Approximation by crystal-refinable functions

MOLTER, URSULA M.; MOURE, MARÍA DEL CARMEN; QUINTERO, ALEJANDRO

GEOMETRIAE DEDICATA

SPRINGER

Lugar: Berlin; Año: 2020 vol. 207 p. 1 - 21

0046-5755

Let 𝛤 be a crystal group in ℝ^𝑑. A function 𝜑:ℝ^𝑑⟶ℂ is said to be crystal-refinable (or 𝛤-refinable) if it is a linear combination of finitely many of the rescaled and translated functions 𝜑(𝛾−1(𝑎𝑥)), where the translations𝛾 are taken on a crystal group 𝛤, and a is an expansive dilation matrix such that 𝑎𝛤𝑎−1⊂𝛤. A 𝛤-refinable function 𝜑:ℝ^𝑑→ℂ satisfies a refinement equation 𝜑(𝑥)=∑_{𝛾∈𝛤}𝑑_𝛾𝜑(𝛾−1(𝑎𝑥)) with 𝑑_𝛾∈ℂ. Let S(𝜑) be the linear span of {𝜑(𝛾−1(𝑥)):𝛾∈𝛤} and S^h=={𝑓(𝑥/ℎ):𝑓∈S(𝜑) }. One important property of S(𝜑) is, how well it approximates functions in 𝐿2(ℝ^𝑑). This property is very closely related to the crystal-accuracy of S(𝜑) , which is the highest degree p such that all multivariate polynomials q(x) of degree(𝑞)