In this paper, an algorithm is discussed for constructing the two-direction Armlet multiwavelet. First, the definition of two-direction Armlet multiwavelet is presented in this paper. A two-direction multiwavelet can be changed to a special multi-wavelet. By Two-scale Similar Transform (TST), a transform can be taken on the two-scale matrix symbols of a two-direction multi-wavelets. This transform keeps the orthogonality of the two-direction multi-wavelets. However, the condition is discussed which the two-direction multi-wavelet corresponding to a two-direction multi-scaling function is an Armlet with order [Formula: see text]. An approach is given for constructing the transform matrix. Finally, an example is given for discussing the two-direction Armlet multi-wavelet with order 2.

In this paper, a special group is used to introduce the two-direction two-scale similarity transform (TDTST) of the two-direction refinable function, and the existence of the solution of the two-direction refinable equation is discussed by the TDTST transform. The approximation order of the two-direction refinable function is discussed by the TDTST transform. Whether the TDTST transform can maintain the orthogonality of the two-direction refinable function is discussed, and it is pointed out that a special real orthogonal matrix can maintain the orthogonality. It is pointed out that the TDTST transform can maintain the symmetry of the two-direction refinable function. Finally, an example is given.

In recent years, the concepts of frame in [Formula: see text]-Hilbert space and [Formula: see text]-frames in [Formula: see text]-Hilbert space have been introduced. The [Formula: see text]-frames in the [Formula: see text]-Hilbert space are more generalized than the ordinary frames. This paper deals with the problem of sum of [Formula: see text]-frames in [Formula: see text]-Hilbert space. First, the sufficient condition for the finite sum of [Formula: see text]-frames and Bessel sequences in [Formula: see text]-Hilbert space is given. Then, some results on the operator [Formula: see text] and pre-frame operator are given, and some new results on the sum of [Formula: see text]-frame are proved. We then discuss several results for the perturbation and stability of the [Formula: see text]-frames in [Formula: see text]-Hilbert space about the sum of the [Formula: see text]-frames.