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Sequences and Series of Functions

Course Material
  1. Sequences of functions may converge pointwise or uniformly; uniform convergence is stronger (see page 6).
  2. The difference between pointwise and uniform convergence lies in the order of quantifiers in the epsilon definition.
  3. The Banach space \(B(X, E)\) consists of bounded functions \(f:X \to E\) with the sup norm.
  4. Uniform limits in \(C[0,1]\) stay in \(C[0,1]\); pointwise limits may not (see page 11).
  5. Lp-convergence (for \(p \geq 1\)) can differ from pointwise or uniform convergence (see pages 13–14); on bounded intervals, uniform implies \(L^p\), but not vice versa.
  6. In Banach spaces, absolute convergence implies convergence.
  7. The Weierstrass M-test ensures uniform convergence of series in \(C[a,b]\) or \(Cb(R)\).
  8. It was used by Weierstrass to construct a nowhere differentiable continuous function.
  9. A series of differentiable functions may converge to a non-differentiable limit unless the derivatives converge absolutely.
02/07 Lecture Slides

Types of Convergence

A sequence of numbers \(\{x_n \}_{n = 1}^\infty \subset \mathbb{R}\) converge to a number \(x\) if:

\[ \text{"for every } \epsilon > 0, \text{ there is a } K(\epsilon) \in \mathbb{N} \text{ such that } |x_n - x| < \epsilon \text{ when } n \geq K \text{."} \]

Definition: A sequence of functions \(f_n: X \to \mathbb{R}\) converges pointwise to \(f\) if for every \(x \in X\) and \(\epsilon > 0\), there is a \(K(x, \epsilon) \in \mathbb{N}\) such that \(|f_n(x) - f(x)| < \epsilon\) when \(n \geq K\).

Definition: A sequence of functions \(f_n: X \to \mathbb{R}\) converges uniformly to \(f\) if for every \(\epsilon > 0\), there is a \(K(\epsilon) \in \mathbb{N}\) such that for every \(x \in X, |f_n(x) - f(x)| < \epsilon\) when \(n \geq K\).

Bounded Functions

Uniform Norm: Let \(X\) be a set, and let \(B(X, \mathbb{R})\) denote the set of bounded real-valued functions on \(X\). The uniform norm is

\[ \lVert f \rVert_\infty = \sup_{x \in X} |f(x)|. \]

Theorem: \((B(X, \mathbb{R}), \lVert \cdot \rVert_\infty)\) is a Banach space.

Let \(X\) be a set and let \(E\) be a Banach space. Then \(B(X, E)\) (bounded \(E\)-valued functions) is a Banach space with the uniform norm. If \(X\) is a metric space, then so is \(C_b(X, E)\) (continuous bounded \(E\)-valued functions).

Let \(\{f _n \}_{n = 1}^\infty\) be a sequences of Riemann integrable functions on an interval \([a, b]\). We say that the sequence **converges in \(L^p\), for some \(p \geq 1\), to an integrable function \(f\) if

\[ \lim_{n \to \infty} \int_a^b |f_n(x) - f(x)|^p \, dx = 0. \]

Series of Functions

If \(\{ x_n \}_{n = 0}^\infty\) is a sequence of numbers, then we write \(\sum_{n = 0}^\infty x_n\) for the corresponding series (meaning sequence of partial sums and/or its limit!).

Similarly, if \(\{f_n(x) \}_{n = 0}^\infty\) is a sequence of functions, we can consider the corresponding series \(\sum_{n = 0}^\infty f_n(x)\).

A series \(\sum_{n = 0}^\infty x_n\) converges absolutely if \(\sum_{n = 0}^\infty |x_n|\) converges. Recall: If a series converges absolutely, then it converges.

Theorem (Absolute convergence implies convergence): Let \(E\) be a Banach space. let \(\{ x_n \}_{n = 0}^\infty\) be a sequence of vectors in \(E\) whose series of norms \(\sum_{n = 0}^\infty \lVert x_n \rVert\) converges (in \(\mathbb{R}\)). Then the series of vectors \(\sum_{n = 0}^\infty x_n\) converges (in \(E\)).

Corollary (Weierstrass M-Test): Let \(\{ f_n \}_{n = 0}^\infty\) be a sequence of real-valued functions on a set \(X\). Suppose that there is a sequence of numbers \(M_n >\geq 0\) such that \(M_n\) is an upper bound for \(f_n\) for each \(n\), and such that the series \(\sum_{n =0 }^\infty M_n\) converges. Then t he series of functions \(\sum_{n = 0}^\infty f_n\) converges uniformly.

Theorem: Let \(\{ f_n \}_{n = 1}^\infty \subseteq C[a, b]\) be a sequence of functions which converges uniformly to \(f\). Then \(\int_a^b f_n(x) dx\) converges to \(\int_a^b f(x) dx\).

Uniform Limits and Differentiation

Theorem: Let \(\{ f_n \}_{n = 1}^\infty \subseteq C[a, b]\) be a uniformly convergent sequence of functions. Suppose that functions \(f_n\) are all differentiable on \((a, b)\), with continuous and bounded derivatives \(f_n'\). Suppose further that the sequence of derivatives \(\{ f_n' \}_{n = 1}^\infty\) converges uniformly on \((a, b)\).

Then the limit function \(f\) is also differentiable on \((a, b)\), and \(f'\) is the uniform limit of the sequence \(\{ f_n' \}_{n = 1}^\infty\).

Corollary: Suppose \(f_n\) is a sequence of continuously differentiable functions on an interval \((a, b)\), and suppose that both the series

\[ f(x) = \sum_{n = 0}^\infty f_n(x) \quad \text{ and } \quad g(x) = \sum_{n = 0}^\infty f_n'(x) \]

converge uniformly. Then \(f(x)\) is differentiable, and we have \(f'(x) = g(x)\).

Let \(\sum_{n = 0}^\infty a_n x^n\) be a power series, and suppose the sequence \(\{|a_n|^{\frac{1}{n}}\}_{n = 0}^\infty\) is bounded. For each \(n \geq 0\), let

\[ b_n = \sup\{|a_n|^{\frac{1}{n}}, |a_{n + 1}|^{\frac{1}{n + 1}}, |a_{n + 2}|^{\frac{1}{n + 2}}, \dots\}. \]

and let

\[ b = \lim_{n \to \infty} b_n \]

(\(b\) is the limit superior or lim sup of the sequence \(\{|a_n|^{\frac{1}{n}}\}_{n = 0}^\infty\)).

Theorem (Cauchy-Hadamard): With notation as above, the power series converges absolutely if \(|x| \cdot b < 1\) and diverges if \(|x| \cdot d > 1\). (True for complex numbers as well!)

Definition: The number \(R = \frac{1}{b}\), for \(b \neq 0\), is called the radius of convergence of the power series. If \(b = 0\) the radius of convergence is said to be \(\infty\). If the sequence \(\{|a_n|^\frac{1}{n}\}_{n = 0}^\infty\) is unbounded, then the radius of convergence is said to be \(0\).

Corollary: Let \(\sum_{n = 0}^\infty a_n x^n\) be a power series, with radius of convergence \(R\). Then the termwise derivative power series \(\sum_{n = 1}^\infty n \cdot a_n x^{n - 1}\) has the same radius of convergence \(R\).

Theorem: Let \(\sum_{n = 0}^\infty a_n x^n\) be a power series, with radius of convergence \(R > 0\). Then the series is differentiable on the interval \((-R, R)\), with the derivative given by \(\sum_{n = 1}^\infty n \cdot a_n x^{n - 1}\).