The Study Of Weyl-Heisenberg Frame Identity

This thesis is divided into six sections. In the first section, the background and development of wavelet analysis are introduced. The frame concept and basic theory of three types of frames on L2(R) are summarized in primarily in the second section: In the third section,the main results of the thesis are stated . Accurately, we haveTheorem 1 (Theorem 3.3 in the thesis). Let g(t) be a measurable function on Rd. The following two conditions are equivalent:(1). g(t) ∈ L2(Rd).(2). For all bounded, compactly supported functions f(t) on Rd, we haveholds and the series of equation (*) converges unconditionally,where, a = (a1, a2, ... ad) G Rd,b = (b1,b2,...bd) ∈ Rd, n = (n1,n2,...,nd) ∈Zd,m = (m1,m2,...,md) ∈ Zd, na = (n1a1,n2a2,...,ndad), mb= (m1b1,m2b2,...,mdbd). Equation (*) is called Weyl-Heisenberg frame identity.Theorem 2 (Theorem 3.4 in the thesis). Let g(t) be a measurable function on Rd. The following two conditions are equivalent:(1). There is a constent B > 0, so that(2). The Weyl-Heisenberg frame identity (*) holds for all compactly supported functions f(t) on Rd and the series converges unconditionally.In order to prove Theorem 1 and Theorem 2, the related auxiliary conclusions are presented in Section 4. The proofs of Theorem 1 and Theorem 2 are put into Section 5and the Section 6, respectively.