$h$-$gamma$ Blossoming, $h$-$gamma$ Bernstein Bases, and $h$-$gamma$ B'{e}zier Curves for Translation Invariant $left(gamma_{1},gamma_{2}right)$ Spaces
arXiv:2604.08697v1 Announce Type: new
Abstract: A $left(gamma_{1}, gamma_{2}right)$ space of order $n$ is a space of univariate functions spanned by $left{gamma_{1}^{n-k}(x), gamma_{2}^{k}(x)right}_{k=0}^{n}$. A $left(gamma_{1}, gamma_{2}right)$ space is said to be translation invariant if $gamma_{1}(x-h)$ and $gamma_{2}(x-h)$ can be expressed as nonsingular linear combinations of $gamma_{1}(x)$ and $gamma_{2}(x)$. Translation invariant $left(gamma_{1}, gamma_{2}right)$ spaces include polynomials $left(gamma_{1}(x)=1, gamma_{2}(x)=xright)$, trigonometric functions $left(gamma_{1}(x)=cos x, gamma_{2}(x)=sin xright)$, hyperbolic functions $left(gamma_{1}(x)=cosh x, gamma_{2}(x)=sinh xright)$, and their discrete analogues. We merge $gamma$-blossoming for $left(gamma_{1}, gamma_{2}right)$ spaces with $h$-blossoming for $h$-Bernstein bases and $h$-B'{e}zier curves to construct a novel $h$-$gamma$ blossom for translation invariant $left(gamma_{1}, gamma_{2}right)$ spaces generated by two continuous, linearly independent functions $gamma_{1}$ and $gamma_{2}$. Based on this $h$-$gamma$ blossom, we define $h$-$gamma$ Bernstein bases and $h$-$gamma$ B'{e}zier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these $h$-$gamma$ Bernstein and $h$-$gamma$ B'{e}zier schemes.
