Transforms look at signals from a domain other than the natural domain. Transforms are essential for understanding some properties of a signal. The Fourier transform is an important transform to begin with.
A Discrete Signal x[n] can be thought of as a vector with countably infinite dimensions.
A Continuous Signal x(t) can be thought of as a vector with uncountably infinite dimensions.
We can define inner products for signals, and thus go on to define eigensignals and eigenvalues for a signal:
for continuous signals
for discrete signals
The Complex Exponential signal is an Eigensignal to
a stable LSI system. The Eigenvalue of this signal can be
represented as an inner product of the impulse response
and the eigensignal
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for continuous signals 
for discrete signals .gif){kind=small}
is an Eigensignal to
a stable LSI system. The Eigenvalue of this signal can be
represented as an inner product of the impulse response
and the eigensignal
