In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spa... more In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spaces will be introduced. Some of its properties will be studied. Also, the relations between two continuous Riesz bases in Hilbert spaces will be clarified through examples.
The conditions for sequences {f k } ∞ k=1 and {g k } ∞ k=1 being Bessel sequences, frames or Ries... more The conditions for sequences {f k } ∞ k=1 and {g k } ∞ k=1 being Bessel sequences, frames or Riesz bases, can be expressed in terms of the so-called cross-Gram matrix. In this paper we investigate the cross-Gram operator, G, associated to the sequence { f k , g j } ∞ j,k=1 and sufficient and necessary conditions for boundedness, invertibility, compactness and positivity of this operator are determined depending on the associated sequences. We show that invertibility of G is not possible when the associated sequences are frames but not Riesz Bases or at most one of them is Riesz basis. In the special case we prove that G is a positive operator when {g k } ∞ k=1 is the canonical dual of {f k } ∞ k=1 .
In this article, we present a constructive method for computing the frame coefficients of finite ... more In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory.
The notation of generalized Bessel multipliers is obtained by a bounded operator on ℓ 2 which is ... more The notation of generalized Bessel multipliers is obtained by a bounded operator on ℓ 2 which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisely in this case multipliers can be easily composed and inverted. Special attention is devoted to the study of invertible generalized multipliers. Sufficient and/or necessary conditions for invertibility are determined. Finally, the behavior of these operators under perturbations is discussed.
An interesting question about the perturbed sequences is: when do they inherit the properties of ... more An interesting question about the perturbed sequences is: when do they inherit the properties of the original one? An elegant relation between frames (fusion frames) and their perturbations is the relation of their redundancies. In this paper, we investigate these relationships. Also, we express the redundancy of frames (fusion frames) in terms of the cosine angle between some subspaces.
In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. ... more In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames gives ordinary weaving K-frames and vice-versa.
In this manuscript, we answer a list of longstanding open problems on weak phase retrieval includ... more In this manuscript, we answer a list of longstanding open problems on weak phase retrieval including: (1) A complete classification of the vectors {x i } 2 i=1 in R 3 that do weak phase retrieval; (2) We show that frames doing weak phase retrieval in R n must span R n ; (3) We give an example of a set of vectors doing phase retrieval but their orthogonal complement hyperplanes fail weak phase retrieval; (4) We give a classification of weak phase retrievable frames -which makes clear the difference between phase retrieval and weak phase retrieval; (5) We classify when weak phase retrievable frames also do norm retrieval. We then introduce the notion of weak phase retrieval by projections and develop their basic properties. We then look at phase (norm) retrieval by projections. We end with some open problems. We provide numerous examples to show that our results are best possible.
In this paper, we investigate Parseval's equality and define the fuzzy frame on Felbin fuzzy Hilb... more In this paper, we investigate Parseval's equality and define the fuzzy frame on Felbin fuzzy Hilbert spaces. For showing the importance of defining fuzzy frame, we know that, in the classical Hilbert space, C(Ω) is not normable, but, in this manuscript, we prove that C(Ω) (the vector space of all continuous functions on Ω) is normable in a Felbin fuzzy Hilbert space and so the defining fuzzy frame on C(Ω) is possible. These consequences of the category of fuzzy frames in Felbin fuzzy Hilbert spaces are wider than the category of the frames in the classical Hilbert spaces. MSC 2010. 54A40, 03E72
Some results of fuzzy frames on fuzzy Hilbert spaces from the point of view of Bag and Samanta ar... more Some results of fuzzy frames on fuzzy Hilbert spaces from the point of view of Bag and Samanta are proved. In this paper, we define dual fuzzy frames and fuzzy Riesz bases and establish some fundamental results via dual fuzzy frames and fuzzy Ries bases. Next, we investigate the relation between fuzzy frame and fuzzy Riesz bases. MSC 2010. 30C45.
In this paper, we introduce the concept of weak fuzzy linear topology on a fuzzy topological vect... more In this paper, we introduce the concept of weak fuzzy linear topology on a fuzzy topological vector space as a generalization of usual weak topology. We prove that this topology consists of all weakly lower semi-continuous fuzzy sets on a given vector space when K (R or C) considered with its usual fuzzy topology. In the case that the topology of K is different from the usual fuzzy topology, we show that the weak fuzzy topology is not equivalent with the topology of weakly lower semi-continuous fuzzy sets.
One of the famous mathematical inequality is Minkowski's inequality. It is an important inequalit... more One of the famous mathematical inequality is Minkowski's inequality. It is an important inequality from both mathematical and application points of view. In this paper, a Minkowski type inequality for fuzzy and pseudo-integrals is studied. The established results are based on the classical Minkowski's inequality for integrals.
In this manuscript, we present several new results in finite and countable dimensional real Hilbe... more In this manuscript, we present several new results in finite and countable dimensional real Hilbert space phase retrieval and norm retrieval by vectors and projections. We make a detailed study of when hyperplanes do norm retrieval. Also, we show that the families of norm retrievable frames {f i } m i=1 in R n are not dense in the family of m ≤ (2n -2)-element sets of vectors in R n for every finite n and the families of vectors which do norm retrieval in ℓ 2 are not dense in the infinite families of vectors in ℓ 2 . We also show that if a Riesz basis does norm retrieval in ℓ 2 , then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.
The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics ... more The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics and differential geometry, arises from the variation problems of ordinary mean curvature vector field. Generally, harmonic submanifolds are biharmonic, but not vice versa. Of course, many examples of biharmonic hypersurfaces are harmonic. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every biharmonic submanifold is harmonic. Although the conjecture has not been proven (in general case), it has been affirmed in many cases, and this has led to its spread to various types of submanifolds. Inspired by the conjecture, we study the Lorentz submanifolds of the Lorentz-Minkowski spaces. We consider an advanced versión of the conjecture (namely, L1-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean 5-space L5 := E15 (i.e. the Minkowski 5-space). We confirm the extended conjecture on Lorentz hypersurfaces with three principal curvatures.
In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis d... more In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis dealing with set-valued functions. The first one is given for a pseudo-integral of set-valued function where pseudo-addition and pseudo-multiplication are constructed by a monotone continuous function Another is given by the semiring ([0, 1], max, ⊙) where pseudo-multiplication is generated by an increasing continuous function.
A new notion in frame theory has been introduced recently that called woven frames. %From the per... more A new notion in frame theory has been introduced recently that called woven frames. %From the perspective of others, Woven and weaving frames are powerful tools for pre-processing signals and distributed data processing. The purpose of introducing fusion frame or frame of subspace is to first construct local components and then build a global frame from these. This type of frame behaves as a generalization of frames. Motivating by the concepts of fusion and weaving frames, we investigate the notion woven-weaving fusion frames and present some of their features. Also, we study some effects of perturbations on woven frames and introduce Riesz decomposition of wovens and then we examine some of behaviors of this type of decomposition.
Controlled frames have been recently introduced in Hilbert spaces to improve the numerical effici... more Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert-Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every , where M is a bijective operator on H. Furthermore, we propose an equivalent accessible condition to the sequence {f k } ∞ k=1 being a (U, C)-controlled Riesz basis.
Improving and extending the concept of dual for frames, fusion frames and continuous frames, the ... more Improving and extending the concept of dual for frames, fusion frames and continuous frames, the notion of dual for continuous fusion frames in Hilbert spaces will be studied. It will be shown that generally the dual of c-fusion frames may not be defined. To overcome this problem, the new concept namely Q-dual for c-fusion frames will be defined and some of its properties will be investigated.
International Journal of Wavelets, Multiresolution and Information Processing
In this paper, we will present several new results in finite and countable dimensional separable ... more In this paper, we will present several new results in finite and countable dimensional separable real Hilbert space phase retrieval and norm retrieval by fusion frames. We will characterize of norm retrieval for fusion frames similar norm retrieval for vectors and we will show that only one direction holds for fusion frames. In similar vector case, we will show that every tight fusion frame can do norm retrieval. Also we will show that the unitary operators preserve phase (norm) retrievability of fusion frames. We will make a detailed study of when hyperplanes do norm retrieval and show a general result about it. We will provide numerous examples to show that our results are best possible.
Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce t... more Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce the invariance of Fréchet frames under perturbation. Also we show that for any Fréchet spaces, there is a Fréchet frame and any element in these spaces has a series expansion.
The concept of (p, q)-pair frames is generalized to (ℓ, ℓ *)-pair frames. Adjoint (conjugate) of ... more The concept of (p, q)-pair frames is generalized to (ℓ, ℓ *)-pair frames. Adjoint (conjugate) of a pair frames for dual space of a Banach space is introduced and some conditions for the existence of adjoint (conjugate) of pair frames are presented.
In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spa... more In this manuscript, the concept of dual and approximate dual for continuous frames in Hilbert spaces will be introduced. Some of its properties will be studied. Also, the relations between two continuous Riesz bases in Hilbert spaces will be clarified through examples.
The conditions for sequences {f k } ∞ k=1 and {g k } ∞ k=1 being Bessel sequences, frames or Ries... more The conditions for sequences {f k } ∞ k=1 and {g k } ∞ k=1 being Bessel sequences, frames or Riesz bases, can be expressed in terms of the so-called cross-Gram matrix. In this paper we investigate the cross-Gram operator, G, associated to the sequence { f k , g j } ∞ j,k=1 and sufficient and necessary conditions for boundedness, invertibility, compactness and positivity of this operator are determined depending on the associated sequences. We show that invertibility of G is not possible when the associated sequences are frames but not Riesz Bases or at most one of them is Riesz basis. In the special case we prove that G is a positive operator when {g k } ∞ k=1 is the canonical dual of {f k } ∞ k=1 .
In this article, we present a constructive method for computing the frame coefficients of finite ... more In this article, we present a constructive method for computing the frame coefficients of finite wavelet frames over prime fields using tools from computational harmonic analysis and group theory.
The notation of generalized Bessel multipliers is obtained by a bounded operator on ℓ 2 which is ... more The notation of generalized Bessel multipliers is obtained by a bounded operator on ℓ 2 which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisely in this case multipliers can be easily composed and inverted. Special attention is devoted to the study of invertible generalized multipliers. Sufficient and/or necessary conditions for invertibility are determined. Finally, the behavior of these operators under perturbations is discussed.
An interesting question about the perturbed sequences is: when do they inherit the properties of ... more An interesting question about the perturbed sequences is: when do they inherit the properties of the original one? An elegant relation between frames (fusion frames) and their perturbations is the relation of their redundancies. In this paper, we investigate these relationships. Also, we express the redundancy of frames (fusion frames) in terms of the cosine angle between some subspaces.
In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. ... more In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames gives ordinary weaving K-frames and vice-versa.
In this manuscript, we answer a list of longstanding open problems on weak phase retrieval includ... more In this manuscript, we answer a list of longstanding open problems on weak phase retrieval including: (1) A complete classification of the vectors {x i } 2 i=1 in R 3 that do weak phase retrieval; (2) We show that frames doing weak phase retrieval in R n must span R n ; (3) We give an example of a set of vectors doing phase retrieval but their orthogonal complement hyperplanes fail weak phase retrieval; (4) We give a classification of weak phase retrievable frames -which makes clear the difference between phase retrieval and weak phase retrieval; (5) We classify when weak phase retrievable frames also do norm retrieval. We then introduce the notion of weak phase retrieval by projections and develop their basic properties. We then look at phase (norm) retrieval by projections. We end with some open problems. We provide numerous examples to show that our results are best possible.
In this paper, we investigate Parseval's equality and define the fuzzy frame on Felbin fuzzy Hilb... more In this paper, we investigate Parseval's equality and define the fuzzy frame on Felbin fuzzy Hilbert spaces. For showing the importance of defining fuzzy frame, we know that, in the classical Hilbert space, C(Ω) is not normable, but, in this manuscript, we prove that C(Ω) (the vector space of all continuous functions on Ω) is normable in a Felbin fuzzy Hilbert space and so the defining fuzzy frame on C(Ω) is possible. These consequences of the category of fuzzy frames in Felbin fuzzy Hilbert spaces are wider than the category of the frames in the classical Hilbert spaces. MSC 2010. 54A40, 03E72
Some results of fuzzy frames on fuzzy Hilbert spaces from the point of view of Bag and Samanta ar... more Some results of fuzzy frames on fuzzy Hilbert spaces from the point of view of Bag and Samanta are proved. In this paper, we define dual fuzzy frames and fuzzy Riesz bases and establish some fundamental results via dual fuzzy frames and fuzzy Ries bases. Next, we investigate the relation between fuzzy frame and fuzzy Riesz bases. MSC 2010. 30C45.
In this paper, we introduce the concept of weak fuzzy linear topology on a fuzzy topological vect... more In this paper, we introduce the concept of weak fuzzy linear topology on a fuzzy topological vector space as a generalization of usual weak topology. We prove that this topology consists of all weakly lower semi-continuous fuzzy sets on a given vector space when K (R or C) considered with its usual fuzzy topology. In the case that the topology of K is different from the usual fuzzy topology, we show that the weak fuzzy topology is not equivalent with the topology of weakly lower semi-continuous fuzzy sets.
One of the famous mathematical inequality is Minkowski's inequality. It is an important inequalit... more One of the famous mathematical inequality is Minkowski's inequality. It is an important inequality from both mathematical and application points of view. In this paper, a Minkowski type inequality for fuzzy and pseudo-integrals is studied. The established results are based on the classical Minkowski's inequality for integrals.
In this manuscript, we present several new results in finite and countable dimensional real Hilbe... more In this manuscript, we present several new results in finite and countable dimensional real Hilbert space phase retrieval and norm retrieval by vectors and projections. We make a detailed study of when hyperplanes do norm retrieval. Also, we show that the families of norm retrievable frames {f i } m i=1 in R n are not dense in the family of m ≤ (2n -2)-element sets of vectors in R n for every finite n and the families of vectors which do norm retrieval in ℓ 2 are not dense in the infinite families of vectors in ℓ 2 . We also show that if a Riesz basis does norm retrieval in ℓ 2 , then it is an orthogonal sequence. We provide numerous examples to show that our results are best possible.
The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics ... more The subject of harmonic and biharmonic submanifolds, with important role in mathematical physics and differential geometry, arises from the variation problems of ordinary mean curvature vector field. Generally, harmonic submanifolds are biharmonic, but not vice versa. Of course, many examples of biharmonic hypersurfaces are harmonic. A well-known conjecture of Bang-Yen Chen on Euclidean spaces says that every biharmonic submanifold is harmonic. Although the conjecture has not been proven (in general case), it has been affirmed in many cases, and this has led to its spread to various types of submanifolds. Inspired by the conjecture, we study the Lorentz submanifolds of the Lorentz-Minkowski spaces. We consider an advanced versión of the conjecture (namely, L1-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean 5-space L5 := E15 (i.e. the Minkowski 5-space). We confirm the extended conjecture on Lorentz hypersurfaces with three principal curvatures.
In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis d... more In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis dealing with set-valued functions. The first one is given for a pseudo-integral of set-valued function where pseudo-addition and pseudo-multiplication are constructed by a monotone continuous function Another is given by the semiring ([0, 1], max, ⊙) where pseudo-multiplication is generated by an increasing continuous function.
A new notion in frame theory has been introduced recently that called woven frames. %From the per... more A new notion in frame theory has been introduced recently that called woven frames. %From the perspective of others, Woven and weaving frames are powerful tools for pre-processing signals and distributed data processing. The purpose of introducing fusion frame or frame of subspace is to first construct local components and then build a global frame from these. This type of frame behaves as a generalization of frames. Motivating by the concepts of fusion and weaving frames, we investigate the notion woven-weaving fusion frames and present some of their features. Also, we study some effects of perturbations on woven frames and introduce Riesz decomposition of wovens and then we examine some of behaviors of this type of decomposition.
Controlled frames have been recently introduced in Hilbert spaces to improve the numerical effici... more Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert-Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every , where M is a bijective operator on H. Furthermore, we propose an equivalent accessible condition to the sequence {f k } ∞ k=1 being a (U, C)-controlled Riesz basis.
Improving and extending the concept of dual for frames, fusion frames and continuous frames, the ... more Improving and extending the concept of dual for frames, fusion frames and continuous frames, the notion of dual for continuous fusion frames in Hilbert spaces will be studied. It will be shown that generally the dual of c-fusion frames may not be defined. To overcome this problem, the new concept namely Q-dual for c-fusion frames will be defined and some of its properties will be investigated.
International Journal of Wavelets, Multiresolution and Information Processing
In this paper, we will present several new results in finite and countable dimensional separable ... more In this paper, we will present several new results in finite and countable dimensional separable real Hilbert space phase retrieval and norm retrieval by fusion frames. We will characterize of norm retrieval for fusion frames similar norm retrieval for vectors and we will show that only one direction holds for fusion frames. In similar vector case, we will show that every tight fusion frame can do norm retrieval. Also we will show that the unitary operators preserve phase (norm) retrievability of fusion frames. We will make a detailed study of when hyperplanes do norm retrieval and show a general result about it. We will provide numerous examples to show that our results are best possible.
Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce t... more Motivating the perturbations of frames in Hilbert and Banach spaces, in this paper we introduce the invariance of Fréchet frames under perturbation. Also we show that for any Fréchet spaces, there is a Fréchet frame and any element in these spaces has a series expansion.
The concept of (p, q)-pair frames is generalized to (ℓ, ℓ *)-pair frames. Adjoint (conjugate) of ... more The concept of (p, q)-pair frames is generalized to (ℓ, ℓ *)-pair frames. Adjoint (conjugate) of a pair frames for dual space of a Banach space is introduced and some conditions for the existence of adjoint (conjugate) of pair frames are presented.
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Papers by Asghar Rahimi