Skip to main content

Fourier Transforms in Two or More Variables

  • Chapter
Integral Transforms and Their Applications

Part of the book series: Applied Mathematical Sciences ((AMS,volume 25))

  • 693 Accesses

Abstract

The theory of Fourier transforms of a single variable may be extended to functions of several variables. Thus, if f(x,y) is a function of two variables, the function F(ξ, η) defined by

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8Nrai % aacIcaiiaacqGF+oaEcaGGSaGaa8NBaiaacMcacqGH9aqpdaWdXaqa % aiaa-rgacaWF4baaleaacqGHsislcqGHEisPaeaacqGHEisPa0Gaey % 4kIipakmaapedabaGaa8hzaiaa-LhacaWFLbWaaWbaaSqabeaacaWF % PbGaaiikamXvP5wqSX2qVrwzqf2zLnharyaqbjxAHXgiv5wAJ9gzLb % sttbacfaGaa0NVdiaa9HhacqGHRaWkcaqF3oGaa0xEaiaacMcaaaGc % caWFMbGaaiikaiaa-HhacaGGSaGaa8xEaiaacMcaaSqaaiabgkHiTi % abg6HiLcqaaiabg6HiLcqdcqGHRiI8aaaa!6586!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$F(\xi ,n) = \int_{ - \infty }^\infty {dx} \int_{ - \infty }^\infty {dy{e^{i(\xi x + \eta y)}}f(x,y)} $$
(1)

is the two-dimensional Fourier transform of f(x,y), and, provided that the inversion formula (7.6) may be applied twice, we have

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8Nzai % aa-HcacaWF4bGaa8hlaiaa-LhacaWFPaGaa8xpamaalaaabaGaaGOm % aaqaaiaacIcacaWFYaaccaGae4hWdaNaaiykamaaCaaaleqabaGaaG % OmaaaaaaGcdaWdXaqaaiaa-rgacqGF+oaEdaWdXaqaaiaa-rgacqGF % 3oaAaSqaaiaa-LgacqaHZoWzdaWgaaadbaGaaGOmaaqabaWccqGHsi % slcqGHEisPaeaacaWFPbGaeq4SdC2aaSbaaWqaaiaa-jdaaeqaaSGa % ey4kaSIaeyOhIukaniabgUIiYdaaleaacaWFPbGaeq4SdC2aaSbaaW % qaaiaa-fdaaeqaaSGaeyOeI0IaeyOhIukabaGaa8xAaiabeo7aNnaa % BaaameaacaWFXaaabeaaliabgUcaRiabg6HiLcqdcqGHRiI8aOGaa8 % xzamaaCaaaleqabaGaeyOeI0Iaa8xAaiaa-HcacqaH+oaEcaWF4bGa % a83kaiabeE7aOjaa-LhacaWFPaaaaOGaa8Nraiaa-HcacqaH+oaEca % WFSaGaeq4TdGMaa8xkaiaa-5caaaa!733D!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$f(x,y) = \frac{2}{{{{(2\pi )}^2}}}\int_{i{\gamma _1} - \infty }^{i{\gamma _1} + \infty } {d\xi \int_{i{\gamma _2} - \infty }^{i{\gamma _2} + \infty } {d\eta } } {e^{ - i(\xi x + \eta y)}}F(\xi ,\eta ).$$
(2)

.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Footnotes

  1. This result is given in I. N. Sneddon, J. Eng. Math. (1974), 8, 177, together with a discussion of the connection with the half-space Dirichlet problem for Laplace’s equation.

    Google Scholar 

  2. See DITKIN & PRUDNIKOV (1970) for more information on double Laplace transforms.

    Google Scholar 

  3. See J. C. Jaeger, Bull. Am. Math. Soc. (1940), 46, 687.

    Article  MathSciNet  Google Scholar 

  4. The application of the double Laplace transform to a more general second-order partial differential equation in the quadrant x 2265; 0, y 2265; 0 is discussed in K. Evans and E. A. Jackson, J. Math. Phys. (1971), 12, 2012.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1978 Springer Science+Business Media New York

About this chapter

Cite this chapter

Davies, B. (1978). Fourier Transforms in Two or More Variables. In: Integral Transforms and Their Applications. Applied Mathematical Sciences, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-5512-1_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-5512-1_11

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-90313-2

  • Online ISBN: 978-1-4757-5512-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics