2.1 Pointwise and uniform convergence of Fourier series

Fourier series convergence comes in two flavors: pointwise and uniform. Pointwise means the series converges at each point, while uniform convergence is stronger, ensuring the same convergence rate everywhere. This distinction is crucial for understanding how Fourier series behave.

Lipschitz conditions and Dini's test help prove uniform convergence for certain functions. The Dirichlet kernel and Riemann-Lebesgue lemma are key tools for studying Fourier series convergence, while Carleson's theorem gives a powerful result on almost everywhere convergence.

Convergence Types

Pointwise and Uniform Convergence

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• Pointwise convergence occurs when a sequence of functions $f_n(x)$ converges to a limit function $f(x)$ at each individual point $x$ in the domain
  • For each fixed $x$, $\lim_{n\to\infty} f_n(x) = f(x)$
  • Does not guarantee uniform behavior or convergence rate across the entire domain
• Uniform convergence is a stronger form of convergence where the sequence of functions $f_n(x)$ converges to the limit function $f(x)$ uniformly across the entire domain
  • For every $\varepsilon > 0$, there exists an $N$ such that $|f_n(x) - f(x)| < \varepsilon$ for all $n \geq N$ and all $x$ in the domain
  • Guarantees that the convergence is uniform and the rate of convergence is the same across the entire domain
• Uniform convergence implies pointwise convergence, but the converse is not always true
  • A sequence of functions may converge pointwise but not uniformly (Fourier series of a square wave)

Lipschitz Condition and Dini's Test

• The Lipschitz condition is a property of functions that limits the rate of change of the function
  • A function $f(x)$ satisfies the Lipschitz condition if there exists a constant $K > 0$ such that $|f(x) - f(y)| \leq K|x - y|$ for all $x$ and $y$ in the domain
  • Functions satisfying the Lipschitz condition are continuous and have bounded derivatives
• Dini's test is a sufficient condition for the uniform convergence of a series of functions
  • If a series of non-negative functions $\sum_{n=1}^\infty f_n(x)$ satisfies $\sum_{n=1}^\infty \sup_{x\in D} f_n(x) < \infty$, then the series converges uniformly on the domain $D$
  • Useful for proving uniform convergence of Fourier series for functions with certain properties (continuous functions with bounded variation)

Fourier Series Kernels

Dirichlet Kernel and Riemann-Lebesgue Lemma

• The Dirichlet kernel $D_n(x)$ is a sequence of functions that plays a crucial role in the convergence of Fourier series
  • Defined as $D_n(x) = \frac{1}{2\pi} \sum_{k=-n}^n e^{ikx} = \frac{1}{2\pi} \frac{\sin((n+\frac{1}{2})x)}{\sin(\frac{x}{2})}$
  • Used to represent the partial sums of Fourier series and study their convergence properties
• The Riemann-Lebesgue lemma states that the Fourier coefficients of an integrable function tend to zero as the index tends to infinity
  • If $f(x)$ is integrable on $[-\pi, \pi]$, then $\lim_{n\to\infty} \hat{f}(n) = 0$, where $\hat{f}(n)$ are the Fourier coefficients of $f(x)$
  • Provides information about the asymptotic behavior of Fourier coefficients and the smoothness of the function

Dirichlet's Test for Convergence

• Dirichlet's test is a criterion for the convergence of a series of the form $\sum_{n=1}^\infty a_n b_n$
  • If the sequence $\{a_n\}$ is decreasing, tends to zero, and has bounded partial sums, and the sequence $\{b_n\}$ is bounded, then the series $\sum_{n=1}^\infty a_n b_n$ converges
  • Can be applied to prove the convergence of certain Fourier series by choosing appropriate sequences $\{a_n\}$ and $\{b_n\}$ (Fourier series of piecewise smooth functions)

Advanced Convergence Theorems

Carleson's Theorem

• Carleson's theorem, also known as the Carleson-Hunt theorem, is a significant result in harmonic analysis concerning the almost everywhere convergence of Fourier series
  • States that the Fourier series of any function in $L^p([-\pi, \pi])$, $1 < p < \infty$, converges almost everywhere to the function
  • Generalizes earlier results on the convergence of Fourier series, such as the Riesz-Fischer theorem and the Kolmogorov-Seliverstov-Plessner theorem
• The proof of Carleson's theorem is highly non-trivial and relies on deep techniques from harmonic analysis and probability theory
  • Involves the study of maximal functions, square functions, and the use of the Carleson-Hunt inequality
• Carleson's theorem has important implications for the study of Fourier series and their convergence properties
  • Provides a strong sufficient condition for the almost everywhere convergence of Fourier series
  • Highlights the connection between the integrability of a function and the convergence of its Fourier series