Non-orthogonal bases

Dr. Matthieu R Bloch

Wednesday, September 29, 2021

Logistics

  • Assignment 4 Assigned later this week
    • Will include a programming components
    • Due October 11, 2021

What’s on the agenda for today?

  • Last time: Non-Orthobases

    • Isomorphism of separable Hilbert space \(\calH\) to \(\ell_2\)
    • Non-orthobases may compromise stability!

  • Today: More on non-orthobases

    • Dual basis
    • Infinite dimension

  • Monday October 04, 2021: Least-square regression

  • Reading: Romberg, lecture notes 7

Recap: 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!