标准正交基

Theorem 7:For j=1,2,〓,we have〓and for any 〓 and 〓,the function family〓is an orthogonal basis of 〓.

定理7对每个j=1，2，…有〓13而且，对于每个〓和k=1，〓，函数族〓14是〓的一个标准正交基。

Corollary 8:For 〓,we have〓Furthermore 〓 isan orthogonal basis of 〓.

推论8对每个j=0，1，2〓〓…而族〓是〓的一个标准正交基。

By applying elementary transformation,this paper obtains a new method to search for a normal orthogonal basis in Euclid space, it also gives a procedure by which the fewer third elementary transformations are enough to realize this method.

导出用初等变换法求Euclid空间的标准正交基的方法，并进一步获得了只需进行较少次数的第三种类型的初等变换就能实现这一方法的结果。

The representation of all the solutions for the equation Lv=f is obtained through constructing a standard orthogonal basis of null space N.

通过构造零空间的一组标准正交基，得到了线性算子方程Lv=f的所有解的表达形式。

On the foundation of the conception of orthonormal basis in finite dimensional Euclidean space,this paper provides the theory of completely orthonormal system in infinite dimensional Euclidean space.

从有限维欧氏空间的标准正交基概念出发，构建了无限维欧氏空间的完全规范正交系理论。

Finite dimensional euclidean spaces Scalar products, norms, orthonormal basis and orthonormalization.

有限维欧几里德空间标量的产品，规范，标准正交基和正交化。

Through the "folded"transform, the "folded"orthonormal wavelet basis for the space L2 [0,1] is got.

对于L 2上的Littlewood-paley 正交小波基进行"折叠"映射，使得折叠小波基为L 2[0，1]的标准正交基。

Firstly, a two-dimension tensor product space is constructed, in which reproducing kernel exists. Secondly a multiresolution analysis is constructed in the reproducing kernel space. Then an orthonormal basis in is obtained. Thus,the reproducing kernel space can be expressed by wavelets spaces. Wavelets approximation formula and sampling formula can also be obtained in the space. Besides, the wavelets approximation formula is simple and easy for calculation. That completes the theory of multiresolution analysis in finite interval fully and provides theoretical base and algorithm for application.

首先，构造二维张量积空间，并证明该空间具有再生核；其次在再生核空间中，建立多尺度分析，获得该空间的一个标准正交基，使得再生核空间可以由小波空间来刻画，进而得到小波级数和相应的采样公式，而且给出的小波级数形式简单易于数值分析，进一步完善了有限区间上的多尺度分析方法，也为实际应用提供了良好的理论基础和算法。

We showed that in noncommutative case, orthonormal base vectors can be chosen to be unitary operators of any fix order.

我们证明了，在非交换条件下，能够选择标准正交基向量成为任何固定序的酉算子。

The characteristic vector of the real symmetric matrix should be found out, which is orthogonalized and normalized to a standard orthogonal base and is used as row vector to construct the transformation matrix P, so the P~(-1)AP can be made into diagonal matrix.

实对称矩阵A经相似变换P-1AP可化为对角矩阵，在x =Py 下，不一定能化A的二次型为标准型；应寻求对称矩阵A的特征向量，将其正交化并单位化作为标准正交基，作为列向量构造变换矩阵P，可使P-1AP=Λ为对角阵，在x =Py 下，要将二次型化为标准型，且二次项系数即为对角阵Λ主对角线上元素。

It has been put forward that there exists single Ball point and double Ball points on the symmetrical connecting-rod curves of equilateral mechanisms.

从鲍尔点的形成原理出发，分析对称连杆曲线上鲍尔点的产生条件，提出等边机构的对称连杆曲线上有单鲍尔点和双鲍尔点。

The factory affiliated to the Group primarily manufactures multiple-purpose pincers, baking kits, knives, scissors, kitchenware, gardening tools and beauty care kits as well as other hardware tools, the annual production value of which reaches US$ 30 million dollars.

集团所属工厂主要生产多用钳、烤具、刀具、剪刀、厨具、花园工具、美容套等五金产品，年生产总值3000万美元，产品价廉物美、选料上乘、质量保证，深受国内外客户的青睐