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

Example

Define $\psi(t)=16u^2(1-u)^2$ , where $u=t-\lfloor t\rfloor\in[0,1)$ . Then $\psi$ is bounded, nonconstant, $1$ -periodic, and $C^1$ . Moreover, $\psi'$ 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 $m \le x < m + 1$ ).

[L2]

Polynomial derivatives follow from Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$ , $(\alpha f)'(c) = \alpha f'(c)$ , $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$ , and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$ , For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$ ; 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 $[k,k+1)$ , $\psi$ is the same quartic in $u=t-k$ , with derivative $32u(1-u)(1-2u)$ . 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 $\psi$ is $1$ -periodic. Step 1.1 and the polynomial formula prove $C^1$ -regularity, and $0\le\psi\le1$ . At fractional parts $u=1/4$ and $u=3/4$ , the derivative formula gives $\psi'=3$ and $\psi'=-3$ , respectively.

step 1.1L1L2algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 92 results over 29 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

A bounded C1C^1C1 periodic oscillator made from a quartic Hermite spline