  We deduce the dynamic frequency-domain-lattice Green's function of a linear
chain with properties (masses and next-neighbor spring constants) of
exponential spatial dependence. We analyze the system as discrete chain as well
as the continuous limiting case which represents an elastic 1D exponentially
graded material. The discrete model yields closed form expressions for the $N
\times N$ Green's function for an arbitrary number $N=2,..,\infty$ of particles
of the chain. Utilizing this Green's function yields an explicit expression for
the vibrational mode density. Despite of its simplicity the model reflects some
characteristics of the dynamics of a 1D exponentially graded elastic material.
As a special case the well-known expressions for the Green's function and
oscillator density of the homogeneous linear chain are contained in the model.
The width of the frequency band is determined by the grading parameter which
characterizes the exponential spatial dependence of the properties. In the
limiting case of large grading parameter, the frequency band is localized
around a single finite frequency where the band width tends to zero inversely
with the grading parameter. In the continuum limit the discrete Green's
function recovers the Green's function of the continuous equation of motion
which takes in the time domain the form of a Klein-Gordon equation.
