A very famous family of functions are the complex exponentials and in the case of real valued functions, the sin and cos functions. They are related by the famous Euler formulas:

$$\exp(i\phi) = \cos(\phi)+i\sin(\phi)$$

For example Discrete Fourier Transform with immense usefulness in science and engineering use basis functions like these:

$$\sum_{n}^{}a_n\exp\left[\phi i n\right]$$ or in the real case: $$\sum_{n}^{}a_n\cos(\phi n) + b_n\sin(\phi n)$$

Where integers $n$ decide how different frequencies these basis functions have.

Another interesting set of functions are the Airy and Bairy functions which are solutions to the following differential equation:


but sadly don't have any simple analytic formula (as far as I know)

They are related in some respects like sin and cos are. By "ocular inspection" (and an engineering mindset) we see that they are "out of phase" with each other, always peaking at the other's zero.

Having observed the above properties, can we define some similar kind of linear transform in which we and up with basis functions consisting of $$Ai(nx), Bi(nx), n\in\left\{0,1,\cdots,\right\}$$


Example: Eigenfunctions and Eigenvalues of the Airy Equation Using Spectral Methods.

Consider the Airy differential equation $u'' = \lambda x u$ where $u\ne 0$, $u(\pm 1) = 0$ and $-1\le x\le 1$ ...
The $n$-th eigenfunction is given by $A_i(\lambda_n^{1/3}x)$ where $A_i$ is the classic Airy function and $\lambda_n$ is the eigenvalue...

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