Orthonormal set for the set of compactly supported functions How to find an orthonormal set in the space of compactly supported smooth functions on $\mathbb{R}$? Moreover, for the operator $\frac{d^2}{dt^2}$, what are the eigenfunctions in the space of compactly supported smooth functions on $\mathbb{R}$?
 A: Orthonormal basis in the space of compactly supported smooth functions
One has to define an inner product first; I assume you mean the $L^2$ inner product, $\langle f,g\rangle = \int_{\mathbb{R}} fg$. Since normalizing the basis vectors is easy, let's focus on orthogonal bases. 
The Daubechies wavelet provides an orthogonal basis for $L^2(\mathbb{R})$ which consists of compactly supported functions of class $C^{k}$, where $k$ can be made arbitrarily large (at the cost of increasing the complexity of the wavelet). See sections 1.3 and 1.4 of these lecture notes. 
There does not exist a $C^\infty$-smooth compactly supported orthogonal wavelet. Thus, the typical options for an explicit orthogonal basis in $L^2(\mathbb{R})$ consist of the following. 


*

*Give up compact support and use Hermite functions or Meyer wavelet.

*Downgrade from infinite smoothness to $C^k$ smoothness and use the Daubechies wavelet or its relatives.

*Give up all smoothness (and even continuity) and use the much simpler Haar wavelet or the functions $f_{m,n}(t) = \exp(2\pi i n t)\chi_{[m,m+1]}$. 


In principle, an orthonormal basis consisting of $C^\infty$ smooth  compactly supported functions can be constructed by applying the Gram-Schmidt process to some complete set of such functions (like monomials multiplied by the shifts of a smooth bump function). But the computation would be intractable, and the resulting basis not even remotely explicit.
Smooth compactly supported eigenfunctions of $d^2/dx^2$
There are none. On the Fourier transform side, applying $d^2/dx^2$ amounts to multiplying by $-\xi^2$. So the Fourier transform of an eigenfunction has to be supported on the set $\{\pm \sqrt{-\lambda}\}$, which means the eigenfunction is of the form $A\cos \sqrt{-\lambda} x+B\sin \sqrt{-\lambda} x$. Not compactly supported, obviously.
