Weierstrass Function: Fractal, Elliptic & Dynamic Analysis

• The Weierstrass function is a continuous yet nowhere differentiable function, illustrating key features of fractal geometry and irregular behavior.
• It exhibits a dichotomy between real analytic and fractal regimes, with precise results in Hölder continuity and Hausdorff dimension based on its parameters.
• Its rich applications span rough path theory, elliptic curve uniformization, and soliton solutions in integrable PDEs, linking diverse fields in mathematics.

The Weierstrass function refers to a broad class of functions constructed to exhibit pathological regularity—specifically, continuity combined with nowhere differentiability—as well as to a family of elliptic and modular functions with deep connections to algebraic geometry, number theory, and the theory of fractals. This entry surveys both the classical real-variable non-differentiable Weierstrass-type functions and the Weierstrass elliptic and sigma functions as they appear in complex analysis and the geometry of algebraic curves.

1. Classical Weierstrass-Type Functions: Definition and Dichotomy

The prototypical Weierstrass function is defined as

$W(x)=\sum_{n=0}^\infty \lambda^n \phi(b^n x),$

where $\phi: \mathbb{R} \to \mathbb{R}$ is a real analytic, $1$-periodic function, $b \geq 2$ is an integer, and $\lambda \in (1/b,1)$. When $\phi(x)=\cos(2\pi x)$, this coincides with the original Weierstrass construction, which was the first explicit example of a continuous everywhere but nowhere differentiable function. The convergence and continuity of $W$ are immediate from the boundedness of $\phi$ and the geometric decay in the summand (Ren et al., 2020).

Ren–Shen establish a dichotomy: for any such $\phi$, $b$, and $\lambda$, exactly one of the following holds:

• $W$ is real analytic,
• the graph of $W$ has Hausdorff dimension

$D = 2 + \frac{\log\lambda}{\log b}.$

Except for finitely many exceptional $\lambda$ (when $\phi$ is nonconstant), the non-analytic case prevails. In this regime, $W$ is Hölder continuous of exponent $-\frac{\log\lambda}{\log b}$ but fails to be Lipschitz (and is nowhere differentiable) (Ren et al., 2020, Johnsen, 2016).

2. Fractal Geometry and Hausdorff Dimension

A central theme is the fractal geometry of Weierstrass-type graphs. For the cosine instance,

$$W_{\lambda, b}(x) = \sum_{n=0}^\infty \lambda^n \cos(2\pi b^n x),$$

the conjecture of Mandelbrot (1977), confirmed in various parameter regimes, asserts

$$\dim_H \operatorname{graph} W_{\lambda, b} = 2 + \frac{\log\lambda}{\log b}\,,$$

with this value interpreted as the “exact” dimension in the sense of Hausdorff measure under suitable entropy and transversality conditions (Barański et al., 2013, Ren et al., 2020). Modern proofs deploy invariant measures under skew-product dynamics, Ledrappier–Young dimension formulae, and entropy growth via Hochman's inverse theorems. The non-integer base, forced random phase, higher-dimensional, and vector-valued generalizations rely on similar mechanisms (Cellarosi et al., 2023).

3. Regularity, Variation, and Differentiability Properties

The regularity analysis of Weierstrass-type functions combines geometric, probabilistic, and functional methods:

• Hölder continuity is sharp, with exponent $\alpha = -\frac{\log\lambda}{\log b}$ (Barczy et al., 12 Jul 2024).
• Nowhere differentiability: For analytic $\phi$ and all $b\geq2$, $1/b<\lambda<1$, $W$ is nowhere differentiable except in finitely many exceptional cases (Ren et al., 2020, Johnsen, 2016).
• Pointwise behaviour: For any cube $Q\subset \mathbb{R}^d$, the set of “slow points” $$D(W) = \{x : \limsup_{h\to0} | W(x+h) - W(x) | / |h| < \infty \}$$ has full Hausdorff dimension $d$ but zero Lebesgue measure (Donaire et al., 2012), even though $W$ is almost surely nowhere differentiable.
• $p$th-variation and Riesz-variation: The critical $p_c$ at which $W$ has non-trivial $p$-variation along $b$-adic refinements is $p_c=1/\alpha$; higher variations vanish, lower ones diverge (Barczy et al., 12 Jul 2024).

A concise summary of regularity regimes for

$$W(x) = \sum_{n=0}^\infty b^{-n\alpha} \cos(b^n x),\quad \alpha\in(0,1)$$

is given below:

Property	Regime	Value
Hölder exponent	Always	$\alpha$
Nowhere diff.	$b>1,~\alpha \in (0,1)$	True
Hausdorff dim.	$b>1,~\alpha \in (0,1)$	$2-\alpha$
$p$-variation	$p = 1/\alpha$	Exists, finite, linear in $t$ ($V^{(p)}_t$)

4. Microlocal and Fourier-Analytic Perspectives

Johnsen's microlocal Fourier approach reformulates the regularity question in terms of the spectral “lacunarity” of the underlying Fourier series. For

$f(t) = \sum_{j=0}^\infty a_j e^{i b_j t}$

with $\liminf b_{j+1}/b_j > 1$ and $a_j b_j \not\rightarrow 0$, or more generally $a_j \Delta b_j \not\rightarrow 0$ where $\Delta b_j$ is the local gap, $f$ is continuous and nowhere differentiable (Johnsen, 2016). This criterion encompasses series with polynomial or even quasi-quadratic gaps, indicating that the speed of frequency growth and amplitude decay together control the differentiability threshold.

5. The Weierstrass Elliptic and Sigma Functions

Independently, the term “Weierstrass function” denotes a distinguished doubly periodic meromorphic function $\wp(z;g_2,g_3)$ on the complex plane, solving

$(\wp'(z))^2 = 4\wp(z)^3 - g_2 \wp(z) - g_3,$

with periods $2\omega_1,~2\omega_2$ and invariants $g_2,~g_3$ determined by the period lattice $$\Lambda = 2\omega_1 \mathbb{Z} + 2\omega_2 \mathbb{Z}$$. These functions underlie the uniformization theorem for elliptic curves and admit the classical addition law, duplication and triplication formulas, and general algebraic addition theorems produced by a determinant-based recipe (Brizard, 2015, Gürel, 16 Apr 2025). The lattice, half-periods, and root structure are classified according to the discriminant $\Delta=g_2^3-27g_3^2$ (Brizard, 2015).

Generalizations include the construction of sigma functions $\sigma(u)$ for higher-genus Riemann surfaces, built from Riemann theta functions, their characteristics, and explicit period matrix data; their logarithmic derivatives recover higher-genus $\wp_{ij}$-functions (Korotkin et al., 2012, Komeda et al., 2022).

6. Applications in Dynamics, Soliton Theory, and Model Theory

The real-variable Weierstrass function is a canonical deterministic model in rough paths theory: its Hölder regularity in the regime $\alpha>1/3$ allows explicit construction of “rough path” iterated integrals and solutions to controlled differential equations with deterministic driving signals of Weierstrass type (Cellarosi et al., 2023). In soliton theory, the Weierstrass elliptic function generates explicit, closed-form solutions (including soliton and cnoidal wave solutions) for integrable PDEs (e.g., mKdV) via projective Riccati expansions and trigonometric/hyperbolic degenerations (Sirendaoreji, 2022).

In mathematical logic and model theory, expansions of the real field by the Weierstrass $\wp$-function and related modular data (zeta, Eisenstein, quasimodular forms) are shown to be strongly uniformly model-complete—every definable set can be existentially defined—under appropriate analytical language extensions (Bianconi, 2014).

7. Extensions, Open Problems, and Significance

Outstanding problems concern the extension of the Ren–Shen dichotomy beyond integer $b$, forced non-analytic phases, or non-analytic $\phi$, as well as the full description of exceptional $\lambda$ parameter loci. Within the fractal dimension program, determination of the exact Hausdorff measure and dimension for general series and parameter values remains partially open (notably for $\lambda \downarrow 1/b$) (Barański et al., 2013). The intersection with fractional calculus reveals that even generalized Weierstrass–Jumarie functions preserve their fractal roughness index, with derivative regularity depending only on the order of differentiation (Ghosh et al., 2015).

The theory of Weierstrass-type functions continues to link core notions in real analysis, harmonic analysis, complex function theory, algebraic geometry, dynamical systems, mathematical logic, and fractal geometry—serving as a unifying subject in modern mathematics.