How can I create a basis using functions of the shape $e^{a_kt} \sin(k\omega_0 t) \ and \ e^{a_kt} \cos(k\omega_0 t)$?
What would be the specific dot-product/scalar-product to be able to construct this basis?
I need it to expand a function as shown below:
$$ f(x)= \sum_{k=0}^\infty e^{c_kt}(a_k \sin(k\omega_0 t) \ + \ b_k \cos(k\omega_0 t)) $$
I am still thinking about how to best formulate the problem.
Maybe it is better expressed as below:
$$ f(x)= \sum_{k=0}^\infty (A_k \sin(k\omega_0 t) \ + \ B_k \cos(k\omega_0 t)) $$ where $$ A_k = a_k e^{c_kt} \ and \ B_k = b_k e^{c_kt} $$
The answer is no!
Or at least, there is no need to.
I asked the question because I wanted to arrive at the Laplace transform.
The reasoning that makes more sense is shown in how-can-i-expand-function-into-a-series-of-exponentially-decaying-cosines
The basis remains the infinite row of (co)sines, the exponential occurs without the need for it to be part of a basis.