A sharper stability bound of Fourier frames

Given a real sequence {lambda(n)}(n is an element of Z). Suppose that {e(i lambda nx)}(n is an element of Z) is a frame for L-2[-pi, pi] with The is t o find a positive constant L such that for any real sequence {mu(n)}(n is a n element of Z) ABSTRACT bounds A, B. The problem is to find a positive con stant L such that for any real sequence {mu(n)}(n is an element of Z) with \mu(n) - lambda(n)\ less than or equal to delta < L, {e(i mu nx)}(n is an e lement of Z) is also a frame for L-2[-pi, pi]. Balan [I] obtained L-R = 1/4 - 1/pi arcsin (1/root 2 (1 - root A/B)). This value ir a good stability bo und of Fourier frames because it covers Kadec's 1/4-theorem (L-r = 1/4 if A = B) and is better than L-DS = 1/pi ln (1+ root A/B) (see Duffin and Schae fer [3]). In this paper; a sharper estimate is given.