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 $\mathbb{R}$?

  • 1
    $\begingroup$ I realize you wrote "an orthonormal set" rather than "basis", but finding just an orthonormal set is too easy: take one smooth compactly supported function, and normalize it. Voilà, an orthonormal set with 1 element. Throw in its translates with disjoint supports to get an infinite orthonormal set. $\endgroup$
    – user357151
    May 10, 2017 at 3:32

1 Answer 1


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.

  • $\begingroup$ thanks a lot for the explanation, yes I was looking for a basis!! $\endgroup$
    – Sanand
    May 11, 2017 at 6:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.