Mathematics#Fourier Analysis, Approximation Theory, Laplace Transforms🔬 ResearchAnalyzed: Jan 3, 2026 16:17
Tight Bounds for Oscillatory Functions via Laplace Transform
Published:Dec 28, 2025 17:01
•1 min read
•ArXiv
Analysis
This paper provides improved bounds for approximating oscillatory functions, specifically focusing on the error of Fourier polynomial approximation of the sawtooth function. The use of Laplace transform representations, particularly of the Lerch Zeta function, is a key methodological contribution. The results are significant for understanding the behavior of Fourier series and related approximations, offering tighter bounds and explicit constants. The paper's focus on specific functions (sawtooth, Dirichlet kernel, logarithm) suggests a targeted approach with potentially broad implications for approximation theory.
Key Takeaways
- •Provides tighter bounds for the approximation of oscillatory functions.
- •Employs Laplace transform representations, particularly of the Lerch Zeta function.
- •Focuses on specific functions like the sawtooth function, Dirichlet kernel, and logarithm.
- •Offers explicit constants in the derived inequalities.
Reference
“The error of approximation of the $2π$-periodic sawtooth function $(π-x)/2$, $0\leq x<2π$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$.”