# Frame-Related Sequences in Chains and Scales of Hilbert Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Known Facts, Definitions, and Notation

#### 2.1. Frames in Hilbert Spaces

- complete (or total) if $\mathsf{span}({\psi}_{k})$, the linear span of $({\psi}_{k})$, is dense in $\mathcal{H}$;
- a frame for $\mathcal{H}$ if there exist $A>0$ and $B<\infty $, such that:$${A\parallel f\parallel}^{2}\le \sum _{k\in \mathbb{N}}|\left(\right)open="\langle "\; close="\rangle ">f\left(\right)open="|"\; close>{\psi}_{k}{|}^{2}\le B{\parallel f\parallel}^{2},\phantom{\rule{2.em}{0ex}}\forall f\in \mathcal{H};$$
- a Bessel sequence in $\mathcal{H}$ if there exists $B>0$, such that the upper inequality in (1) holds true. It is called an upper semi-frame [42] if the Bessel sequence is also complete (this is equivalent to $0<\sum _{k\in \mathbb{N}}|\left(\right)open="\langle "\; close="\rangle ">f\left(\right)open="|"\; close>{\psi}_{k}{|}^{2}\le {B||f||}^{2},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall f\ne 0$);
- a Riesz basis for $\mathcal{H}$ if there exist an orthonormal basis $({e}_{k})$ for $\mathcal{H}$ and a bounded bijective operator $T:\mathcal{H}\to \mathcal{H}$, such that ${\psi}_{k}=T{e}_{k}$ for all $k\in \mathbb{N}$.

- a reproducing pair if the cross-frame operator, defined by:$$\left(\right)open="\langle "\; close="\rangle ">{S}_{\psi ,\varphi}f,g\left(\right)open="\langle "\; close="\rangle ">{\varphi}_{k},g$$

#### 2.2. Rigged Hilbert Spaces

#### 2.3. Scales of Hilbert Spaces

## 3. Hilbert Chains

#### 3.1. Hilbert Triplets

#### 3.2. Duality by Pivot Spaces

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### 3.3. Hilbert Chains

**Remark**

**4.**

#### 3.3.1. Different Adjoints

**Remark**

**5.**

**Remark**

**6.**

- 1.
- The double pivot adjoint of an operator $A\in \mathcal{B}({\mathcal{H}}_{p},{\mathcal{H}}_{q})$, $p,q\in \mathbb{Z}$, is:$${A}^{\u2605\u2605}={({A}^{\u2605})}^{\u2605}=A\phantom{\rule{1.em}{0ex}}and\phantom{\rule{1.em}{0ex}}\parallel {A}^{\u2605\u2605}\parallel =\parallel A\parallel ;$$
- 2.
- Let $A\in \mathcal{B}({\mathcal{H}}_{0},{\mathcal{H}}_{p})$ and $B\in \mathcal{B}({\mathcal{H}}_{m},{\mathcal{H}}_{0})$, $p,m\in \mathbb{N}$; then ${(AB)}^{\u2605}={B}^{\u2605}{A}^{\u2605}.$ Indeed, if $x\in {\mathcal{H}}_{m}$ and $y\in {\mathcal{H}}_{p}$, by (11):$${\left(\right)}_{ABx}00$$

**Remark**

**7.**

#### 3.3.2. Putting It All Together

**Theorem**

**1.**

**Remark**

**8.**

**Remark**

**9.**

#### 3.4. Generator of a Scale and Shifting of the Central Space

**Corollary**

**1.**

**Proof.**

**Remark**

**10.**

## 4. Main Results: Frame-Related Properties on Hilbert Scales

**Corollary**

**2.**

**Remark**

**11.**

#### 4.1. Completeness

**Lemma**

**1.**

- (i)
- If $({\psi}_{k})$ is complete in ${\mathcal{H}}_{p}$, then it is also complete in ${\mathcal{H}}_{r}$ for $r\le p\le m$;
- (ii)
- If $({\psi}_{k})$ is complete in ${\mathcal{H}}_{r}$$r\le m$, then $({I}_{r,p}{\psi}_{k})$ is a complete sequence in ${\mathcal{H}}_{p}$ for any p.

**Proof.**

#### 4.2. Unbounded Frame-Related Operators on Hilbert Chains

#### 4.3. Frame Properties of ${I}_{r,p}$$\psi $

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Corollary**

**3.**

- 1.
- If $({\psi}_{k})$ is a Bessel sequence in ${\mathcal{H}}_{p}$, then $({I}_{p,r}{\psi}_{k})$ is a Bessel sequence in ${\mathcal{H}}_{r}$;
- 2.
- If $({\psi}_{k})$ is a semi-frame in ${\mathcal{H}}_{p}$, then $({I}_{p,r}{\psi}_{k})$ is a semi-frame in ${\mathcal{H}}_{r}$ with the same bounds;
- 3.
- If $({\psi}_{k})$ is a frame in ${\mathcal{H}}_{p}$, then $({I}_{p,r}{\psi}_{k})$ is a frame in ${\mathcal{H}}_{r}$ with the same bounds;
- 4.
- If $({\psi}_{k})$ and $({\varphi}_{k})\subset {\mathcal{H}}_{p}$ are a reproducing pair, then $({I}_{p,r}{\varphi}_{k})$ and $({I}_{p,r}{\psi}_{k})$ are a reproducing pair in ${\mathcal{H}}_{r}$ with the same bounds;
- 5.
- If $({\varphi}_{k})\subset {\mathcal{H}}_{p}$ is a dual sequence of $({\psi}_{k})$ in ${\mathcal{H}}_{p}$, then $({I}_{p,r}{\varphi}_{k})$ is a dual sequence of $({I}_{p,r}{\psi}_{k})$ in ${\mathcal{H}}_{r}$;
- 6.
- If $({\psi}_{k})$ is an orthonormal basis of ${\mathcal{H}}_{p}$, then $({I}_{p,r}{\psi}_{k})$ is an orthonormal basis of ${\mathcal{H}}_{r}$;
- 7.
- If $({\psi}_{k})$ is a Riesz basis of ${\mathcal{H}}_{p}$ and $T\in \mathcal{B}({\mathcal{H}}_{p})$ is the bijective operator such that $T{e}_{k}={\psi}_{k}$, for every k with $\{{e}_{k}\}$, is an orthonormal basis of ${\mathcal{H}}_{p}$, then $({I}_{p,r}{T}^{-1}{\psi}_{k})$ is an orthonormal basis of ${\mathcal{H}}_{r}$;
- 8.
- If $({\psi}_{k})$ is a Riesz basis of ${\mathcal{H}}_{p}$, then $({I}_{p,r}{\psi}_{k})$ is a Riesz basis of ${\mathcal{H}}_{r}$ with the same bounds.

#### 4.4. Frame-Related Operators for the Original Sequence $\psi $

**Lemma**

**6.**

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

#### 4.4.1. Frames

**Corollary**

**4.**

#### 4.4.2. Duality

**Proposition**

**1.**

#### 4.4.3. A Negative Result

**Proposition**

**2.**

**Proof.**

**Remark**

**12.**

**Remark**

**13.**

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Casazza, P.G.; Kutyniok, G. Finite Frames. Theory and Applications; Applied and Numerical Harmonic Analysis; Birkhäuser: Boston, MA, USA, 2013. [Google Scholar]
- Christensen, O. An Introduction to Frames and Riesz Bases; Applied and Numerical Harmonic Analysis; Birkhäuser: Boston, MA, USA, 2016. [Google Scholar]
- Heil, C. A Basis Theory Primer; Applied and Numerical Harmonic Analysis; Birkhäuser: Boston, MA, USA, 2011. [Google Scholar]
- Ali, S.T.; Antoine, J.-P.; Gazeau, J.P. Continuous frames in Hilbert space. Ann. Phys.
**1993**, 222, 1–37. [Google Scholar] [CrossRef] - Cotfas, N.; Gazeau, J.P. Finite tight frames and some applications. J. Phys. A Math. Theor.
**2010**, 43, 193001. [Google Scholar] [CrossRef] - Benedetto, J.J.; Li, S. The theory of multiresolution analysis frames and applications to filter banks. Appl. Comput. Harmon. Anal.
**1998**, 5, 389–427. [Google Scholar] [CrossRef][Green Version] - Bölcskei, H.; Hlawatsch, F.; Feichtinger, H.G. Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process.
**1998**, 46, 3256–3268. [Google Scholar] [CrossRef] - Dahlke, S.; Fornasier, M.; Raasch, T. Adaptive Frame Methods for Elliptic Operator Equations. Adv. Comput. Math.
**2007**, 27, 27–63. [Google Scholar] [CrossRef][Green Version] - Stevenson, R. Adaptive solution of operator equations using wavelet frames. SIAM J. Numer. Anal.
**2003**, 41, 1074–1100. [Google Scholar] [CrossRef][Green Version] - Balazs, P.; Laback, B.; Eckel, G.; Deutsch, W.A. Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking. IEEE Trans. Audio. Speech Lang. Process.
**2010**, 2010 18, 34–49. [Google Scholar] [CrossRef] - Bellomonte, G. Continuous frames for unbounded operators. Adv. Oper. Theory
**2021**, 6, 1–28. [Google Scholar] [CrossRef] - Casazza, P.G.; Kutyniok, G.; Li, S. Fusion frames and distributed processing. Appl. Comput. Harmon. Anal.
**2008**, 254, 114–132. [Google Scholar] [CrossRef][Green Version] - Christensen, O.; Stoeva, D. p-frames in separable Banach spaces. Adv. Comput. Math.
**2003**, 18, 117–126. [Google Scholar] [CrossRef] - Găvruţa, L. Frames and operators. Appl. Comp. Harmon. Anal.
**2012**, 32, 139–144. [Google Scholar] [CrossRef][Green Version] - Bellomonte, G.; Corso, R. Frames and weak frames for unbounded operators. Adv. Comput. Math.
**2020**, 46, 1–21. [Google Scholar] [CrossRef] - Duffin, J.; Schaeffer, A.C. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc.
**1952**, 72, 341–366. [Google Scholar] [CrossRef] - Antoine, J.-P. Quantum mechanics beyond Hilbert space. In Irreversibility and Causality Semigroups and Rigged Hilbert Spaces. Lecture Notes in Physics; Bohm, A., Doebner, H.D., Kielanowski, P., Eds.; Springer: Berlin/Heidelberg, Germany, 1998; Volume 504, pp. 1–33. [Google Scholar]
- Balazs, P.; Gröchenig, K.; Speckbacher, M. Kernel theorems in coorbit theory. Trans. Am. Math. Soc. Ser. B
**2019**, 6, 346–364. [Google Scholar] [CrossRef] - Gröchenig, K.; Heil, C. Modulation spaces and pseudodifferential operators. Integral Equ. Oper. Theory
**1999**, 34, 439–457. [Google Scholar] [CrossRef] - Ullrich, T.; Rauhut, H. Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type. J. Funct. Anal.
**2011**, 11, 3299–3362. [Google Scholar] - Bellomonte, G.; Trapani, C. Riesz-like bases in rigged Hilbert spaces. Zeitschrift Analysis Anwendung
**2016**, 35, 243–265. [Google Scholar] [CrossRef][Green Version] - Trapani, C.; Triolo, S.; Tschinke, F. Distribution Frames and Bases. J. Fourier Anal. Appl.
**2019**, 2019 25, 2109–2140. [Google Scholar] [CrossRef][Green Version] - Tschinke, F. Riesz-Fischer Maps, Semi-frames and Frames in Rigged Hilbert Spaces. In Operator Theory, Functional Analysis and Applications, 625–645; Operator Theory: Advances and Applications, 282, Bastos, M.A., Castro, L., Karlovich, A.Y., Eds.; Birkhäuser: Cham, Switzerland, 2021. [Google Scholar]
- Feichtinger, H.G. Modulation Spaces: Looking Back and Ahead. Sampl. Theory Signal Image Process.
**2006**, 5, 109–140. [Google Scholar] [CrossRef] - Cordero, E.; Nicola, F. Kernel theorems for modulation spaces. J. Fourier Anal. Appl.
**2017**, 19, 131–144. [Google Scholar] [CrossRef][Green Version] - Gel’fand, I.M.; Shilov, G.E. Generalized Functions, Volume 2: Spaces of Fundamental and Generalized Functions; AMS Chelsea Publishing/American Mathematical Society: Providence, RI, USA, 1968; Volume 378. [Google Scholar]
- Gel’fand, I.M.; Vilenkin, N.Y. Generalized Functions, Volume 4: Applications of Harmonic Analysis; AMS Chelsea Publishing/American Mathematical Society: Providence, RI, USA, 1964; Volume 380. [Google Scholar]
- Ehler, M. The multiresolution structure of pairs of dual wavelet frames for a pair of Sobolev spaces. Jaen J. Approx.
**2010**, 2, 193–214. [Google Scholar] - Daubechies, I. Ten Lectures on Wavelets; SIAM: Philadelphia, PA, USA, 1992. [Google Scholar]
- Albeverio, S.; Kurasov, P. Singular Perturbations of Differential Operators: Solvable Schrödinger-Type Operators; London Mathematical Society, Lecture Note Series 271; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Brenner, S.; Scott, L. The Mathematical Theory of Finite Element Methods; Springer: New York, NY, USA, 2002. [Google Scholar]
- Schneider, R. Multiskalen- und Wavelet-Matrixkompression; Vieweg/Teubner Verlag: Wiesbaden, Germany, 1998. [Google Scholar]
- Balazs, P.; Gröchenig, K. A guide to localized frames and applications to Galerkin-like representations of operators. In Novel Methods in Harmonic Analysis with Applications to Numerical Analysis and Data Processing; Applied and Numerical Harmonic Analysis Series (ANHA); Pesenson, I., Mhaskar, H., Mayeli, A., Gia, Q.T.L., Zhou, D.-X., Eds.; Birkhauser/Springer: Basel, Switzerland, 2017; pp. 47–79. [Google Scholar]
- Fornasier, M.; Gröchenig, K. Intrinsic localization of frames. Constr. Approx.
**2005**, 22, 395–415. [Google Scholar] [CrossRef] - Balazs, P.; Harbrecht, H. Frames for the solution of operator equations in Hilbert spaces with fixed dual pairing. Numer. Funct. Anal. Optim.
**2019**, 40, 65–84. [Google Scholar] [CrossRef][Green Version] - Harbrecht, H.; Schneider, R.; Schwab, C. Multilevel frames for sparse tensor product spaces. Numer. Math.
**2008**, 110, 199–220. [Google Scholar] [CrossRef][Green Version] - Antoine, J.-P.; Inoue, A.; Trapani, C. Partial *-Algebras and Their Operator Realizations; Mathematics and its Applications, 553; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2002. [Google Scholar]
- Schmüdgen, K. Unbounded Self-Adjoint Operators on Hilbert Space; Graduate Texts in Mathematics 265; Springer: Dordrecht, The Netherlands, 2012. [Google Scholar]
- Antoine, J.-P.; Trapani, C. Partial Inner Product Spaces: Theory and Applications; Lecture notes in Mathematics 1986; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Grossmann, A. Elementary properties of nested Hilbert spaces. Commun. Math. Phys.
**1966**, 2, 1–30. [Google Scholar] [CrossRef] - Picard, R.; McGhee, D. Partial Differential Equations. A Unified Hilbert Space Approach; De Gruyter: Berlin, Germany, 2011. [Google Scholar]
- Antoine, J.-P.; Balazs, P. Frames and semi-frames. J. Phys. A-Math. Theor.
**2011**, 44, 205201. [Google Scholar] [CrossRef] - Balazs, P.; Stoeva, D.; Antoine, J.-P. Classification of General Sequences by Frame-Related Operators. Sampl. Theory Signal Image Process.
**2011**, 10, 151–170. [Google Scholar] [CrossRef] - Speckbacher, M.; Balazs, P. Reproducing pairs and the continuous nonstationary Gabor transform on lca groups. J. Phys. A Math. Theor.
**2015**, 48, 395201. [Google Scholar] [CrossRef] - Feichtinger, H.G.; Zimmermann, G. A Banach space of test functions for Gabor analysis. In Gabor Analysis and Algorithms—Theory and Applications; Series: Applied and Numerical Harmonic Analysis; Feichtinger, H.G., Strohmer, T., Eds.; Birkhäuser: Boston, MA, USA, 1998; pp. 123–170. [Google Scholar]
- Koshmanenko, V.; Dudkin, M.; Koshmanenko, N. The Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators; Operator Theory: Advances and Applications 253; Birkhäuser: Boston, MA, USA, 2016. [Google Scholar]
- Berezanskii, J.M. Expansions in Eigenfunctions of Selfadjoint Operators; Translations of Mathematical Monographs, 17; American Mathematical Society: Providence, RI, USA, 1968. [Google Scholar]
- Conway, J.B. A Course in Functional Analysis, 2nd ed.; Graduate Texts in Mathematics; Springer: New York, NY, USA, 1990. [Google Scholar]
- Casazza, P.; Christensen, O.; Li, S.; Lindner, A. Riesz-Fischer sequences and lower frame bounds. Zeitschrift Analysis Anwendung
**2002**, 21, 305–314. [Google Scholar] - Beylkin, G.; Coifman, R.; Rokhlin, V. Fast Wavelet Transforms and Numerical Algorithms I. Comm. Pure Appl. Math.
**1991**, 44, 141–183. [Google Scholar] [CrossRef] - Feichtinger, H.G.; Kozek, W. Quantization of TF lattice-invariant operators on elementary LCA groups. In Gabor Analysis and Algorithms—Theory and Applications; Series: Applied and Numerical Harmonic Analysis; Feichtinger, H.G., Strohmer, T., Eds.; Birkhäuser: Boston, MA, USA, 1998; pp. 233–266. [Google Scholar]

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**MDPI and ACS Style**

Balazs, P.; Bellomonte, G.; Hosseinnezhad, H.
Frame-Related Sequences in Chains and Scales of Hilbert Spaces. *Axioms* **2022**, *11*, 180.
https://doi.org/10.3390/axioms11040180

**AMA Style**

Balazs P, Bellomonte G, Hosseinnezhad H.
Frame-Related Sequences in Chains and Scales of Hilbert Spaces. *Axioms*. 2022; 11(4):180.
https://doi.org/10.3390/axioms11040180

**Chicago/Turabian Style**

Balazs, Peter, Giorgia Bellomonte, and Hessam Hosseinnezhad.
2022. "Frame-Related Sequences in Chains and Scales of Hilbert Spaces" *Axioms* 11, no. 4: 180.
https://doi.org/10.3390/axioms11040180