Abstract—The motivation of this dissertation mainly is which affine frame wavelet systems span Lebesgue spaces have been investigated less. The technique we used is analogous to technique of Calderón-Zygmund operators, but we rely on Calderón-Zygmund decomposition theorem. We prove our main results without smoothness assumption on frame wavelets. We prove that affine tight frame wavelets span Lebesgue spaces under the condition, we also show that the affine tight frame operator extends from L²(R) to a bounded, linear and bijective operator on  L^p (R) , for 1< p < . Under such condition, the affine orthonormal basis of L²(R) is also an unconditional basis for 1< p < .

Index Terms—Bijective, frames, orthogonal basis, unconditional basis, wavelets.

Kai-Cheng Wang is with Ph.D. Program in Mechanical and Aeronautical Engineering, Feng-Chia University, 40724 Tai-Chung, Taiwan (e-mail: gtotony98@gmail.com).
Chi-I. Yang and Kuei-Fang Chang are with Department of Applied Mathematics, Feng-Chia University, 40724 Tai-Chung, Taiwan (e-mail: yesheslhamo@gmail.com, kfchang@math.fcu.edu.tw).

Cite: Kai-Cheng Wang, Chi-I. Yang, and Kuei-Fang Chang, "The Bijectivity of the Tight Frame Operators in Lebesgue Spaces," International Journal of Applied Physics and Mathematics vol. 4, no. 1, pp. 46-50, 2014.