We study shift invariant spaces generated by refinable distributions. We classify the summation and the intersection of shift invariant spaces generated by refinable distributions,and prove that they are also shift invariant spaces generated by refinable distributions.
As a typical family of mono-component signals,the nonlinear Fourier basis {eikθa(t)}k∈Z,defined by the nontangential boundary value of the M¨obius transformation,has attracted much attention in the field of nonlinear and nonstationary signal processing in recent years.In this paper,we establish the Jackson's and Bernstein's theorems for the approximation of functions in Xp(T),1 p ∞,by the nonlinear Fourier basis.Furthermore,the analogous theorems for the approximation of functions in Hardy spaces by the finite Blaschke products are established.
The V-system is a complete orthogonal system of functions defined on the interval [0, 1], generated by finite Legendre polynomials and the dilation and translation of a function generator, which consists of a finite number of continuous and discontinuous functions. The V-system has interesting properties, such as orthogonality, symmetry, completeness and short compact support. It is shown in this paper that the V-system is essentially a special multi-wavelet basis. As a result, some basic properties of the V-system are established through the well-developed theory of multi-wavelets. From this point of view, more other V-systems are constructed.
