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A bounded $C^{1}$ periodic oscillator made from a quartic Hermite spline

Example

Define $ψ (t) = 16 u^{2} (1 - u)^{2}$ , where $u = t - ⌊ t ⌋ \in [0, 1)$ . Then $ψ$ is bounded, nonconstant, $1$ -periodic, and $C^{1}$ . Moreover, $ψ^{'}$ takes the values $3$ and $- 3$ in every period.

Facts & Assumptions

Given: The displayed definition.

[L1]

The integer-part lemma supplies the unique $u \in [0, 1)$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[L2]

Polynomial derivatives follow from Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, and Integer powers $a^{m}$ .

Verification

technique · direct

1.1

On every interval $[k, k + 1)$ , $ψ$ is the same quartic in $u = t - k$ , with derivative $32 u (1 - u) (1 - 2 u)$ . Its values and first derivatives at $u = 0$ and $u = 1$ are all $0$ , so adjacent pieces and their derivatives agree continuously at every integer.

L1L2algebra

2.1

Translation by an integer leaves the fractional part unchanged, so $ψ$ is $1$ -periodic. Step 1.1 and the polynomial formula prove $C^{1}$ -regularity, and $0 \leq ψ \leq 1$ . At fractional parts $u = 1 / 4$ and $u = 3 / 4$ , the derivative formula gives $ψ^{'} = 3$ and $ψ^{'} = - 3$ , respectively.

step 1.1L1L2algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)

A bounded C1C1 periodic oscillator made from a quartic Hermite spline