Sequences sometimes extend to good examples of continuous functions with special properties e.g. interesting limsups and liminfs. For example, you may want construct a continuous function $$h(x)$$ that continuously alternates between $$f(x)=x$$ and $$g(x)=0$$ so that it has a limsup of infinity and liminf of 0. By observation we could choose $$h(x)=\vert x \sin x\vert$$. We would try to generalise this.

## Definitions

We call a sequence $$(a_n)$$ $$k$$-periodic for some positive integer $$k$$ if there exists continuous functions $$f_i : \mathbb{R} \to \mathbb{R}$$ for $$i \in \{0, 1, \dots , k-1\}$$ such that $$a_n = f_i (n) \text{ if } n \equiv i \; (\text{mod } k)$$

For example, $$a_n = (-1)^n$$ is 2-periodic with $$f_0(x) = 1, f_1(x) = -1$$.

## 2-periodic sequences

Proposition: If $$(a_n)$$ is 2-periodic, then there exists a continuous function $$f: \mathbb{R} \to \mathbb{R}$$ such that for all $$n \in \mathbb{N}$$, $$f(n) = a_n$$

Proof: Suppose $$(a_n)$$ 2-periodic with associated $$f_0$$ and $$f_1$$ functions. Let

$f(x) = \cos^2 \bigg(\frac{\pi x}{2}\bigg) f_0(x) + \sin^2 \bigg(\frac{\pi x}{2}\bigg) f_1(x)$

We could see that if $$x$$ is an even integer then $$f(x)=f_0(x)$$ and if $$x$$ is an odd integer then $$f(x) = f_1(x)$$. Clearly $$f$$ continuous as trignometric functions, $$f_0$$ and $$f_1$$ are continuous.

$\blacksquare$

For example, we can extend $$a_n=(-1)^n$$ to $$f(x) = \cos^2 (\frac{x \pi}{2}) - \sin^2 (\frac{x \pi}{2}) = \cos (\pi x)$$.

## k-periodic sequences

Proposition: If $$(a_n)$$ is k-periodic, then there exists a continuous function $$f: \mathbb{R} \to \mathbb{R}$$ such that for all $$n \in \mathbb{N}$$, $$f(n) = a_n$$

Proof: Suppose $$(a_n)$$ k-periodic with associated $$f_0, \dots , f_k$$ functions. Let

$g_j(x) = \sin \bigg(\frac{\pi}{k} (x-j) \bigg)$

for $$j \in \{0, 1, \dots k - 1\}$$

Note that for all $$j \in \{0, 1, \dots k - 1\}$$ and integers $$i$$, we have $$i \equiv j \; (\text{mod } k) \iff g_j(i) = 0$$. Now we consider

$f(x) = \sum_{i=0}^{k-1} \bigg[ \bigg(\prod_{j=0, j \neq i}^k \frac{g_j(x)}{g_j(i)} \bigg) f_i(x) \bigg]$

We can the verify that $$f$$ is continuous and satisfies $$f(n) = a_n \; \forall n \in \mathbb{N}$$. The idea is that each of the $$g_j$$ acts a sieve and annihilates the necessary terms.

$\blacksquare$

For example we can extend the sequence $$-1, 1, 0, -1, 1, 0, -1, \dots$$ (assuming the sequence starts at $$0$$) which is 3-periodic with $$f_0(x) = -1, f_1(x) = 0, f_2(x) = -1$$ into

\begin{align*} f(x) &= -\frac{\sin(\frac{\pi}{3}(x-1)) \sin(\frac{\pi}{3}(x-2)) }{\sin(\frac{\pi}{3}(0-1)) \sin(\frac{\pi}{3}(0-2))} \\ &\phantom{=}+\frac{\sin(\frac{\pi}{3}(x-0)) \sin(\frac{\pi}{3}(x-2)) }{\sin(\frac{\pi}{3}(1-0)) \sin(\frac{\pi}{3}(1-2))} \\ &= \frac{231}{200} \sin\bigg( \frac{2\pi}{3} (x- \frac{1}{2})\bigg) \end{align*}

Here’s a graph of the function, Next maths post Previous maths post

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