Duration and Bandwidth Limiting pp 129-151 | Cite as

# Time and Band Limiting of Multiband Signals

## Abstract

When \( {\mathcal a} \) = 2Ω*T*, the operator *P*_{Ω}*Q*_{ T } corresponding to single time and frequency intervals has an eigenvalue λ_{∟} \( {}_{\mathcal a} \) _{∟} *≈*1/2, as Theorem 4.1.2 below will show. The norm λ_{0}(\( {\mathcal a} \) = 1) of the operator *PQ*_{1/2} satisfies λ_{0}(\( {\mathcal a} \) = 1) ≥ ∥sinc _{[−1/2, 1/2]}∥ > 0*.*88. The trace of *PQ*_{1/2} is equal to \( {\mathcal a} \) = 1, on the one hand and to Σ^{λ}_{ n } on the other, so λ_{1}(\( {\mathcal a} \) = 1) ≤ 1*−*λ_{0}(\( {\mathcal a} \) = 1) *<* 1/2. Suppose that *T* = 1 and Σ is a finite, pairwise disjoint union of \( {\mathcal a} \) frequency intervals *I*_{1}, … , *I*\(_{} {\mathcal a} \) each of unit length. Then *P*_{Σ}*Q* should have on the order of \( {\mathcal a} \) eigenvalues of magnitude *at least* 1/2. Consider now the limiting case in which the frequency intervals become *separated at infinity*. Any function ψ_{ j } that is concentrated in frequency on *I*_{ j } will be almost orthogonal over [*−T,T*], in the separation limit, to any function ψ_{ k } that is frequency-concentrated on *I*_{ k } when *j ≠ k*.

## Keywords

Pairwise Disjoint Partition Element Stirling Number Frequency Product Frequency Support## Preview

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