ar optimal, s˚a ¨ar kx0k1 = kx`1k1. D¨armed ¨ar x0 en l¨osning till BP, och med beaktande av entydighetsantagandet ser man att x0 = x`1. Detta bevisar, att varje delf¨oljd (xγk) konvergerar mot punkten x`1, och detsamma g¨aller f¨or hela f¨oljden (xγ).
A.4 Allm¨ an os¨ akerhetsprincip
F¨oljande sats ¨ar en generalisering av Donoho–Starks os¨akerhetsprincip. Be-viset ¨ar fr˚an [30], men det gavs ursprungligen av A. Pinkus.
Sats A.4 L˚at Ψ och Φ vara tv˚a olika baser f¨or Cnoch antag att deras ¨ omsesi-diga koherens ¨ar µ(Ψ, Φ). Om en vektor x har framst¨allningarna α respektive β i dessa baser, s˚a ¨ar
vilket vidare ger, med beaktande av f¨orh˚allandet mellan geometriskt och aritmetiskt medeltal, att
1
µ(Ψ, Φ) ≤p|S| · |T | ≤ |S| + |T |
2 . (A.19)
Som en konsekvens av denna sats f˚as ett entydighetsvillkor f¨or glesa l¨ os-ningar till systemet (1.1) i s˚adana fall, d˚a matrisen har 2-orto-struktur, och matrisens koherens ¨ar lika med basernas ¨omsesidiga koherens. Betrakta n¨ am-ligen systemet [Ψ|Φ]x = y och antag att x och x0 ¨ar dess tv˚a l¨osningar. resul-tatet av Gribonval och Nielsen (se s. 9).
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