3.4 Dini's test and Jordan's test for convergence

Dini's test and Jordan's test are powerful tools for determining the convergence of Fourier series. These tests help us understand when a series converges uniformly or at specific points, which is crucial for analyzing functions in harmonic analysis.

By examining conditions like monotonicity, bounded variation, and Lipschitz continuity, we can predict how Fourier series behave. This knowledge is key for working with complex functions and their representations in real-world applications.

Dini's Test and Uniform Convergence

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 point $x$ in the domain
• Uniform convergence is a stronger condition that requires the sequence of functions to converge to the limit function uniformly across the entire domain
  • For uniform convergence, the maximum difference between $f_n(x)$ and $f(x)$ must approach zero as $n$ approaches infinity, regardless of the choice of $x$
  • Example: The sequence of functions $f_n(x) = x^n$ on the interval $[0, 1]$ converges pointwise to the function $f(x) = 0$ for $x \in [0, 1)$ and $f(1) = 1$, but the convergence is not uniform
• Uniform convergence implies pointwise convergence, but the converse is not always true

Dini's Test for Uniform Convergence

• Dini's test provides a sufficient condition for the uniform convergence of a sequence of continuous functions
• If a sequence of continuous functions $f_n(x)$ defined on a compact interval $[a, b]$ satisfies the following conditions, then it converges uniformly to a continuous limit function $f(x)$:
  1. $f_n(x) \leq f_{n+1}(x)$ for all $x \in [a, b]$ and all $n$ (monotonically increasing)
  2. The sequence $f_n(x)$ converges pointwise to $f(x)$ on $[a, b]$
• Example: Consider the sequence of functions $f_n(x) = 1 - \frac{1}{n}e^x$ on the interval $[0, 1]$. It can be shown that $f_n(x)$ satisfies the conditions of Dini's test, and thus converges uniformly to the limit function $f(x) = 1$

Lipschitz Condition and Uniform Convergence

• The Lipschitz condition is a property of functions that limits the rate at which the function can change with respect to changes in its input
• A function $f(x)$ is said to satisfy the Lipschitz condition on an interval $[a, b]$ if there exists a constant $L > 0$ such that $|f(x) - f(y)| \leq L|x - y|$ for all $x, y \in [a, b]$
• If a sequence of functions $f_n(x)$ converges pointwise to a function $f(x)$ on $[a, b]$, and each $f_n(x)$ satisfies the Lipschitz condition with the same Lipschitz constant $L$, then the convergence is uniform
• Example: The sequence of functions $f_n(x) = \frac{x}{n}$ on the interval $[0, 1]$ satisfies the Lipschitz condition with $L = \frac{1}{n}$, and converges uniformly to the limit function $f(x) = 0$

Jordan's Test and Bounded Variation

Jordan's Test for Convergence

• Jordan's test provides a sufficient condition for the convergence of a Fourier series of a function $f(x)$ at a point $x_0$
• If a function $f(x)$ is of bounded variation on an interval $[a, b]$ containing $x_0$, then the Fourier series of $f(x)$ converges to $\frac{1}{2}[f(x_0^+) + f(x_0^-)]$ at $x_0$, where $f(x_0^+)$ and $f(x_0^-)$ denote the right-hand and left-hand limits of $f(x)$ at $x_0$, respectively
• Example: The function $f(x) = x$ on the interval $[-\pi, \pi]$ is of bounded variation, and its Fourier series converges to $0$ at $x_0 = 0$, which is equal to $\frac{1}{2}[f(0^+) + f(0^-)]$

Bounded Variation and Total Variation

• A function $f(x)$ is said to be of bounded variation on an interval $[a, b]$ if its total variation on $[a, b]$ is finite
• The total variation of a function $f(x)$ on $[a, b]$ is defined as the supremum of $\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|$ over all partitions $a = x_0 < x_1 < \ldots < x_n = b$ of the interval
• Functions of bounded variation include monotonic functions, piecewise monotonic functions, and functions with a finite number of discontinuities
• Example: The function $f(x) = \sin(x)$ on the interval $[0, 2\pi]$ is of bounded variation, as its total variation is equal to $4$

Discontinuities in Fourier Series

• The behavior of a Fourier series at discontinuities of the function depends on the type of discontinuity
• If a function $f(x)$ has a jump discontinuity at a point $x_0$, then the Fourier series of $f(x)$ converges to the average of the left-hand and right-hand limits at $x_0$, i.e., $\frac{1}{2}[f(x_0^+) + f(x_0^-)]$
• If a function $f(x)$ has a removable discontinuity or an infinite discontinuity at a point $x_0$, then the Fourier series of $f(x)$ may not converge at $x_0$
• Example: The function $f(x) = \begin{cases} 1, & 0 \leq x < \pi \\ -1, & \pi \leq x < 2\pi \end{cases}$ has a jump discontinuity at $x_0 = \pi$, and its Fourier series converges to $0$ at $x_0$, which is equal to $\frac{1}{2}[f(\pi^+) + f(\pi^-)]$