Math 2.0

Harmonic analysis on Euclidean spaces Convergence of Fourier series Theorem

Riemann localization principle

Builds on Fourier seriesRiemann–Lebesgue lemmaDirichlet kernelPeriodic function

Statement

Let f,g ⁣:ℝ→ℂf, g \colon \R \to \C be periodic functions of period 2π2\pi that are integrable on [−π,π][-\pi, \pi]. Let SNfS_N f and SNgS_N g be the partial sums of their Fourier series. Let x0∈ℝx_0 \in \R and δ>0\delta > 0, and suppose that f(x)=g(x)f(x) = g(x) for almost every xx in (x0−δ,x0+δ)(x_0 - \delta, x_0 + \delta). Then lim⁡N→∞(SNf(x0)−SNg(x0))=0.\lim_{N \to \infty} \big(S_N f(x_0) - S_N g(x_0)\big) = 0. In particular the Fourier series of ff converges at x0x_0 if and only if that of gg does, and then to the same limit.

Remarks

Put h=f−gh = f - g; then hh vanishes near x0x_0, and SNh(x0)=12π∫δ≤∣t∣≤πh(x0−t)DN(t) dtS_N h(x_0) = \frac{1}{2\pi} \int_{\delta \le \abs{t} \le \pi} h(x_0 - t) D_N(t) \, dt with the Dirichlet kernel DND_N. On δ≤∣t∣≤π\delta \le \abs{t} \le \pi the factor 1sin⁡(t/2)\frac{1}{\sin(t/2)} is bounded, so t↦h(x0−t)sin⁡(t/2)t \mapsto \frac{h(x_0 - t)}{\sin(t/2)} is integrable there. The Riemann–Lebesgue lemma applied to sin⁡((N+12)t)\sin((N + \frac{1}{2}) t) gives SNh(x0)→0S_N h(x_0) \to 0. The same argument shows that SNh→0S_N h \to 0 uniformly on every closed interval inside the interval where hh vanishes. Example: a function equal to x2x^2 near x0=1x_0 = 1 has a Fourier series converging to 11 at x0x_0, whatever its values far from x0x_0. By contrast, each Fourier coefficient depends on the values of ff on the whole period.

Going deeper

For a student

Localization reduces convergence questions to local behaviour, which is why the Dini and Dirichlet–Jordan tests involve only values of ff near xx. It contrasts with the global nature of the coefficients.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptDirichlet kernel, concept, no evidence levelDirichlet kernelconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFourier series, concept, no evidence levelFourier seriesconceptIntegrable function, concept, no evidence levelIntegrable functionconceptLebesgue integral, concept, no evidence levelLebesgue integralconceptOuter measure, concept, no evidence levelOuter measureconceptPeriodic function, concept, no evidence levelPeriodic functionconceptLebesgue measure, object, no evidence levelLebesgue measureobjectRiemann–Lebesgue lemma, statement, no evidence levelRiemann–Lebesgue lemmastatementRiemann localization principle, statement, no evidence levelRiemann localization …statement
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.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Dirichlet kernel concept
  3. Euclidean space object
  4. Fourier series concept
  5. Integrable function concept
  6. Lebesgue integral concept
  7. Outer measure concept
  8. Periodic function concept
  9. Lebesgue measure object
  10. Riemann–Lebesgue lemma statement
  11. Riemann localization principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Convergence_of_Fourier_series
About this page
TagHC3T4EY6WZ
URIm2://HC3T4EY6WZ
Kindstatement, theorem
WritingW0 stub
MSC42A20, 42A63
StatusThis copy of the site has no build data. Run m2 status HC3T4EY6WZ.
AboutFourier series
UsesFourier series, Riemann–Lebesgue lemma, Dirichlet kernel, Periodic function

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