Math 2.0

Harmonic analysis on Euclidean spaces Fourier transform Example

Rectangular function

Builds on Fourier transformConvolution theoremBand-limited function

Example

The rectangular function is rect⁡ ⁣:ℝ→ℝ\operatorname{rect} \colon \R \to \R with rect⁡(x)=1\operatorname{rect}(x) = 1 for ∣x∣<12\abs{x} < \frac12, rect⁡(x)=12\operatorname{rect}(x) = \frac12 for ∣x∣=12\abs{x} = \frac12, and rect⁡(x)=0\operatorname{rect}(x) = 0 for ∣x∣>12\abs{x} > \frac12. It is integrable, so its Fourier transform is defined by the integral. For ξ≠0\xi \ne 0, rect⁡^(ξ)=∫−1/21/2e−2πixξ dx=eπiξ−e−πiξ2πiξ=sin⁡(πξ)πξ,\widehat{\operatorname{rect}}(\xi) = \int_{-1/2}^{1/2} e^{-2\pi i x \xi}\, dx = \frac{e^{\pi i \xi} - e^{-\pi i \xi}}{2\pi i \xi} = \frac{\sin(\pi \xi)}{\pi \xi}, and rect⁡^(0)=1\widehat{\operatorname{rect}}(0) = 1. This function is the normalized sinc function, sinc⁡(ξ)=sin⁡(πξ)/(πξ)\operatorname{sinc}(\xi) = \sin(\pi\xi)/(\pi\xi).

Remarks

The transform is continuous and tends to 00 at infinity, as for every integrable function. It decays only like 1/∣ξ∣1/\abs{\xi} and is not integrable, which reflects the jumps of rect⁡\operatorname{rect}. So the inversion formula does not hold as an absolutely convergent integral. It holds in L2(ℝ)L^2(\R). As a symmetric improper integral it gives 12\frac12 at x=±12x = \pm\frac12, the average of the two one-sided limits, which is why the value 12\frac12 is used there.

By the convolution theorem, the triangle function rect⁡∗rect⁡(x)=max⁡(1−∣x∣,0)\operatorname{rect} * \operatorname{rect}(x) = \max(1 - \abs{x}, 0) has transform sinc⁡2\operatorname{sinc}^2, which is integrable. Read in the other direction, sinc⁡\operatorname{sinc} is a band-limited function whose transform is rect⁡\operatorname{rect}.

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To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Band-limited function, concept, no evidence levelBand-limited functionconceptConvolution, concept, no evidence levelConvolutionconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFourier transform, concept, no evidence levelFourier transformconceptFunction, concept, no evidence levelFunctionconceptLp space, concept, no evidence levelLp spaceconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptIterated integral, definition, no evidence levelIterated integraldefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDouble integral, definition, no evidence levelDouble integraldefinitionFubini's theorem, statement, no evidence levelFubini's theoremstatementConvolution theorem, statement, no evidence levelConvolution theoremstatementRectangular function, example, no evidence levelRectangular functionexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Band-limited function concept
  2. Convolution concept
  3. Euclidean space object
  4. Fourier transform concept
  5. Function concept
  6. Lp space concept
  7. Function of several variables concept
  8. Iterated integral definition
  9. Limit of a function of several variables concept
  10. Continuity in R^n definition
  11. Double integral definition
  12. Fubini's theorem statement
  13. Convolution theorem statement
  14. Rectangular function example

Further reading

  1. Web https://en.wikipedia.org/wiki/Rectangular_function
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MSC42A38, 94A12
IllustratesFourier transform
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