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An interval map is a continuous function $F:I\to I$ from a real interval to itself. Its iterates form a one-dimensional discrete dynamical system.

A Borel probability measure is invariant for a Markov kernel semigroup if every transition preserves it. For a Feller semigroup on a compact metric space, it suffices to test the displayed identity on continuous functions. Kernel Jensen inequality gives contraction on $L^p(\mu )$, and sup-norm strong continuity followed by density gives strong continuity there for $1\le p<\infty$.

A left-topological semigroup requires only that every right translation $x\mapsto xa$ be continuous.

Between two prescribed values, linear interpolation assigns $(1-t)f(a)+tf(b)$ at $(1-t)a+tb$, for $0\le t\le 1$. Joining successive samples produces a piecewise-linear function. For a continuous function on a compact interval, interpolants along grids with mesh tending to zero converge in the uniform norm; their error is bounded by the modulus of continuity at the mesh size.

For a continuous function $f:{\mathbb{Z}}_p\to {\mathbb{Q}}_p$, its nth Mahler coefficient isThis is the explicit iterate of the forward difference operator. These coefficients are exactly those of its Mahler expansion.

If $f\in L^1(\mathbb{R})$ and $x^{k}f\in L^1(\mathbb{R})$ for $0\le k\le r$, then its Fourier transform is $r$ times continuously differentiable and the displayed formula holds. The dominated convergence theorem permits each derivative under the integral. Thus $r+1$ vanishing moments give a zero Taylor polynomial through degree $r$ at the origin. Unweighted $L^1$ membership only guarantees a continuous function as the Fourier transform.