Explicit bounds of complex exponential frames

We discuss the stability of complex exponential frames[inline-graphic not available: see fulltext] in[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext]. Specifically, we improve the[inline-graphic not available: see fulltext]-theorem and obtain explicit upper and lower bounds for some complex exponential frames perturbed along the real and imaginary axes, respectively. Two examples are given to show that the bounds are best possible. In addition, the growth of the entire functions of exponential type[inline-graphic not available: see fulltext][inline-graphic not available: see fulltext] on the integer sequence is estimated.