Overview
This lecture analyzes Weierstrass's famous example of a continuous but nowhere differentiable function, discussing generalizations, key theorems, and related non-differentiable functions.
Introduction to Weierstrass's Function
- Weierstrass defined ( f(x) = \sum a^n \cos(b^n \pi x) ), with ( 0 < a < 1 ), ( b ) odd integer, and ( ab > 1 ).
- This function is continuous everywhere but lacks a derivative at any point.
- Generalizations involve series of the form ( \sum a_n \cos(b_n x) ) and ( \sum a_n \sin(b_n x) ), with specific convergence conditions.
- Previous results gave various artificial conditions for non-differentiability.
Main Results and Theorems
- Theorem 1.31: ( C(x) = \sum a^n \cos(b^n \pi x) ) and ( S(x) = \sum a^n \sin(b^n \pi x) ), with ( 0 < a < 1 ), ( b > 1 ), have no finite derivative if ( ab \geq 1 ).
- Theorem 1.32: The above fails if only infinite derivatives are considered.
- If ( ab > 1 ), ( f(x+h) - f(x) = O(|h|^\alpha) ) for all ( x ), but not for higher orders.
Lemmas and Proof Sketch (b Integer)
- Lemmas establish behavior of relevant harmonic and Fourier series functions near the boundary.
- Main argument shows contradiction if a finite derivative exists, using behaviors of associated series and their bounds.
- Exceptional points (e.g., ( x = p/b^q )) require special handling but do not lead to differentiability.
Extension: b Not an Integer
- Similar methods work for non-integer ( b ) by adapting integral representations (Dirichlet series).
- No exceptional values for ( x ); conclusions remain: nowhere differentiable under same parameter conditions.
Other Non-Differentiable Functions and Related Results
- Example provided: a Fourier series that fails any Lipschitz condition of order ( \alpha > 0 ).
- Bernstein's theorem: if a function obeys a Lipschitz condition with ( \alpha > 1/2 ), its Fourier series converges absolutely (and ( 1/2 ) is the best possible).
- Riemann's function ( \sum \frac{\sin(n^2 x)}{n^2} ) is shown to be non-differentiable for any irrational ( x ).
Key Terms & Definitions
- Nowhere Differentiable — a function continuous everywhere but with no point where the derivative exists.
- Lipschitz Condition — a function satisfies it of order ( \alpha ) if ( |f(x+h) - f(x)| \leq K|h|^\alpha ) for all ( x ).
- Fourier Series — an expansion of a function in sines and cosines.
- Harmonic Function — a function satisfying Laplace's equation, often appearing in analysis of series.
Action Items / Next Steps
- Review proof details for the main lemmas and theorems in the lecture.
- Study related functions (like Riemann's) and their differentiability properties.
- Practice verifying if a given Fourier series satisfies a Lipschitz condition.