We show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces $ V_{p,j} $ of $ L_{2}(\mathbb{R}) $ which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters they even satisfy all those properties. Moreover we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from $\mathcal{S}'(\mathbb{R})$. Finally we also verify that quarks and quarklets can be used to define sequences of subspaces $ W_{p,j} $ of $ L_{2}(\mathbb{R}) $ that yield non-orthogonal decompositions of $ L_{2}(\mathbb{R}) $.
翻译：我们显示B- spline 夸克和相关的夸克子体符合双振多波子理论。 夸克矢量用于定义满足多种分辨率分析几乎所有条件的子空间序列 $ V ⁇ p, j}$ $ ⁇ 2} (\ mathbb{R}) 美元。 在参数的某些特殊条件下, 它们甚至满足所有这些特性。 此外, 我们证明 夸克和 qurklets 拥有满足完美重建条件的调制矩阵。 此外, 我们显示存在通用的双夸克和夸克子, 已知它们至少由$\ mathcal{S} (\mathb{R}$ ) 以压实方式支持的温度分布 。 最后, 我们还核实, 夸克和 ⁇ 可以用来定义子空间序列 $ W ⁇ p, j} $ L ⁇ 2} (\mathb{R} $ 产生 $ L ⁇ 2} (mathrobb} $ 。
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