Talk:Weierstrass function

message: Is this the one known as the "blancmage"? Can't remember... Dysprosia 05:32, 27 Jan 2004 (UTC)
replies: text: Perhaps we can believe the MathWorld page on Blancmange function... --Pt 22:56, 9 Oct 2004 (UTC)

text: Of course you can. Just the derivative doesn't (usually --Pt) converge as the function itself does. Pt 21:44, 6 Sep 2004 (UTC)
thread_title:  Blancmarge 

message: :probably the second known was also one of Weierstrass : sigma cos(n²x)/n² and the third its generalization cos(n^c x)/n^c.
I moved the sentence here because it certainly needs some discussion before going to the encyclopedia article. --Pt 23:18, 9 Oct 2004 (UTC)
thread_title:  An edit by [[User:217.136.97.96|217.136.97.96]] at 00:49 [[EET]], 10 Oct 2004 

message: I just added the requirement that b is an odd integer.  This is how I learned it, and is verified by (for instance) , , .  I don't know how to verify its necessity myself, but this requirement is, at least, common. LWizard @ 01:25, 27 April 2006 (UTC)
thread_title:  Odd integer requirement 

message: Would it be possible to get a graphic showing the construction of this function, just to make it clearer? -- 314159 01:05, 27 August 2006 (UTC)
thread_title:  Picture? 

message: It says in the "elementary" proof that
: Since the terms of the infinite series which defines it are bounded by \pm a^n and this has finite sum for 0 < a < 1, convergence of the partial sums to the function is uniform
in the article. But why does a finite sum imply uniform convergence? It seems to me that since, for |b| < 1,
: \left|\sum_{k=0}^\infty b^k - \sum_{k=0}^n b^k\right| = \frac{|b|^{n+1}}{|1-b|},
which blows up as b \to 1, we have that
: \sup_{|b| < 1} \left|\sum_{k=0}^\infty b^k - \sum_{k=0}^n b^k\right| = \infty
for all n, thus it is not going to converge to anything as n goes to infinity and so it doesn't look like uniform convergence to me... or did I do something wrong? -- 129.78.64.102 11:39, 18 July 2007 (UTC)
replies: text: * Use the Weierstrass M-test with M_n=a^n.  This will show uniform convergence; continuity is preserved through convergence since each partial sum is continuous. Neocapitalist 23:38, 4 November 2007 (UTC)
thread_title:  Proof of continuity of the function... 

message: In the section "Construction of the Weierstrass function" is:
:\lim \inf
meant to be:
: \lim_{x \to \infty}
and is the:
: \lim \sup
meant to be something else?
</br>RMFan1 (talk) 15:36, 2 January 2008 (UTC)
thread_title: Formating the formulae

message: Nice picture. But what choices of a and b were made in such plot?  —Preceding unsigned comment added by 67.83.187.113 (talk) 00:44, 30 March 2008 (UTC)
thread_title:  The picture on the article 

message: The discrepancy with the inequality is because the requirement ab > 1 + \frac{3}{2}\pi is purely an historic artifact. G. Hardy showed in 1916 that the only true requirement is that ab > 1.
See for example, *
MMmpds 16:59, 19 January 2022 (UTC)
thread_title:  The Historic Inequality Is Not the Tightest Lower Bound on the Product: <math>\,ab</math> 

message: When I try to get the derivative using the differentiation rules, I get this function:
f(x)'=\sum_{n=0}^\infty -a^n b^n \pi \sin(b^n\pi x)
With most values for x, the number changes completely on every iteration of the sum operator.
It doesn't converge to a number, which is expectable for a function which hasn't a derivative.
But, if you use an integer for x, the argument of the sine function is a multiplycation of pi.
In that case the sine function returns 0, which causes the whole function to return 0.
This means the Weierstrass function is differentiable at least some places.
These places are where x is an integer, the derivative equals 0 at those places.
Do you agree with me?
Paul Breeuwsma (talk) 23:40, 16 October 2008 (UTC)
replies: text: You cannot in general differentiate an infinite series by differentiating each of the terms. Eric119 (talk) 16:31, 19 October 2008 (UTC)

text: I'm guessing that you are also letting b be an integer. You shouldn't. If b is not an integer, this won't happen since an integer multiple of a non-integer is not an integer, and
b^n will not be an integer in most cases. You COULD still choose x such that xb^n is an integer (e.g. x = m*b^(-n), m an integer), but obviously xb^(n+1) = b*(xb^n) won't be.
Thus: choose b a non-integer and the argument for sin will be a non-integer almost always for any x and any n (and for every x, there will be n such that it is non-integer).
Also, note the comment from Eric119. In your case, the function could well have a derivative, but it is not proven just by applying a derivative formula for each term.84.238.115.158 (talk) 21:11, 16 March 2009 (UTC)
thread_title:  Differentiable at integers 

message: From Hardy's true condition, namely, the requirement that ab > 1, one finds that the Dini derivatives even at zeros of the function are all infinite. This means the usual derivative cannot exist. So while our minds may wander and confuse us with the lingering thought that the constant function f(x) = 0 is the only smooth analytic function with compact support, this has no bearing on whether the derivative of a function exists at a zero of the function. In the case of zeros of the function, the derivative does not suddenly become defined. In fact, it suffices to check that none of the Dini derivatives are finite, and therefore the ordinary derivative cannot exist. Since this is true at every point, the function is nowhere differentiable. Another obvious way to see this conclusion is to realize that if one were to translate any W-function up by an amount y_+ = \min(W) + \varepsilon or down by y_- = -\max(W) - \varepsilon with \varepsilon > 0, and then ask if one believes that the points formerly crossing the x-axis suddenly gained or lost differentiability, purely due to translation of the graph, the answer is clearly that no such derivative can exist. To make this more logically precise, consider the weak derivative D_w and note that if W(x) is a Weierstrass like-function then D_w W(x) = D_w [ W(x) + y_\pm ] since both y_\pm are constants.  — Preceding unsigned comment added by MMmpds (talk • contribs) 17:47, 19 January 2022 (UTC)
thread_title:  A More Precise Perspective 

message: The wolfram reference is to a different function by the same name.  Probably not a good reference and should be removed.
Dannyboytward (talk) 15:22, 4 November 2008 (UTC)
replies: text: Above is discussed that MathWorld article is more likely about Riemann function. Perhaps it is better to mention it than to completely remove the reference. --EnJx (talk) 15:58, 8 November 2008 (UTC)
thread_title:  references 

message: I've just been reading a book called "Fractals, Chaos, Power Laws", by Manfred Schroeder.  It mentions a very interesting property of these functions that I think should be mentioned.
If this function is taken as a timeseries, it's waveform can be played as a musical note.  If beta = 2^(13/12) we have (I don't know how to type equations, I appologise profusely)
f(t) = Sum_k[cos(2^(13/12)*t)]
If we double the frequency
f(2*t) = Sum_k[cos(2^(13/12 + 1)*t)] = Sum_k'[cos(2^(13/12)*2^(-1/12)*t)]
if the sum is over the entire audible range, a human perceives
f(2*t) = f(2^(-1/12)*t), or a doubling of frequency results in a paradoxical one semitone decrase in pitch (as opposed to an expected one octave increase)
Dannyboytward (talk) 15:28, 4 November 2008 (UTC)
thread_title:  Weierstrass Functions and a Musical Paradox 

message: The definition given in the article really looks like a Fourier series and in fact, the coefficients would be something like
\dots,0,0,a^n,0,0,\dots,0,a^{n+1},0,0,\dots
I think mentioning the connection would improve the article, but i'm a bit hesitant to add it myself since i'm not an expert on the subject.
-Actually never mind, b is not an integer in general.130.234.5.136 (talk) 14:19, 6 October 2010 (UTC)
thread_title: Fourier series

message: Would it be useful to put in a mention of the distributional derivative of this function? I don't remember seeing a specific reference about it, but it seems that it would be a good example...I imagine that the singular support of the derivative would be equal to the support, but that's just a guess. Anyone have a reference for something like this? -- Spireguy (talk) 04:16, 16 November 2010 (UTC)
thread_title:  Distributional derivative 

message: In the article is said that one just proves this:
:\lim \inf \frac{f(x_n) - f(x)}{x_n - x} > \lim \sup \frac{f(x'_n) - f(x)}{x'_n - x}
But I think it is enough to prove:
:\lim \sup \frac{f(x_n) - f(x)}{x_n - x} > \lim \inf \frac{f(x'_n) - f(x)}{x'_n - x}
This is a much weaker condition and therefore probably easier.
I'm of course also interested in the whole proof. Why is the rest omitted? --Jobu0101 (talk) 08:58, 18 December 2010 (UTC)
thread_title:  Sketch of the proof 

message: Recently a section was written about the fractional differentiability of the function, and then again removed "as an undue promotion of a result. The edit summary further says: "An equivalent result has been known for many decades already." I'd say the fractional differentiability is quite an interesting property of the function, and a natural generalization of usual differentiability. That is: it is natural and interesting to ask whether a non-differentiable function is fractionally differentiable. The cited article has been cited already 40 times according to Scopus. In short, I think the fractional differentiability deserves at least a sentence or a few in the article. If you disagree, then what is the "equivalent result" known for many decades?
This equivalence by itself seems encyclopedia-worthy! — Pt (T) 16:43, 18 December 2010 (UTC)
thread_title:  Fractional differentiability 

message: The last section left me a little confused, as I don't have any background with topology. I was wondering if the last section implied that the set of everywhere differentiable functions on [0,1] is countable.  — Preceding unsigned comment added by 2001:18E8:2:1093:41A8:3C62:6615:9534 (talk) 15:20, 26 November 2012 (UTC)
thread_title:  Density of nowhere differential functions 

message: The external link "Weierstrass function in the complex plane", while pretty, is not a graph of the Riemann-Weierstrass function.  — Preceding unsigned comment added by 128.59.193.197 (talk) 16:18, 23 January 2013 (UTC)
thread_title:  Linked Fractal 

message: Article says ab must be > 1+3/2*pi = 5.712, but the standard values of a and b, like used in , , and the example image at the top of the page, are a = 0.5 and b = 3, so ab = 1.5.  What is this condition for? — Omegatron (talk) 15:24, 14 May 2013 (UTC)
thread_title:  Contradiction 

message: The discrepancy with the inequality is because the requirement ab > 1 + \frac{3}{2}\pi is purely an historic artifact. G. Hardy showed in 1916 that the only true requirement is that ab > 1.
See for example, *
MMmpds 17:04, 19 January 2022 (UTC)
thread_title:  The Historic Inequality Is Not the Tightest Lower Bound on the Product: <math>\,ab</math> 

message: In the section Construction, the following paragraph appears:
"The computation of the Hausdorff dimension D of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that D = 2 + \log_b(a) < 2. That D is strictly less than 2 follows from the conditions on a and b from above. Only after more than 30 years was this proved rigorously."
It is not clear what the word "this" refers to.
Is it only that D < 2 ?
Or is it the full statement that D = 2 + \log_b(a) < 2 ? 2601:200:C000:1A0:C0A5:7B4A:921E:7CA (talk) 23:29, 5 July 2022 (UTC)
thread_title:  Unclear statement 

message: The redirect Nowhere differentiable function has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached.  Randi Moth TalkContribs 22:23, 16 May 2023 (UTC)
thread_title:  "[[:Nowhere differentiable function]]" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]