平移不变$(γ_1,γ_2)$空间的$h$-$γ$开花、$h$-$γ$伯恩斯坦基与$h$-$γ$贝塞尔曲线
$h$-$γ$ Blossoming, $h$-$γ$ Bernstein Bases, and $h$-$γ$ Bézier Curves for Translation Invariant $\left(γ_{1},γ_{2}\right)$ Spaces
AI总结：
本文针对平移不变$(γ_1,γ_2)$空间，融合$γ$-开花与$h$-开花方法构建$h$-$γ$开花，定义$h$-$γ$伯恩斯坦基和贝塞尔曲线并推导其核心算法与恒等式。
摘要：
n阶$(γ_1,γ_2)$空间是由$\{γ_1^{n-k}(x), γ_2^{k}(x)\}_{k=0}^{n}$张成的一元函数空间。若$γ_1(x-h)$与$γ_2(x-h)$可表示为$γ_1(x)$和$γ_2(x)$的非奇异线性组合，则称该$(γ_1,γ_2)$空间具有平移不变性。平移不变$(γ_1,γ_2)$空间涵盖多项式（$γ_1(x)=1, γ_2(x)=x$）、三角函数（$γ_1(x)=\cos x, γ_2(x)=\sin x$）、双曲函数（$γ_1(x)=\cosh x, γ_2(x)=\sinh x$）及其离散对应形式。本文将适用于$(γ_1,γ_2)$空间的$γ$-开花方法，与适用于$h$-伯恩斯坦基、$h$-贝塞尔曲线的$h$-开花方法相融合，针对由两个连续线性无关函数$γ_1$、$γ_2$生成的平移不变$(γ_1,γ_2)$空间，构建了全新的$h$-$γ$开花方法。基于该$h$-$γ$开花，本文定义了$h$-$γ$伯恩斯坦基与$h$-$γ$贝塞尔曲线并研究其性质，推导了这类$h$-$γ$伯恩斯坦格式与$h$-$γ$贝塞尔格式的递归求值算法、细分流程、Marsden恒等式，以及升阶公式与插值公式。
英文摘要：
A $\left(γ_{1}, γ_{2}\right)$ space of order $n$ is a space of univariate functions spanned by $\left\{γ_{1}^{n-k}(x), γ_{2}^{k}(x)\right\}_{k=0}^{n}$. A $\left(γ_{1}, γ_{2}\right)$ space is said to be translation invariant if $γ_{1}(x-h)$ and $γ_{2}(x-h)$ can be expressed as nonsingular linear combinations of $γ_{1}(x)$ and $γ_{2}(x)$. Translation invariant $\left(γ_{1}, γ_{2}\right)$ spaces include polynomials $\left(γ_{1}(x)=1, γ_{2}(x)=x\right)$, trigonometric functions $\left(γ_{1}(x)=\cos x, γ_{2}(x)=\sin x\right)$, hyperbolic functions $\left(γ_{1}(x)=\cosh x, γ_{2}(x)=\sinh x\right)$, and their discrete analogues. We merge $γ$-blossoming for $\left(γ_{1}, γ_{2}\right)$ spaces with $h$-blossoming for $h$-Bernstein bases and $h$-Bézier curves to construct a novel $h$-$γ$ blossom for translation invariant $\left(γ_{1}, γ_{2}\right)$ spaces generated by two continuous, linearly independent functions $γ_{1}$ and $γ_{2}$. Based on this $h$-$γ$ blossom, we define $h$-$γ$ Bernstein bases and $h$-$γ$ Bézier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these $h$-$γ$ Bernstein and $h$-$γ$ Bézier schemes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
