Scalability of Generalized Frames for Operators

Kumar, Varinder; Malhotra, Sapna; Khanna, Nikhil

doi:https://doi.org/10.1155/2024/8358987

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 8358987 | https://doi.org/10.1155/2024/8358987

Scalability of Generalized Frames for Operators

Varinder Kumar,¹Sapna Malhotra,²and Nikhil Khanna³

Academic Editor: Calogero Vetro

Received06 Feb 2024

Revised20 Mar 2024

Accepted26 Mar 2024

Published16 Apr 2024

Abstract

In this paper, the Parseval --frames are constructed from a given --frame by scaling the elements of the --frame with the help of diagonal operators, and these frames are named scalable --frames. Also, we prove some properties of scalable --frames and construct new scalable --frames from a given --frame. The necessary and sufficient conditions for a --frame to be scalable are given. Further, equivalent conditions for the scalability of --frames and the -frames induced by --frames are obtained. Finally, it is shown that the direct sum of two scalable --frames is again a scalable --frame for some suitable bounded linear operator .

1. Introduction

Duffin and Schaeffer [1] initiated the study of frames in Hilbert spaces in 1952 while working on some profound problems in non-Harmonic Fourier series and came up with the following definition of frames.

A sequence in a separable Hilbert space is said to be a frame for , if there exist constants such that for all ,

The constants and are called lower and upper frame bounds, respectively. If , then the frame is said to be a tight frame. If , then the frame is said to be a Parseval frame. If only the right hand inequality in (1) holds, then is known as a Bessel sequence. In 1986, Daubechies et al. [2] presented their remarkable work in which they proved that frames can also be used to provide series expansion of functions in . Since then, frames have been a point of fascination among researchers. For more information on frames, see [3, 4].

Tight frames are more flexible than the orthonormal basis in the sense that every vector can be expressed as a linear combination of frame elements where frame coefficients need not be unique; that is, if is a tight frame for with bound , then every vector can be expressed as . Therefore, tight frames have direct applications in expanding data as they admit optimally stable reconstruction and hence have numerous applications in sampling theory, signal processing, filtering, smoothing, compression, image processing, etc. With this objective, Kutyniok et al. [5] generated a tight frame from a given general frame by scaling the frame vectors with the help of nonnegative scalars and termed such frames as scalable frames. They proved several important properties of scalable frames and provided geometrical interpretation of scalability in canonical surfaces.

In recent years, several generalizations of frames in Hilbert and Banach spaces have been proposed and studied like fusion frames in Hilbert spaces [6], -frames by Sun [7, 8], fusion Banach frames in Banach spaces by Kaushik and Kumar [9–11], -frames by Găvruta [12, 13], --frames by Xiao et al. [14], and scalable -frames by Ramesan and Ravindran [15]. The concept of -frames was put forward to study the atomic systems with respect to a bounded linear operator . These frames allowed to reconstruct elements from the range of bounded linear operator and were more flexible than the classical frames. It was observed that in the case of frames, the frame operator turned out to be invertible on , whereas in the case of -frames, the -frame operator need not be invertible on . But, if the operator has a closed range, then the -frame operator is invertible on . Further, it has been seen that --frames are the generalization of -frames and have better practical applications. For more information on --frames, one may refer to [14, 16–18]. Tight -frames are similar to tight frames and are helpful in reconstructing signals. Recently, Ahmadi and Rahimi [19] came up with the idea of scalability of -frames based on nonnegative diagonal operators instead of nonnegative scalars for scale change and obtained its various characterizations.

The Parseval --frames are the generalization of the Parseval -frames and help in restoring data loss in signal processing. Motivated by the work in [5, 15, 19], in this paper, we study the problem of determining whether a --frame for is scalable or not, like -frames. Therefore, we construct a Parseval --frame from a given --frame by scaling the --frame elements with the help of nonnegative diagonal operators and introduce the notion of scalable --frames. Some examples are provided to show the existence of such frames. Further, we determine its various characterizations and construct new scalable --frames from a given scalable --frame. The necessary and sufficient conditions for a --frame to be scalable are proven. At last, we prove a result relating the direct sum of two scalable --frames.

2. Preliminaries

Throughout this paper, is used to denote a separable Hilbert space. and are countable indexing sets, and is a sequence of separable Hilbert spaces. denotes the collection of all bounded linear operators from into . is the collection of all bounded linear operators on . For , Dom, , , and denote the domain of , range of , identity operator on the range of , and adjoint of , respectively. An operator is said to be nonnegative if , for all , and positive if , for all . For more details on positive operators, see [20]. Further, the sequence space with the inner product given by is a separable Hilbert space.

Definition 1 (see [12]). Let . A sequence in is said to be a -frame for , if there exist constants such that for all , The constants and are called lower and upper frame bounds of -frame , respectively. In particular, if , for all then the -frame is said to be a tight -frame for . If , then it is called a Parseval -frame for .

Definition 2 (see [15]). Let . A -frame for is said to be a scalable -frame for if there exist nonnegative scalars, such that is a Parseval -frame for .

Definition 3 (see [7]). A sequence is said to be a -frame for with respect to , if there exist constants such that for all , We call the constants and to be the lower and upper frame bounds of -frame, respectively. A -frame is said to be a tight -frame if in (4) and a Parseval -frame, if . If only the right hand inequality in (4) holds, then is called a -Bessel sequence for with respect to .
The synthesis operator for -frame is given by such that , and the analysis operator is given by such that . The -frame operator given by , for all is bounded, self-adjoint, and invertible.

Definition 4 (see [14]). Let . A sequence is said to be a - frame for with respect to , if there exist constants such that for all , We call the constants and to be the lower and upper frame bounds of the --frame , respectively, and called a -Bessel sequence if only the right hand in (5) holds.
If , for all , then the --frame is called a tight --frame; moreover, if , then it is called a Parseval --frame.

Remark 5. For , --frames are just the ordinary -frames.

Definition 6 (see [16]). Let . Suppose and be two - frames. Then, is a --dual of if .

Definition 7 (see [19]). An operator defined on a closed linear span of basis in a normed space is called a diagonal operator, whenever , where and ’s are complex numbers. If is a continuous operator, then Since a positive diagonal operator is invertible, so there exist two constants and such that for all ,

Definition 8 (see [16]). Assume that is a diagonal operator on for . We say that the operator on is a block diagonal operator with as its diagonal whenever , Dom, where Dom. For the block diagonal operator , .

Definition 9 (see [19]). A sequence of positive diagonal operators is called seminormalized, whenever where and are the lower and upper bounds of , respectively.

Definition 10 (see [19]). A -frame is said to be a scalable -frame for with respect to , whenever there exists a sequence of nonnegative diagonal operators such that is a Parseval -frame. Further, if ’s are positive operators, then the -frame is called a strictly scalable -frame.

Definition 11 (see [21]). An operator is said to be bounded below if there exists such that , for all .

Theorem 12 (see [22] (Douglas Factorization Theorem)). Let and . Then, the following are equivalent: (i)(ii)there exists such that (iii)there exists such that

3. Some Properties of Scalable --Frames

In this section, we introduce the concept of scalable --frames and discuss some of their properties.

Definition 13. Let . A --frame of with respect to is called a scalable --frame for with respect to whenever there exists a sequence of nonnegative diagonal operators such that is a Parseval --frame, i.e., for all , If ’s are positive operators, then the --frame is called strictly scalable --frame.

It is obvious that every --frame is not scalable as is evident from the following example.

Example 14. Let be a separable Hilbert space with an orthonormal basis and define , for . For each , define as , for and as , and , for . Then, , , , for . Then, is a --frame which is not scalable because there does not exist a sequence of nonnegative diagonal operators such that .

The next example illustrates the existence of scalable --frames.

Example 15. Let be its standard orthonormal basis and . Define as , as , as , and as . Then, Consider the operators for given by Then for each , and . Therefore, is a Parseval --frame, and hence, is a scalable --frame.

Since scaling of a --frame by a sequence of diagonal operators does not need to always result in a --frame, but if we scale the --frame by a sequence of seminormalized diagonal operators, then it always gives rise to a new --frame as shown in the next theorem.

Theorem 16. Let and be a --frame for with respect to with --dual . If is a sequence of seminormalized positive diagonal operators, then is a --frame for with respect to with as its --dual.

Proof. Let and be bounds of the --frame , and and be the lower and upper bounds, respectively, for the diagonal operators , for . Then, for each , Also, for each , Hence, it follows that is a --frame for with respect to with as its --dual.

Next, we examine the relationship between a scalable --frame and a scalable -frame. For , every scalable --frame is a scalable -frame. In the following example, we observe that a scalable --frame can be a scalable -frame even if .

Example 17. Let be its standard orthonormal basis and , , and . Define as , as , as , and as , and for , as Here Then, and . Thus, is a scalable -frame and scalable --frame for with respect to , where .

In the next theorem, we prove the necessary and sufficient condition on the operator under which a scalable --frame becomes a scalable -frame. Further, we find a relationship between the and , where denotes the synthesis operator of .

Theorem 18. Let and be a scalable --frame for with respect to with nonnegative diagonal operators . Then, the following holds: (i) is a scalable -frame with nonnegative diagonal operators if and only if (ii), where denotes the synthesis operator of

Proof. (i)By definition of scalable -frame and scalable --frame, for all , and . Thus, we get , for each . Hence, . Conversely, if , then and as is a scalable --frame for , we get (ii)As is a scalable --frame, and is a sequence of nonnegative diagonal operators; therefore, is a -Bessel sequence, and so its synthesis operator is well defined. For each Using Theorem 12, we get .

Next, we construct new scalable --frames from a given scalable --frame.

Theorem 19. Let and be a scalable --frame for with respect to . Then, the following holds: (i)If is an isometry that commutes with , then is a scalable --frame for with respect to (ii) is a scalable --frame for with respect to (iii) be a scalable --frame for with respect to , where denote the frame operator of

Proof. Since is a scalable --frame for with respect to , therefore, there exists a sequence of nonnegative diagonal operators such that for all , (i)Let be an isometry that commutes with . Then for all , we have(ii)Replacing by in the definition of scalable --frame, we obtain for all , (iii)By definition of scalable --frame, for all . Thus, , for all and . Hence, is a scalable --frame for with respect to

Theorem 20. Let and . If is a scalable --frame for with respect to and is an isometry satisying , then is a scalable --frame for with respect to .

Proof. By definition of scalable --frame, there exists a sequence of nonnegative diagonal operators such that for all . Hence, for . As and is an isometry, we have for all . Hence, is a scalable --frame for with respect to .

Theorem 21. Let and be a --frame for with respect to . If for , is a scalable -frame of with respect to by a sequence of nonnegative diagonal operators . Then, is a scalable --frame for with respect to by the sequence of nonnegative diagonal operators if and only if is a Parseval --frame for with respect to .

Proof. For each , is a scalable -frame of with respect to by a sequence of nonnegative diagonal operators , therefore for all , First, assume that is a scalable --frame for with respect to by the sequence of nonnegative diagonal operators . Then, for each As, for each , therefore, using (19) and (20), we obtain Thus, is a Parseval --frame for with respect to .
Conversely, assume that is a Parseval --frame for with respect to . As for each , is a scalable -frame of with respect to by a sequence of nonnegative diagonal operators . Therefore,

Now, we prove a necessary and sufficient condition under which a scalable --frame generates a scalable --frame for any .

Theorem 22. Let . Then, is a scalable --frame for if and only if is a scalable --frame for with respect to .

Proof. Suppose is a scalable --frame for , then there exists a sequence of nonnegative diagonal operators such that for all , This implies that for , we have Hence, it follows that is a scalable --frame for with respect to . The converse part can be proved in the similar manner.

Next, we determine the conditions under which a --frame can be rescaled to obtain another --frame.

Theorem 23. Let and be a --frame for with respect to with analysis operator , and be a sequence of nonnegative diagonal operators. Then, the following conditions are equivalent: (i) is a --frame for with respect to (ii)There exists a diagonal operator on such that Dom, and is bounded and . In this case, the frame operator of is given by

Proof. Suppose that is a --frame for with respect to with bounds and , respectively and synthesis operator . Define as . Then, is a bounded block diagonal operator on with as its diagonal blocks and for any , Thus, and Dom. Also, for , , for some . Then, . Therefore, is bounded. As, , therefore by Theorem 12, .
Conversely, suppose there exists a diagonal operator on such that Dom, is bounded and . By Theorem 12, we have for all , for some . As is bounded, there exists some such that , for all . Thus, for , we have Thus, from equations (26) and (27), it follows that is a --frame for with respect to . Also, .

Theorem 24. Let and be a --frame for with respect such that ’s are bounded below and . If is a sequence of nonnegative diagonal operators, then the following statements are equivalent: (i) is a --frame(ii)There exists a bounded diagonal operator on with ’s as its diagonal blocks and Dom

Proof. Suppose that is a --frame for with respect to with bounds and , respectively. Since for each , is bounded below, therefore there exist such that , for each . As , so there exist and such that , . Also, for each , Therefore, we have , for each . Hence, there exists such that , for all , and thus, is bounded on . Using Theorem 23, we obtain Dom. Conversely, assume that the diagonal operator on with as its diagonal blocks is bounded and Dom. By Theorem 23, is a --frame.

In the following result, we present an equivalent condition for the scalability of --frames.

Theorem 25. Let and be a --frame for with respect to with analysis operator . Then, the following are equivalent: (i) is a scalable --frame for with respect to (ii)There exists a nonnegative bounded block diagonal operator on such that

Proof. First, let us assume that is a scalable --frame. Then, there exists a sequence of nonnegative diagonal operators such that for all , Define as .
Then, is a bounded block diagonal operator on with ’s as its diagonal blocks. By Theorem 23, the frame operator of is . So, we have .
Conversely, assume that there exists a nonnegative bounded block diagonal operator with diagonal blocks on such that .
Then, for all , we have Hence, the result holds.

Xiang [17] proved that if is a --frame for with respect to , then the -frame induced by this --frame takes over the desired properties of the --frame. In the next result, we prove another property that the -frame induced by --frame inherits in terms of scalability.

Theorem 26. Let and be a --frame for with respect to . If is an orthonormal basis of , for each , then the following are equivalent: (i) is a scalable --frame for with respect to (ii) is a scalable -frame for

Proof. Let be a scalable --frame for with respect to ; then, there exists a sequence of nonnegative diagonal operators such that for all , . By definition of diagonal operators, there exist nonnegative scalars satisfying , where is an orthonormal basis for . Then, we have for all , Hence, is a scalable -frame for .
Conversely, if is a scalable -frame for , then there exists a sequence of nonnegative scalars satisfying For each , define as . Then, is a sequence of nonnegative diagonal operators such that , for all .

If and are any two Hilbert spaces, then the direct sum of and is denoted by is a Hilbert space with pointwise operations and inner product given by

If and are Hilbert spaces and and , then and Next, for , we construct scalable --frame on Hilbert space . For more information on direct sums, one can refer to [23–26].

Theorem 27. Let and . If is a scalable --frame for with respect to and is a scalable --frame for with respect to , then is a scalable --frame for with respect to .

Proof. By definition of scalable --frame, , there exist nonnegative diagonal operators , for such that and For , define . Then is a sequence of nonnegative diagonal operators and for all ,

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have contributed equally to manuscript writing, editing, and conceptualization. All authors reviewed the manuscript and confirmed that it met the ICMJE criteria.

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Copyright © 2024 Varinder Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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