

[edit] Hankel transform

Hankel transform

[edit] Z transforms

Z-transform

[edit] See Also

• Spectral theory
• Laplace transform
• Fourier transform
• Transform (mathematics)

[edit] References

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