New Criteria On Concentration Factors And Affine Frames

In the first part of this paper, we discuss the concentration factor methods that make certain the jumps of periodic integral functions. Set σ(x) is a continuous function on [0,1] and (S_n~σ|～)(f, x) is the n-th partial sum of conjugate Fourier series with the factor a(k/n), we consider the convergence of (S_n~σ|～)(f, x) at the ordinary discontinuity ζ. As we make some improvements of the theorem of A.Gelb and E.Tadmor, we get the sufficient and necessary conditions of σ(k/n) which is the concentration factor of / at the ordinary discontinuity ζ. Let μ is a continuous function on [0, ∞) and (P_r~u|～)(f, x) is the Abel-Poisson means of conjugate Fourier series with the factor μ((1 - r)k). Then we consider the convergence of (P_r~u|～)(f, x) at the ordinary discontinuity ζ and establish the relevant theorem. We proved all the theorems established in the paper. In addition,we estimate the vanishing rate of (S_n~σ|～)(f, x) and (P_r~u|～)(f, x) at the arbitrary ordinary discontinuity of f(x). We also establish the parallel theorem in cosine series.In the other part of this paper, we mainly discuss the criterion on affine frames. Affine frame is a very basic concept in wavelet theory. In the well-known Daubechies Criterion on affine frames, the quantities which are calculated in terms of absolute values are applied. In the new criterion established in this paper, we use new criterion quantities which are calculated via algebric sums of suitable combinations of When Ψ is an even function, the new criterion is more advanced than Daubechies Criterion.