Explicit approximate representations of sequences as suborbits in banach space universal and function space approaches

HIGHLIGHTS

• What: The authors explore approximate representations {𝑓𝑘 }∞ 𝜑}𝑘=1 for certain types of 𝑘=1 ≈ {𝑇 sequences {𝑓𝑘 }∞ 𝑘=1. The aim of this section is to derive approximate representations of sequences {𝑓𝑘 }∞ 𝑘=1 in a separable Banach space ∞ 𝛼(𝑘) 𝑋 of the form {𝑇 𝜑}𝑘=1 for suitable choices of a vector 𝜑 ∈ 𝑋, a bounded operator 𝑇: 𝑋 → 𝑋, and the powers 𝛼(𝑘), 𝑘 ∈ ℕ. The aim of Sections 2.4 - 2.5 is to apply the results in Sections 2.1-2.3 to the setting of atomic decompositions and Banach frames: the authors show how to design the ∞ approximations such that the sequence {𝑇 𝛼(𝑘) 𝜑}𝑘=1 keeps essential features of the given sequence {𝑓𝑘 }∞ 𝑘=1 . . .
  

   

  

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