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Weierstrass function

From Wikipedia, the free encyclopedia
Function that is continuous everywhere but differentiable nowhere
Not to be confused with theWeierstrass elliptic function (℘{\displaystyle \wp }), theWeierstrass–Mandelbrot function, or theWeierstrass sigma, zeta, or eta functions.

Plot of Weierstrass function over the interval [−2, 2]. Like some otherfractals, the function exhibitsself-similarity: every zoom (red circle) is similar to the global plot.

Inmathematics, theWeierstrass function, named after its discoverer,Karl Weierstrass, is an example of a real-valuedfunction that iscontinuous everywhere butdifferentiable nowhere. It is also an example of afractal curve.

The Weierstrass function has historically served the role of apathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points.[1] Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions ofsmoothness. These types of functions were disliked by contemporaries:Charles Hermite, on finding that one class of function he was working on had such a property, described it as a "lamentable scourge".[disputed –discuss][2] The functions were difficult to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models ofBrownian motion necessitated infinitely jagged functions (nowadays known as fractal curves).[3]

Construction

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Animation based on the increasing of the b value from 0.1 to 5.

In Weierstrass's original paper, the function was defined as aFourier series:

f(x)=∑n=0∞ancos⁡(bnπx),{\displaystyle f(x)=\sum _{n=0}^{\infty }a^{n}\cos(b^{n}\pi x),}

where0<a<1{\textstyle 0<a<1},b{\textstyle b} is a positive odd integer, and

ab>1+32π.{\displaystyle ab>1+{\frac {3}{2}}\pi .}

The minimum value ofb{\textstyle b} for which there exists0<a<1{\textstyle 0<a<1} such that these constraints are satisfied isb=7{\textstyle b=7}. This construction, along with the proof that the function is not differentiable at any point, was first delivered by Weierstrass in a paper presented to theKönigliche Akademie der Wissenschaften on 18 July 1872.[4][5][6]

Despite being differentiable nowhere, the function is continuous: Since the terms of the infinite series which defines it are bounded by±an{\textstyle \pm a^{n}} and this has finite sum for0<a<1{\textstyle 0<a<1}, convergence of the sum of the terms isuniform by theWeierstrass M-test withMn=an{\textstyle M_{n}=a^{n}}. Since each partial sum is continuous, by theuniform limit theorem, it follows thatf{\textstyle f} is continuous. Additionally, since each partial sum isuniformly continuous, it follows thatf{\textstyle f} is also uniformly continuous.

It might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be countably infinite or finite. According to Weierstrass in his paper, earlier mathematicians includingGauss had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points. Analogous results for better behaved classes of continuous functions do exist, for example theLipschitz functions, whose set of non-differentiability points must be aLebesgue null set (Rademacher's theorem). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved. Moreover, the fact that the set of non-differentiability points for amonotone function ismeasure-zero implies that the rapid oscillations of Weierstrass' function are necessary to ensure that it is nowhere-differentiable.

The Weierstrass function was one of the firstfractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone.

The computation of theHausdorff dimensionD{\textstyle D} of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed thatD=2+logb⁡(a)<2{\textstyle D=2+\log _{b}(a)<2}.[7][8] ThatD is strictly less than 2 follows from the conditions ona{\textstyle a} andb{\textstyle b} from above. Only after more than 30 years was this proved rigorously.[9]

The term Weierstrass function is often used inreal analysis to refer to any function with similar properties and construction to Weierstrass's original example. For example, the cosine function can be replaced in the infinite series by apiecewise linear "zigzag" function.G. H. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions0<a<1,ab≥1{\textstyle 0<a<1,ab\geq 1}.[10]

Riemann function

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The Weierstrass function is based on the earlier Riemann function, claimed to be differentiable nowhere. Occasionally, this function has also been called the Weierstrass function.[11]f(x)=∑n=1∞sin⁡(n2x)n2.{\displaystyle f(x)=\sum _{n=1}^{\infty }{\frac {\sin(n^{2}x)}{n^{2}}}.}

WhileBernhard Riemann strongly claimed that the function is differentiable nowhere, no evidence of this was published by Riemann, and Weierstrass noted that he did not find any evidence of it surviving either in Riemann's papers or orally from his students.

In 1916,G. H. Hardy confirmed that the function does not have a finite derivative in any value ofπx{\textstyle \pi x} wherex is irrational or is rational with the form of either2A4B+1{\textstyle {\frac {2A}{4B+1}}} or2A+12B{\textstyle {\frac {2A+1}{2B}}}, whereA andB are integers.[10] In 1969,Joseph Gerver found that the Riemann function has a defined differential on every value ofx that can be expressed in the form of2A+12B+1π{\textstyle {\frac {2A+1}{2B+1}}\pi } with integerA andB, that is, rational multipliers ofπ{\displaystyle \pi } with an odd numerator and denominator. On these points, the function has a derivative of−12{\textstyle -{\frac {1}{2}}}.[12] In 1971, J. Gerver showed that the function has no finite differential at the values ofx that can be expressed in the form of2A2B+1π{\textstyle {\frac {2A}{2B+1}}\pi }, completing the problem of the differentiability of the Riemann function.[13]

As the Riemann function is differentiable only on anull set of points, it is differentiablealmost nowhere.

Hölder continuity

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It is convenient to write the Weierstrass function equivalently as

Wα(x)=∑n=0∞b−nαcos⁡(bnπx){\displaystyle W_{\alpha }(x)=\sum _{n=0}^{\infty }b^{-n\alpha }\cos(b^{n}\pi x)}

forα=−ln⁡(a)ln⁡(b){\textstyle \alpha =-{\frac {\ln(a)}{\ln(b)}}}. ThenWα(x){\textstyle W_{\alpha }(x)} isHölder continuous of exponent α, which is to say that there is a constantC such that

|Wα(x)−Wα(y)|≤C|x−y|α{\displaystyle |W_{\alpha }(x)-W_{\alpha }(y)|\leq C|x-y|^{\alpha }}

for allx{\textstyle x} andy{\textstyle y}.[14] Moreover,W1{\textstyle W_{1}} is Hölder continuous of all ordersα<1{\textstyle \alpha <1} but notLipschitz continuous.

Density of nowhere-differentiable functions

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It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions:

See also

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Notes

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  1. ^At least two researchers formulated continuous, nowhere differentiable functions before Weierstrass, but their findings were not published in their lifetimes.Around 1831,Bernard Bolzano (1781–1848), a Czech mathematician, philosopher, and Catholic priest, constructed such a function; however, it was not published until 1922. See:
    • Martin Jašek (1922),"Funkce Bolzanova" (Bolzano's function),Časopis pro Pěstování Matematiky a Fyziky (Journal for the Cultivation of Mathematics and Physics), vol. 51, no. 2, pages 69–76 (in Czech and German).
    • Vojtěch Jarník (1922),"O funkci Bolzanově" (On Bolzano's function),Časopis pro Pěstování Matematiky a Fyziky (Journal for the Cultivation of Mathematics and Physics), vol. 51, no. 4, pages 248–264 (in Czech). See alsoEnglish translation.
    • Karel Rychlík (1923),"Über eine Funktion aus Bolzanos handschriftlichem Nachlasse" (On a function from Bolzano's literary remains in manuscript),Sitzungsberichte der königlichen Böhmischen Gesellschaft der Wissenschaften (Prag) (Proceedings of the Royal Bohemian Society of Philosophy in Prague) (for the years 1921–1922), Class II, no. 4, pages 1-20. (Sitzungsberichte was continued as:Věstník Královské české společnosti nauk, třída matematicko-přírodovědecká (Journal of the Royal Czech Society of Science, Mathematics and Natural Sciences Class).)
    Around 1860, Charles Cellérier (1818–1889), a professor of mathematics, mechanics, astronomy, and physical geography at the University of Geneva, Switzerland, independently formulated a continuous, nowhere differentiable function that closely resembles Weierstrass's function. Cellérier's discovery was, however, published posthumously:
  2. ^Hermite, Charles;Stieltjes, Thomas (1905). "Letter 374, 20 May 1893". In Baillaud, Benjamin; Bourget, Henri (eds.).Correspondance d'Hermite et de Stieltjes (in French). Vol. 2. Gauthier-Villars. pp. 317–319.
  3. ^Kucharski, Adam (26 October 2017)."Math's Beautiful Monsters: How a destructive idea paved the way for modern math". Retrieved11 October 2023.
  4. ^Onpage 560 of the 1872Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (Monthly Reports of the Royal Prussian Academy of Science in Berlin), there is a brief mention that on 18 July,"Hr. Weierstrass las über stetige Funktionen ohne bestimmte Differentialquotienten" (Mr. Weierstrass read [a paper] about continuous functions without definite [i.e., well-defined] derivatives [to members of the Academy]). However, Weierstrass's paper was not published in theMonatsberichte.
  5. ^Karl Weierstrass,"Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen" [On continuous functions of a real argument which possess a definite derivative for no value of the argument], in:Königlich Preußische Akademie der Wissenschaften,Mathematische Werke von Karl Weierstrass (Berlin, Germany: Mayer & Mueller, 1895), vol. 2, pages 71–74.
  6. ^See also: Karl Weierstrass,Abhandlungen aus der Functionenlehre [Treatises from the Theory of Functions] (Berlin, Germany: Julius Springer, 1886),page 97.
  7. ^Kenneth Falconer,The Geometry of Fractal Sets (Cambridge, England: Cambridge University Press, 1985), pages 114, 149.
  8. ^See also: Brian R. Hunt (1998),"The Hausdorff dimension of graphs of Weierstrass functions",Proceedings of the American Mathematical Society, vol. 126, no. 3, pages 791–800.
  9. ^Shen Weixiao (2018). "Hausdorff dimension of the graphs of the classical Weierstrass functions".Mathematische Zeitschrift.289 (1–2):223–266.arXiv:1505.03986.doi:10.1007/s00209-017-1949-1.ISSN 0025-5874.S2CID 118844077.
  10. ^abHardy G. H. (1916) "Weierstrass's nondifferentiable function",Transactions of the American Mathematical Society, vol. 17, pages 301–325.
  11. ^Weisstein, Eric W."Weierstrass Function".MathWorld.
  12. ^Gerver, Joseph (1969)."The Differentiability of the Riemann Function at Certain Rational Multiples of π".Proceedings of the National Academy of Sciences of the United States of America.62 (3):668–670.Bibcode:1969PNAS...62..668G.doi:10.1073/pnas.62.3.668.PMC 223649.PMID 16591735.
  13. ^Gerver, Joseph (1971). "More on the Differentiability of the Riemann Function".American Journal of Mathematics.93 (1):33–41.doi:10.2307/2373445.JSTOR 2373445.S2CID 124562827.
  14. ^Zygmund, A. (2002) [1935].Trigonometric Series. Cambridge Mathematical Library. Vol. I, II (3rd ed.).Cambridge University Press. p. 47.ISBN 978-0-521-89053-3.MR 1963498.
  15. ^Mazurkiewicz, S.. (1931)."Sur les fonctions non-dérivables".Studia Math.3 (3):92–94.doi:10.4064/sm-3-1-92-94.
  16. ^Banach, S. (1931)."Über die Baire'sche Kategorie gewisser Funktionenmengen".Studia Math.3 (3):174–179.doi:10.4064/sm-3-1-174-179.

References

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External links

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