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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 44: Integral transforms, operational calculus

Introduction

Integral transforms include the Fourier transform (see section 43) as well as the transforms of Laplace, Radon, and others. (The general theory of transformations between function spaces is part of Functional Analysis, section 46) Also includes convolution operators and operational calculi.

History

Applications and related fields

For fractional derivatives and integrals, See 26A33. For Fourier transforms, See 42A38, 42B10. For integral transforms in distribution spaces, See 46F12. For numerical methods, See 65R10

Subfields

There is only one division (44A) but it is subdivided:

• 44A05: General transforms, See also 42A38
• 44A10: Laplace transform
• 44A12: Radon transform, See also 92C55
• 44A15: Special transforms (Legendre, Hilbert, etc.)
• 44A20: Transforms of special functions
• 44A30: Multiple transforms
• 44A35: Convolution
• 44A40: Calculus of Mikusinski and other operational calculi
• 44A45: Classical operational calculus
• 44A55: Discrete operational calculus
• 44A60: Moment problems
• 44A99: Miscellaneous topics

This is one of the smallest areas in the Math Reviews database.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Widder, D. V.: "What is the Laplace transform?" Amer. Math. Monthly 52, (1945). 419--425. MR7,155c

Software and tables

Other web sites with this focus

• Here are the AMS and Goettingen resource pages for area 44.

Selected topics at this site

• What's an intuitive description of the Laplace transform?
• Relationship between Laplace and Fourier transforms?
• Finding the inverse Laplace transform
• Pointers to Laplace Transform data
• Geometric interpretation of Laplace transform as a linear map
• Riesz transforms and others as generalizations of the Hilbert Transform
• What is convolution (and more general integration kernels)?
• Applications of kernel functions to the solutions of PDEs.
• What is Van der Corput's Lemma (on boundedness of integral transforms)