Harmonic analysis on Euclidean spaces Frames and time-frequency analysis Theorem

Feichtinger conjecture

Statement

Let $H$ be a separable Hilbert space, and let $(f_i)_{i \in \N}$ be a frame of $H$ with $\inf_{i} \norm{f_i} > 0$ . That is, there are constants $0 < A \le B < \infty$ with $A \norm{f}^2 \le \sum_{i} \abs{\ip{f}{f_i}}^2 \le B \norm{f}^2$ for all $f \in H$ . Then $\N$ is the union of finitely many sets $I_1, \dots, I_r$ such that each family $(f_i)_{i \in I_j}$ is a Riesz sequence. Here a Riesz sequence is a family that is a Riesz basis of its closed linear span.

Remarks

The question is due to Hans Feichtinger, who came to it from time–frequency analysis. In the 2000s Peter Casazza and coauthors showed that it is equivalent to the Kadison–Singer problem. The positive solution of the Kadison–Singer problem by Marcus, Spielman and Srivastava therefore proves the statement. The hypothesis $\inf_i \norm{f_i} > 0$ is necessary: a Riesz sequence has norms bounded below, so a frame containing vectors of norm tending to $0$ cannot be split into finitely many Riesz sequences.

Example: the union of two orthonormal bases $(e_n)$ and $(e'_n)$ of $H$ is a frame with $\norm{f_i} = 1$ ; it splits into the two Riesz sequences $(e_n)$ and $(e'_n)$ .

Going deeper

For a student

The statement lets one replace a redundant frame by finitely many non-redundant pieces, each of which behaves like an orthonormal basis of its span. The equivalence with the Kadison–Singer problem passes through Anderson's paving conjecture and Weaver's discrepancy conjecture. The proof of Marcus, Spielman and Srivastava works with sums of random rank-one matrices and interlacing families of polynomials.

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Tag	`QYQ983Z0BK`
URI	`m2://QYQ983Z0BK`
Kind	statement, theorem
Writing	W0 stub
MSC	42C15, 46C05, 46L05
Status	This copy of the site has no build data. Run `m2 status QYQ983Z0BK`.
About	Riesz basis
Uses	Hilbert space, Riesz basis