Non-orthogonal bases

Dr. Matthieu R Bloch

Monday, September 27, 2021


  • Assignments 2/3: Grading ongoing
  • Midterm 1
    • Grading ongoing
    • General logistics of exams
  • Assignment 4 assigned today or tomorrow

What’s on the agenda for today?

  • Last time: Orthobases

    • Things work in infinite dimension too with orthobases (without thinking too much)
    • Last little bit to prove: isomorphism to \(\ell_2\)
  • Today: What happens with non-orthogonal bases

  • Wednesday September 29, 2021: Least-square regression

  • Reading: Romberg, lecture notes 7

Recap - Orthobases in infinite dimension

  • The following properties of an orthonormal set \(\set{e_i}_{i\geq 1}\) are equivalent:

    1. Finite linear combinations of elements in \(\calH\) are dense in \(\calH\);
    2. If \(v\in\calH\) and \(\forall j\geq 1\) \(\dotp{v}{e_j}=0\) then \(v=0\);
    3. If \(v\in\calH\) then \(\sum_{i=1}^n\dotp{v}{e_i}e_i\) converges to \(v\) (in the norm \(\norm{\cdot}\)) as \(n\to\infty\);
    4. If \(v\in\calH\) and \(\forall k\) \(a_k\eqdef \dotp{v}{e_k}\) then \(\norm{v}^2=\sum_{k\geq 1}\abs{a_k}^2\).
  • This does not say anything about the existence of such nice orthornormal set

  • A space is separable if it contains a countable dense subset.
  • Separability is the key property to deal with sequences instead of collections

  • Any separable Hilbert space has an orthonormal basis.
  • Most useful Hilbert spaces are separable! We won’t worry about non-separable Hilbert spaces

  • Any separable Hilbert space is isomorphic to \(\ell_2\)

Non-orthogonal bases in finite dimension

  • Let \(\set{v_i}_{i=1}^n\) be a linearly independent set in a Hilbert space \(\calH\) of dimension \(n\). Then, for any \(x\in\calH\), \(x=\sum_{i=1}^n\alpha_iv_i\) for some \(\bfalpha\in\bbR^n\). In addition, there exists \(A,B>0\) such that \[ A\norm[2]{\bfalpha}^2 \leq \norm[\calH]{x}^2\leq B\norm[2]{\bfalpha}^2 \]
  • Inequality is tight on both sides

  • For orthobases, \(A=B=1\)

  • Interpretation:

    • The values of \(A\) and \(B\) govern the stability of the representation
  • Examples

Non-orthogonal bases in finite dimension: dual basis

  • Recall from orthobases:
    • perfectly stable representation \(A=B=1\)
    • Efficient computation of representations: \(\alpha_i=\dotp{x}{v_i}\)
  • For any \(x\in\calH\) with basis \(\set{v_i}_{i=1}^n\)we have \[ x=\sum_{i=1}^n\alpha_iv_i\qquad\text{with}\qquad\bfalpha = \matG^{-1}\mat{c}{\dotp{x}{v_1}\\\dotp{x}{v_2}\\\vdots\\\dotp{x}{v_n}} \] There also exists a basis \(\set{\tilde{v}_i}_{i=1^n}\) such that \(\alpha_i=\dotp{x}{\tilde{v}_i}\)

Non-orthogonal bases in infinite dimension

  • \(\set{v_i}_{i=1}^\infty\) is a Riesz basis for Hilbert space \(\calH\) if \(\text{cl}(\text{span}(\set{v_i}_{i=1}^\infty))=\calH\) and there exists \(A,B>0\) such that \[ A\sum_{i=1}^\infty\alpha_i^2\leq \norm[\calH]{\sum_{i=1}^n\alpha_iv_i}^2\leq B\sum_{i=1}^\infty\alpha_i^2 \] uniformly for all sequences \(\set{\alpha_i}_{i\geq 1}\) with \(\sum_{i\geq 1}\alpha_i^2<\infty\).
  • In infinite dimension, the existence of \(A,B>0\) is not automatic.

  • Examples

Non-orthogonal bases in finite dimension: dual basis

  • Computing expansion on Riesz basis not as simple in infinite dimension: Gram matrix is “infinite”

  • The Grammiam is a linear operator \[ \calG:\ell_2(\bbZ)\to\ell_2(\bbZ): \bfx\mapsto \bfy\text{ with }[\calG(\bfx)]_n\eqdef y_n=\sum_{\ell=-\infty^\infty}\dotp{v_\ell}{v_n}x_\ell \]

  • Fact: there exists another linear operator \(\calH:\ell_2(\bbZ)\to\ell_2(\bbZ)\) such that

    \[ \calH(\calG(\bfx)) = \bfx \] We can replicate what we did in finite dimension.