The purpose of the paper is to analyze frames {f k } k∈Z having the form {T k f 0 } k∈Z for some linear operator T : span{f k } k∈Z → span{f k } k∈Z . A key result characterizes boundedness of the operator T in terms of shift-invariance of a certain sequence space. One of the consequences is a characterization of the case where the representation {f k } k∈Z = {T k f 0 } k∈Z can be achieved for an operator T that has an extension to a bounded bijective operator T : H → H. In this case we also characterize all the dual frames that are representable in terms of iterations of an operator V ; in particular we prove that the only possible operator is V = ( T * ) −1 . Finally, we consider stability of the representation {T k f 0 } k∈Z ; rather surprisingly, it turns out that the possibility to represent a frame on this form is sensitive towards some of the classical perturbation conditions in frame theory. Various ways of avoiding this problem will be discussed. Throughout the paper the results will be connected with the operators and function systems appearing in applied harmonic analysis, as well as with general group representations.

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The purpose of this paper is to study frames for a Hilbert space H, having the form {T n ϕ} ∞ n=0 for some ϕ ∈ H and an operator T : H → H. We characterize the frames that have such a representation for a bounded operator T, and discuss the properties of this operator. In particular, we prove that the image chain of T has finite length N in the overcomplete case; furthermore {T n ϕ} ∞ n=0 has the very particular property that {T n ϕ} N −1 n=0 ∪{T n ϕ} ∞ n=N +ℓ is a frame for H for all ℓ ∈ N 0 . We also prove that frames of the form {T n ϕ} ∞ n=0 are sensitive to the ordering of the elements and to norm-perturbations of the generator ϕ and the operator T. On the other hand positive stability results are obtained by considering perturbations of the generator ϕ belonging to an invariant subspace on which T is a contraction.

We consider sequences in a Hilbert space H of the form (T n f0)n∈I , with a linear operator T , the index set being either I = N or I = Z, a vector f0 ∈ H, and answer the following two related questions: (a) Which frames for H are of this form with an at least closable operator T ? and (b) For which bounded operators T and vectors f0 is (T n f0)n∈I a frame for H? As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form (T n f0) n∈N with a bounded operator T . The corresponding problem for I = Z remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form (T n f0) n∈N with a bounded operator T is significantly larger than what could be expected from the examples known so far.2010 Mathematics Subject Classification. 94A20, 42C15, 30J05.

The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let $$\mathcal {H}$$ H and $$\mathcal {K}$$ K be real Hilbert spaces, $$b \in \mathcal {K}$$ b ∈ K and $$T \in \mathcal {B} (\mathcal {H},\mathcal {K})$$ T ∈ B ( H , K ) a linear operator with closed range and Moore–Penrose inverse $$T^\dagger $$ T † . Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $$\mathrm {Prox}:\mathcal {K}\rightarrow \mathcal {K}$$ Prox : K → K the operator $$T^\dagger \, \mathrm {Prox}( T \cdot + b)$$ T † Prox ( T · + b ) is a proximity operator on $$\mathcal {H}$$ H equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $$\mathrm {Prox}= S_{\lambda }:\ell _2 \rightarrow \ell _2$$ Prox = S λ : ℓ 2 → ℓ 2 and any frame analysis operator $$T:\mathcal {H}\rightarrow \ell _2$$ T : H → ℓ 2 that the frame shrinkage operator $$T^\dagger \, S_\lambda \, T$$ T † S λ T is a proximity operator on a suitable Hilbert space. The concatenation of proximity operators on $$\mathbb R^d$$ R d equipped with different norms establishes a PNN. If the network arises from tight frame analysis or synthesis operators, then it forms an averaged operator. In particular, it has Lipschitz constant 1 and belongs to the class of so-called Lipschitz networks, which were recently applied to defend against adversarial attacks. Moreover, due to its averaging property, PNNs can be used within so-called Plug-and-Play algorithms with convergence guarantee. In case of Parseval frames, we call the networks Parseval proximal neural networks (PPNNs). Then, the involved linear operators are in a Stiefel manifold and corresponding minimization methods can be applied for training of such networks. Finally, some proof-of-the concept examples demonstrate the performance of PPNNs.