CAJ | 학술논문

具有切向边界的无散度小波在向量场的数值模拟中扮演着重要的角色。鉴于 Hardin-Marasovich 小波函数的零边值性质和简单结构，主要研究一类利用 Hardin-Marasovich 小波函数构造的具有切向边界的三维各向同性无散度多小波。首先，基于 Hardin-Marasovich 小波函数的微分关系，证明了具有切向边界的无散度向量场在对应的向量尺度空间上的双正交投影还是无散度的。其次，利用无散度空间的刻画给出了各向同性无散度尺度函数的定义，并证明对应的无散度尺度函数空间构成了一个无散度多尺度分析。最后，定义各向同性无散度多小波，给出切向边界无散度向量在无散度小波基下分解系数与经典小波基下分解系数的关系，从而说明无散度向量的小波分解系数可快速计算。
구유절향변계적무산도소파재향량장적수치모의중분연착중요적각색。감우 Hardin-Marasovich 소파함수적령변치성질화간단결구，주요연구일류이용 Hardin-Marasovich 소파함수구조적구유절향변계적삼유각향동성무산도다소파。수선，기우 Hardin-Marasovich 소파함수적미분관계，증명료구유절향변계적무산도향량장재대응적향량척도공간상적쌍정교투영환시무산도적。기차，이용무산도공간적각화급출료각향동성무산도척도함수적정의，병증명대응적무산도척도함수공간구성료일개무산도다척도분석。최후，정의각향동성무산도다소파，급출절향변계무산도향량재무산도소파기하분해계수여경전소파기하분해계수적관계，종이설명무산도향량적소파분해계수가쾌속계산。
Divergence-free wavelets with tangential boundary plays an important role in numerical simu-lation of vector fields.In view of the zero boundary and the simple structure of Hardin-Marasovich wave-lets,a class of three-dimensional isotropic divergence-free wavelets with tangential boundary are studied. Firstly,based on differential relations of Hardin-Marasovich wavelet functions,it is proved that the bi-orthogonal projection of divergence-free vector fields is still divergence-free.Then,the definition of iso-tropic divergence-free scale functions are given based on the characterization of divergence-free space, and the corresponding divergence-free scale spaces are proved to form a multiresolution analysis.Finally, the isotropic divergence-free multiwavelets are defined,and the relation between the decomposition coef-ficients of the divergence-free wavelets and the classical wavelets is given,which shows that the diver-gence-free decomposition coefficients can be fastly computed.