Continuous Schauder Frames for Banach Spaces
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Continuous Schauder Frames for Banach Spaces Joseph Eisner1 · Daniel Freeman2,3 Received: 1 January 2019 / Revised: 25 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in this general context. Keywords Schauder frames · Continuous frames · Shrinking · Boundedly complete Mathematics Subject Classification 42C15 · 81R30 · 46B10
1 Introduction Frames and orthonormal bases give discrete ways to represent vectors in a Hilbert space using series, and continuous frames and coherent states give continuous ways to represent vectors using integrals.(x j ) j∈J ⊂H for which there exists constants 0 < A ≤ B such that for any x ∈ H , Ax2 ≤ j∈J |x, x j |2 ≤ Bx2 . Given any frame (x j ) j∈J for a Hilbert space H , there exists a frame ( f j ) j∈J for H , called a dual frame, such that
Communicated by Hans G. Feichtinger. Daniel Freeman was supported by Grant 353293 from the Simon’s foundation. This paper forms a portion of the Joseph Eisner masters thesis which was prepared at St Louis University.
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Daniel Freeman [email protected] Joseph Eisner [email protected]
1
Department of Mathematics, University of Virginia, Charlottesville, VA 22901, USA
2
Department of Mathematics and Statistics, St Louis University, St Louis, MO 63103, USA
3
Department of Mathematics, Duke University, Durham, NC 27708, USA 0123456789().: V,-vol
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Journal of Fourier Analysis and Applications
x=
x, f j x j
for all x ∈ H .
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(1.1)
j∈J
The equality in (1.1) allows the reconstruction of any vector x in the Hilbert space from the sequence of coefficients (x, f j ) j∈J . Continuous frames and coherent states are a generalization of frames in that instead of summing over a discrete set, we integrate over a measure space. Coherent states were invented by Schrödinger [27] and were generalized to continuous frames by Ali et al. [2]. The short time Fourier transform and the continuous wavelet transform are two particularly important examples of continuous frames. Let (M, , μ) be a σ -finite measure space and let H be a separable Hilbert space. A weakly measurable function ψ : M → H is a continuous frame of H withrespect to μ if there exists constants A, B > 0 such that for all x ∈ H , Ax2 ≤ M |x, ψ(t)|2 dμ(t) ≤ Bx2 . If A = B = 1, then the continuous frame is called a Parseval frame. As is the case with frames, any continuous frame may be used to reconstruct vectors using a dual frame. That is, if ψ : M → H is a continuous frame, then there exists a dual frame φ : M → H such that x, φ(t)ψ(t)dμ(t) for all x ∈ H . (1.2) x= M
Equation (1.2) involv
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