Functional Series

Fourier series is an example of a functional series. Here we look at important tools in our later discussion of Fourier series.

There are three main results:

• Uniform convergence of a sequence of continuous functions gives us a continuous function as a limit
• Weierstrass’ Majorant Theorem, gives a condition that guarantees that a functional series converges to a continuous function
• Integrals of a sequence of functions which converges uniformly to a limit function \(f(x)\) also converge with the limit being the integral of \(f(x)\)

Pointwise and uniform convergence

Pointwise convergence

Let \((f^{(n)})_{n=1}^\infty\) be a sequence of functions from one metric space \((X,d_X)\) to another \((Y,d_Y)\), and let \(\{f : X \to Y \}\) be another function. We say that \((f^{(n)})_{n=1}^\infty\) converges pointwise to \(f\) on \(X\) if we have

$$\lim\limits_{n\rightarrow\infty} f^{(n)}(x) = f(x)$$ for all \(x\in X\), i.e. $$\lim\limits_{n\rightarrow\infty} d_Y(f^{(n)}(x),f(x)) = 0$$
We call the function \(f\) the pointwise limit of the functions \(f^{(n)}\).
Note that \(f^{(n)}(x)\) and \(f(x)\) are points in \(Y\), rather than functions.

Uniform convergence

Let \((f^{(n)})_{n=1}^\infty\) be a sequence of functions from one metric space \((X,d_X)\) to another \((Y,d_Y)\), and let \(\{f : X \to Y\}\) be another function. We say that \((f^{(n)})_{n=1}^\infty\) converges uniformly to \(f\) on \(X\) if for every \(\varepsilon \gt 0\) there exists \(N \gt 0\) such that \(d_Y(f^{(n)}(x),f(x)) \lt \varepsilon\) for every \(n \gt N\) and \(x\in X\).
We call the function \(f\) the uniform limit of the functions \(f^{(n)}\).
Note that in the definition of pointwise convergence, \(N\) was allowed to depend on \(x\); here uniform convergence it is not.

Uniform convergence and continuity

Suppose \((f^{(n)})_{n=1}^\infty\) is a sequence of continuous functions on an interval \(I\) and suppose also that \((f^{(n)})_{n=1}^\infty\) converges uniformly to \(f\) on the interval \(I\). Then the limit function \(f\) is also continuous.

Functional series

A functional series is a series \(\sum\limits_{k=0}^\infty u_k(x)\) where each term of the series \(u_k(x)\) is a function on an interval \(I\).

Pointwise convergent

The functional series \(\sum\limits_{k=0}^\infty u_k(x)\) is pointwise convergent for each \(x \in I\) if the limit \(\sum\limits_{k=0}^\infty u_k(x) = \lim\limits_{N\rightarrow\infty}\sum\limits_{k=0}^N u_k(x)\) exists for each \(x \in I\).

Weierstrass’ Majorant Theorem

Suppose that the functional series \(\sum\limits_{k=0}^\infty u_k(x)\) is defined on an interval \(I\) and that there is a sequence of positive constants \(M_k\) so that \(|u_k(x)| \le M_k, k = 0,1,2,\dotsc\) for all \(x \in I\). If \(\sum\limits_{k=0}^\infty M_k\) converges, then \(\sum\limits_{k=0}^\infty u_k(x)\) converges uniformly on \(I\).
So fortunately, while most continuous functions are not as well behaved as polynomials, they can always be uniformly approximated by polynomials. Polynomials are always very well behaved, e.g. being always differentiable.

If
\((i)\) the functional series \(u_k(x)\) converges uniformly on interval \(I\);
\((ii)\) \(u_k(x)\) is a continuous function on \(I\) for each \(k = 0,1,2,\dotsc\).
then
\(S(x) = \sum\limits_{k=0}^\infty u_k(x)\) is continuous on \(I\);

$$\int_a^x (\sum\limits_{k=0}^\infty u_k(t))\,dt = \sum\limits_{k=0}^\infty (\int_a^x u_k(t)\,dt)$$;

Power series

A formal power series centered at \(a\) is any series of the form

$$\sum\limits_{n=0}^\infty c_n(x-a)^n$$ where the coefficients \(c_0, c_1, c_2, \dotsc\) are real or complex numbers.

Radius of convergence

Let \(\sum\limits_{n=0}^\infty c_n(x-a)^n\) be a formal power series. The radius of convergence R of this series to be the quantity

$$R := \dfrac{1}{\limsup_{n\rightarrow\infty} |c_n|^{1/n}}$$

Real analytic functions

A function \(f(x)\) which is lucky enough to be representable as a power series has a special name, it is a real analytic function.
Let \(E\) be a subset of \(\mathbb{R}\), and let \(\{ f : E \rightarrow\mathbb{R} \}\) be a function. If \(a\) is an interior point of \(E\), we say that \(f\) is real anlytic at \(a\) if there exists an open interval \((a-r, a+r)\) in \(E\) for some \(r \gt 0\) such that there exists a power series \(\sum\limits_{n=0}^\infty c_n(x-a)^n\) centered at \(a\) which has a radius of convergence greater than or equal to \(r\), and which converges to \(f\) on \((a-r, a+r)\). If \(E\) is an open set, and \(f\) is real analytic at every point \(a\) of \(E\), we say that \(f\) is real analytic on \(E\).

Reference:

1. Terence Tao. (2009). Analysis I. Analysis II (Hindustan Book Agency)
2. TATA57 Transform Theory. Linköping University (Sweden: http://courses.mai.liu.se/GU/TATA57/)

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