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13: The " saw-tooth " function φ(x) on [−3, 3].  

13: The " saw-tooth " function φ(x) on [−3, 3].  

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Figure 3.13: The " saw-tooth " function φ(x) on [−3, 3].  
Figure 3.14: Petr's function in a 4-adic system.  
Figure 3.10: The functions f 1 (dashed) and f 2 (whole).  
Figure 3.19: Line segment with band neighborhoods for Lynch's function.  
+4 Figure 3.5: Darboux's function D(x) on [0, 3].  
Continuous Nowhere Differentiable Functions (MS Thesis)
Article
Full-text available
  • Jan 2003
Abstract In the early nineteenth century, most mathematicians believed that a contin- uous function has derivative at a significant set of points. A. M. Amp` ere even tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July...
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... In the event that p h (t, 0) = 0, Proposition 3 makes it easy to construct functions that are nowhere p-differentiable on the whole real axis simply by choosing, in particular, any function with p h (t, 0) = 1. For a fascinating historical survey of classical nowhere differentiable functions, the reader is encouraged to look at [22] At this point one may think that p-differentiability is normal and that most functions have a zero p-derivative if p h (t, 0) = 0. This motivates the next question: Does there exist a function p satisfying (20) such that it is continuous and nowhere p-differentiable on R? The answer is yes and is in the next theorem. ...
... Since b < 1 we must have a ≥ 3 so that ab > α √ ab. The stronger hypothesis (22) forces both α √ ab > 1 and the term in the parentheses in (29) to be positive. Since h m → 0 as m → ∞, the left hand side of (29) tends to infinity, so that the resulting p-derivative cannot exist at x. Since x is arbitrary, the conclusion follows. ...
On the Basic Theory of Some Generalized and Fractional Derivatives
Article
Full-text available
  • Nov 2022
We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle’s theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which provides simple proofs of the corresponding version for the so-called conformable fractional derivatives considered by many. Finally, we show that for each α>1 there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line.
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... If all the level sets of a function are well-behaved, this does not imply that the function is Lipschitz continuous. In Section D.1 we present an example of a function on [0, ∞) (the Schwarz function [34]), which is strictly monotonically increasing, locally Holder continuous with exponent γ ∈ (0, 1), and is not locally Holder continuous with any exponent γ > γ on any interval. Using such a function as a building block, one can create a function with well-behaved level sets in the sense of Lemma 4. Due to the limited smoothness of such H 0 , the derivation in Sections 3, 5 for the original DTB cannot guarantee convergence faster than O( γ/2 ), which is slower than the conjectured rate. ...
Resolution of 2D reconstruction of functions with nonsmooth edges from discrete Radon transform data
Preprint
  • Dec 2021
Let f be an unknown function in R2\mathbb R^2, and fϵf_\epsilon be its reconstruction from discrete Radon transform data, where ϵ\epsilon is the data sampling rate. We study the resolution of reconstruction when f has a jump discontinuity along a nonsmooth curve Sϵ\mathcal S_\epsilon. The assumptions are that (a) Sϵ\mathcal S_\epsilon is an O(ϵ)O(\epsilon)-size perturbation of a smooth curve S\mathcal S, and (b) Sϵ\mathcal S_\epsilon is Holder continuous with some exponent γ∈(0,1]\gamma\in(0,1]. We compute the Discrete Transition Behavior (or, DTB) defined as the limit DTB(xˇ):=lim⁡ϵ→0fϵ(x0+ϵxˇ)\text{DTB}(\check x):=\lim_{\epsilon\to0}f_\epsilon(x_0+\epsilon\check x), where x0x_0 is generic. We illustrate the DTB by two sets of numerical experiments. In the first set, the perturbation is a smooth, rapidly oscillating sinusoid, and in the second - a fractal curve. The experiments reveal that the match between the DTB and reconstruction is worse as Sϵ\mathcal S_\epsilon gets more rough. This is in agreement with the proof of the DTB, which suggests that the rate of convergence to the limit is O(ϵγ/2)O(\epsilon^{\gamma/2}). We then propose a new DTB, which exhibits an excellent agreement with reconstructions. Investigation of this phenomenon requires computing the rate of convergence for the new DTB. This, in turn, requires completely new approaches. We obtain a partial result along these lines and formulate a conjecture that the rate of convergence of the new DTB is O(ϵ1/2ln⁡(1/ϵ))O(\epsilon^{1/2}\ln(1/\epsilon)).
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... Hernández-Verón et al. [25] studied semilocal convergence of this Secant type method for non-differentiable operators. In [26], one can find a number of very good examples of continuous nowhere differentiable functions with their properties are also discussed there in. In this paper, we obtain a new semilocal convergence result for a three step Kurchatov-type method in which the function G is not required to be differentiable, so that we can extend the applicability of the method to solve non-differentiable systems. ...
Three Step Kurchatov Method for Nondifferentiable Operators
Article
Full-text available
  • Dec 2017
In this paper we find the order of convergence and semilocal convergence of three step Kurchatov-type method. We also analyze the efficiency index and computational efficiency of this method. The semilocal convergence analysis of method has been established by using recurrence relations under the assumption of first order divided difference operators satisfy Lipschitz condition. The convergence theorem and domain of parameters of the method has also been included. The applicability of the proposed convergence analysis is illustrated by solving some numerical examples.
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... Around 1860 the Swiss mathematician, C. Cellérier discovered such a function but unfortunately it was not published then and could be published only in 1890 after his death. To know more about the interesting history and details about such functions, the reader is referred to the excellent Master's thesis of J. Thim [22]. Riemann, as mentioned in the earlier paragraph, opined that the function, ...
Quadratic Reciprocity and Some “Non-differentiable” Functions
Chapter
Full-text available
  • Sep 2017
Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. In [28] (Proc Int Conf–Number Theory 1, 107–116, 2004), Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta transformation formula and Gerver (Amer J Math 92, 33–55, 1970) [12] was the first to give a proof of differentiability/non-differentiability of Riemann’s function. The aim here is to survey some of the work done in these two directions and concentrates more onto a recent work of the first author along with Kanemitsu and Li (Res Number Theory 1, 14, 2015) [5]. In that work (Kanemitsu and Li, Res Number Theory 1, 14, 2015) [5], an integrated form of the theta function was utilised and the advantage of that is that while the theta function Θ (τ) is a dweller in the upper half-plane, its integrated form F(z) is a dweller in the extended upper half-plane including the real line, thus making it possible to consider the behaviour under the increment of the real variable, where the integration is along the horizontal line.
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... In the context of nowhere differentiable functions followed by Weierstrass function several functions where studied in [2, 10-18, 20, 21]. Apart from the mentioned functions, several other functions of nowhere differentiable are given in [22]. ...
... Dini in [12][13][14][15] and Knopp in [18] defined the continuous everywhere and differentiable nowhere functions which satisfies the conditions given in [22] (Remark 1, pp 25-26 in [22]). ...
... Dini in [12][13][14][15] and Knopp in [18] defined the continuous everywhere and differentiable nowhere functions which satisfies the conditions given in [22] (Remark 1, pp 25-26 in [22]). ...
Sumudu transform of Weierstrass function
Conference Paper
Full-text available
  • Mar 2016
Weierstrass function, Weierstrass­ Mandelbrot function and its variants of fractal functions are Sumudu transformed to show that non regular curves can be smoothed.
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... The graph of f was studied as a fractal curve in the plane by Besicovitch and Ursell [4]. A elementary and readable account of the history of nowhere differentiable functions, is [16]; it includes the construction by Bolzano (1830) of one of the earliest examples of such a function. In Section 3 it is established that, if g is continuous, then f can be selected to be continuous. ...
Approximation of Rough Functions
Article
Full-text available
  • Dec 2014
  • J APPROX THEORY
For given p∈[1,∞]p\in\lbrack1,\infty] and g∈Lp(R)g\in L^{p}\mathbb{(R)}, we establish the existence and uniqueness of solutions f∈Lp(R)f\in L^{p}(\mathbb{R)}, to the equation \begin{equation*} f(x)-af(bx)=g(x), \end{equation*} where a∈Ra\in\mathbb{R}, b∈R,b\in\mathbb{R}, b≠0,b\neq0, and ∣a∣≠∣b∣1/p\left\vert a\right\vert \neq\left\vert b\right\vert ^{1/p}. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.
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... The construction of function F is very similar to the constructions of nowhere-differentiable real functions like Weierstrass function (many of such functions could be found in [3]) in the sense that function F is an infinite sum of simple functions with certain properties. A similar construction can also be found in [1] (in the examples regarding complexity of derivatives of polynomial time computable real functions). ...
Computable real function F such that F is not polynomial time computable on [0,1]
Article
  • Apr 2014
A computable real function F on [0,1] is constructed such that there exists an exponential time algorithm for the evaluation of the function on [0,1] on Turing machine but there does not exist any polynomial time algorithm for the evaluation of the function on [0,1] on Turing machine (moreover, it holds for any rational point on (0,1))
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... Although the first published example is certainly due to Weierstrass, already in 1830 the Czech mathematician B. Bolzano exhibited a continuous nowhere differentiable function. Let us give a brief overview of the appearance throughout history of " Weierstrass' Monsters " (see, e.g., [264] for a thorough study of the citations below): After 1872 many other mathematicians also constructed similar functions. Just to cite a partial list of these, we have: H. A. Schwarz (1873), M. G. Darboux (1874), U. Dini (1877), K. Hertz (1879), G. Peano (1890), D. Hilbert (1891), T. Takagi (1903), H. von Koch (1904), W. Sierpi´nskiSierpi´nski (1912), G. H. Hardy (1916), A. S. Besicovitch (1924), B. van der Waerden (1930), S. Mazurkiewicz (1931), S. Banach (1931), S. Saks (1932) , and W. Orlicz (1947). ...
Linear subsets of nonlinear sets in topological vector spaces
Article
Full-text available
  • Jan 2014
  • B AM MATH SOC
For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.
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... where F 1 (x) = F 1 (x + 1) is a continuous, periodic, and nowhere differentiable function (see also [127,128]). His expression for F 1 is as follows; see Figure 5 for a plot of −F 1 (x) and its first few approximations by 1 2 log 2 n − E(X n ). ...
Distribution of the sum-of-digits function of random integers: A survey
Article
Full-text available
  • Dec 2012
  • Probab Surv
We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein's method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.
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... century, a number of different geometrical and analytic constructions of continuous nowhere differentiable functions have been obtained. See [17] for a historical survey on the subject. For 0 < α ≤ 1, let Lip α (R d ) be the Hölder class of bounded functions f : R d → R for which there exists a constant C = C(f ) > 0 such that |f (x) − f (y)| < C|x − y| α for any x, y ∈ R d . ...
Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions
Article
  • Feb 2012
  • REV MAT IBEROAM
We study differentiability properties of Zygmund functions and series of Weierstrass type in higher dimensions. While such functions may be nowhere differentiable, we show that, under appropriate assumptions, the set of points where the incremental quotients are bounded has maximal Hausdorff dimension.
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