1. Introduction
Discrete frame theory on Hilbert spaces is relatively recent. This theory was introduced by Duffin and Schaeffer in 1952 [
1] as a tool to study problems related to non-harmonic Fourier series. Almost 30 years later, it was developed by Daubechies, Grossman and Meyer [
2]; they used frames to find expansions in series of functions on
similar to the expansion in series that is done using orthonormal bases. As a result, the main advantage is that a good discrete frame can behave almost like an orthonormal basis; with an additional, they do not require the uniqueness of the coefficients when writing a Hilbert space vector as a linear combination of the frame elements (frame decomposition theorem). For this reason, in the literature, they are often called over-complete bases. Consequently, frames have turned out to be a powerful tool in signal processing, image processing, data understanding, sampling theory, and wavelet analysis [
3] (see also [
4,
5,
6]). Frame theory has been extended to the context of Krein’s spaces, which are a generalization of Hilbert spaces that have a wide variety of applications in physics (see [
7,
8]).
Soft set theory was introduced by Molodtsov [
9] in 1999 as a new mathematical tool for dealing with uncertainties while modeling problems in engineering, physics, computer science, economics, social science, and medical sciences. This theory began to receive special attention in 2002 when Maji et al. [
10] applied the soft sets to decision-making problems, using rough mathematics, and later in 2003, with the definitions of various operations of soft sets [
11]. Since then, research works in soft sets theory and its applications in various fields have been progressing rapidly because this theory is free from the many difficulties that have troubled the usual theoretical approaches. This is how research related to soft sets has been carried out in several directions among which we can mention the following: information systems, decision making, nonlinear neutral differential equations and algebraic structures, fuzzy sets, and rough sets, as we can see in the papers [
12,
13,
14,
15,
16]. In particular, regarding mathematical analysis and its applications, the concepts and results of soft real sets, soft real numbers, soft complex sets, soft complex numbers, soft linear spaces, soft metric spaces, soft normed spaces, soft inner product, soft Hilbert spaces, soft linear operator, soft linear functional, soft Banach algebra, soft topology, etc., have originated (see [
11,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26]). In the book [
27], various applications of soft sets are discussed in problems related to data filling in incomplete information systems, and as is well known, frames are also a suitable tool to deal with these same types of problems, for which it is interesting to address frame theory together with soft sets theory to propose a new mathematical tool (soft frames) that is more efficient in the absence of partial information. This theoretical framework motivates the study of certain topics based on operator theory in spaces with a complete inner product (Hilbert spaces), which is potentially applicable in signal processing, where the frame coefficients serve to model the data packets to be transmitted; hence, the elimination of an element of the frame is equivalent to a packet that is not delivered in the communication, but this could be recovered using the frame decomposition theorem.
This manuscript was designed as follows.
Section 2 corresponds to the preliminaries, where we cover all the theory of soft sets necessary to study soft frames, in addition to introducing the notions of soft inner product, soft operators, soft Hilbert spaces, etc. In
Section 3, we develop the definition of soft frame on soft Hilbert spaces, and state the most important result: the soft frame decomposition theorem.
2. Preliminaries
Throughout this paper, X denotes a non-empty set (possibly without algebraic structure), the power set of X, and A a non-empty set of parameters.
Definition 1 ([
9]).
A soft set on X is a pair where F is a mapping given by . In this way, we can see a soft set as the following:
where
is the graph of
F with respect to
A.
Note that a soft set is determined by knowing for all . Therefore, it is common to find in the literature that is called a soft set on X, but it should not be a cause for confusion.
Example 1. Let and . If describes the generating elements of the cyclic group. Then, it is easy to see that . Hence, is a soft set on X seen as follows: Definition 2 ([
23]).
A soft set on X is said to be a null soft set if for all , and in this case we write If for some , then the soft set is said to be a non-null soft set on X.
Definition 3 ([
23]).
A soft set on X is said to be an absolute soft set if for all ; in this case, we write . This convention of absolute soft set is adopted throughout the present work. Definition 4 ([
23]).
A soft element on X is a function . Now if for all , the soft element ϵ is said to belong to the soft set on X, which we denote by . Proposition 1 ([
23]).
Let be a soft set on X and ϵ be a soft element on X, which belongs to . Given , we have We denote the collection of all the soft elements of a soft set
by
; this is
Definition 5 ([
23]).
Let or , and let A be a non-empty set of parameters. Consider the setthen, a mapping is called a soft -set. This is denoted by . Furthermore, if for all it is satisfied that is a singleton, then by identifying with its corresponding soft element, we call this soft element a soft -number. We denote the set of all the soft -numbers or soft real numbers by . In addition, we use the symbols , etc., to denote soft -numbers such that they behave as constants, that is for all . Similarly, we symbolize the set of all soft -numbers or soft complex numbers by .
Definition 6 ([
23]).
The set is called the set of all nonnegative soft real numbers and is denoted by . Definition 7 ([
23]).
For two soft real numbers and we define the following:- 1.
, if , for all
- 2.
, if , for all
- 3.
, if , for all
- 4.
, if , for all .
Note that if we assign to X a structure of vector space, it is interesting to think of the possible structure of as a subset of X for all . This motivates the following important definition.
Definition 8 ([
18,
24]).
Let X be a -vector space (typically or ), A be a nonempty set of parameters and be a soft set on X. The soft set is said to be a soft -vector space on X if is a vector subspace of X for all . The importance of the above definition is that it allows us to relate the usual linear algebra to soft set theory. Furthermore, it gives us the tools to define important concepts in the classical functional analysis, such as norm, inner product, Banach space, and Hilbert space, among others, but from this context.
Definition 9 ([
18,
24]).
Let be soft -vector space. A soft element of is said to be a soft vector of . Similarly, a soft element is said to be a soft scalar, where is the scalar field. Definition 10 ([
18,
24]).
Let be a soft -vector space on X. A soft vector x of is said to be a null soft vector if , where θ is the zero element of X. This is denoted by Θ. Definition 11 ([
18,
24]).
Let be a soft -vector space on X and be two soft vectors of and k be a soft scalar. Then, the addition of x with and scalar multiplication of k and x are defined by . Evidently, are soft vectors of . Theorem 1 ([
18,
24]).
Let X be a -vector space, A be a nonempty set of parameters and be a soft -vector space on X. Then, we have the following:- 1.
, for all ,
- 2.
, for all soft scalar k,
- 3.
, for all
Definition 12 ([
19]).
Let be an absolute soft -vector space, then a mapping is said to be a soft norm on if satisfies the following conditions:- (N1)
for all ,
- (N2)
if and only if ,
- (N3)
for all and for all soft scalar α,
- (N4)
for all
The soft -vector space with a soft norm on is said to be a soft -normed space and is denoted by or .
Lemma 1 ([
19]).
Let be a soft -normed space, then for all and for , we have if and only if . Proof. Consider the soft scalar defined by if , if Then, note that whenever ; also for . Now, by (N3), we have ⟺⟺⟺⟺. □
Lemma 2 ([
19]).
Every soft norm satisfies the following condition:- (N5)
For and , is a singleton.
The two lemmas above allow us to prove the following important theorem about soft normed spaces.
Theorem 2 ([
19] Decomposition theorem).
If is a soft normed space, then for ech , the mapping defined by where x is such that is a norm on X for all . Definition 13 ([
24]).
If is an absolute soft vector space, then a binary operation is said to be a soft inner product on if it satisfies the following conditions:- (I1)
for all and if and only if ,
- (I2)
for all ,
- (I3)
for all and for all soft scalar α,
- (I4)
for all .
The soft vector space with a soft inner product on is said to be a soft inner product space and is denoted by or .
Lemma 3 ([
24]).
Every soft inner product on satisfies the following condition:- (I5)
For and is a singleton.
Theorem 3 ([
24]).
If is a soft inner product space, then for all , the mapping is defined by where are such that and is an inner product on X. Theorem 4 ([
17] Cauchy-Schwarz inequality).
Let be a soft inner product space. Then, for all . Definition 14 ([
17]).
A soft inner product space is said to be complete if it is complete with respect to the soft metric defined by the soft inner product. A complete soft inner product space is said to be a soft Hilbert space. Definition 15 ([
18,
24]).
Let be an operator. Then, T is said to be soft linear if the following holds:- (L1)
for all ,
- (L2)
for all soft scalar c and all .
Lemma 4 ([
18,
24]).
Every soft linear operator where are absolute soft normed spaces, and satisfies the following condition:- (L3)
For and the set is a singleton.
Theorem 5 ([
18,
24]).
Every soft linear operator can be decomposed into a parameterized family of linear operators. This is, if is a soft linear operator, then the family where is defined by for all and with , is a family of linear operators. Theorem 6 ([
18,
24]).
Let be a parameterized family of linear operators from X to Y. Then, the operator given by if , is soft linear. Definition 16 ([
18,
24]).
Let be a soft linear operator, where are absolute soft normed spaces. The operator T is called bounded if there exists such that Definition 17 ([
18,
24]).
Let be a bounded soft linear operator. Then, the norm of the operator T, denoted by , is a soft real number defined as follows: Theorem 7 ([
18,
24]).
Suppose that are soft normed spaces and is a bounded soft linear operator. Then, for all . Definition 18 ([
18]).
Let be a soft linear operator, where are absolute soft normed spaces. The operator T is said to be continuous at if for every sequence of soft elements of with as , we have as . If T is continuous at each soft element of , then T is said to be a continuous operator. Theorem 8 ([
18]).
Let be a soft linear operator, where are soft normed spaces. Then, T is bounded if and only if T is continuous. Theorem 9 ([
18,
24]).
Let and be soft normed spaces and be a continuous soft linear operator. Then, is continuous on X for all . Theorem 10 ([
18,
24]).
Let and be soft normed spaces and be a bounded soft linear operator. Then, for all , where is the norm of the linear operator . Theorem 11 ([
24]).
Suppose that and are soft normed spaces and is a family of continuous linear operators such that for all λ. Then, the soft linear operator defined by , is a continuous soft linear operator. Definition 19 ([
24]).
Let , be soft Hilbert spaces and be a bounded soft linear operator. The operator is called the adjoint operator of T, if , for all and all . Theorem 12 ([
24]).
Let be a soft Hilbert space, be a continuous soft linear operator and be the adjoint operator of T. Then, the following properties hold:- 1.
is unique;
- 2.
is a soft linear operator;
- 3.
is a continuous soft operator with ;
- 4.
;
- 5.
;
- 6.
where are continuous soft linear operators;
- 7.
- 8.
- 9.
, where are continuous soft linear operators.
Proposition 2 ([
24]).
Let be a soft Hilbert space and be a continuous soft linear operator and be the adjoint operator of T. Then defined by is the adjoint operator of . Proposition 3 ([
24]).
Let be a soft Hilbert space and be a continuous soft linear operator. Let be a family of adjoint operators of . Then, the soft linear operator defined by is the adjoint operator of T. Definition 20 ([
24]).
A continuous soft linear operator is called a self-adjoint soft linear operator if . Definition 21 ([
17]).
Let be a soft Hilbert space. Then, a collection of soft vectors of is said to be orthonormal if for all , the following holds:If the soft set contains only a countable number of soft vectors, then we arrange it in a sequence of soft vectors and call it an orthonormal sequence.
Theorem 13 ([
17] Bessel inequality).
Let be an orthonormal sequence on a soft Hilbert space . Then for every , . Theorem 14 ([
17]).
Let be an orthonormal sequence on a soft Hilbert space having a finite set of parameters A. Then, the infinite serieswhere are soft scalars, is convergent if and only if the series is convergent. Theorem 15 ([
17]).
Let be an orthonormal sequence on a soft Hilbert space having a finite set of parameters A. Then, for any , Definition 22 ([
17]).
Let be a non-null collection of orthonormal soft elements of . Then, is said to be complete orthonormal if there exists a non-orthonormal set such that is a proper subset of . If the set contains only a countable number of soft elements then we call it a complete orthonormal sequence. In the following theorem, we consider
S the collection of all soft vectors
x of
such that
for all
, together with the null soft vector
. In symbols,
Theorem 16 ([
17]).
Let be an orthonormal sequence in a soft Hilbert space having a finite set of parameters A. Then, the following conditions are equivalent:- 1.
is complete,
- 2.
If for all and for all we have then
- 3.
For all , with ,
- 4.
For all , with , .
Corollary 1. Let G be a soft Banach algebra. If and , then there exists and .
3. Soft Frames in Soft Hilbert Spaces
Definition 23 ([
25]).
Let be a soft Hilbert space. We say that a self-adjoint soft linear operator T is positive if for all . In notation, we write . Also, means that for all and all soft scalar . Proposition 4. Let be a soft Hilbert space and be a self-adjoint soft linear operator. Then, for all and all .
Lemma 5. Let be a Hilbert space and A be a finite set of parameters. Then, the mappinggiven by for all and all defines a soft inner product on . Theorem 17. Let be a Hilbert space. If A is a finite set of parameters, we have that is a soft Hilbert space, where is given by for all and all
Proof. Let be a Hilbert space. Let be a Cauchy sequence in , where d is the soft metric on induced by . We must show that is convergent in . Indeed, we assert that for all , is a Cauchy sequence, because given —note that is a Cauchy sequence—there exists such that for all , that is, for all and all , which proves the assertion. Thus, since is complete, we have that is convergent for all for example, as Now, given , there exists such that for all . Hence, since A is finite, taking and making , the theorem is satisfied. □
Example 2. Let . It is well known that X is a Hilbert space with respect to the inner product for , en Now, let be soft elements of the absolute soft vector space Then, , are elements of Thus, the mapping , defined by for all is a soft inner product on . Therefore, , with A being a finite set of parameters, is a soft Hilbert space.
Next, we introduce the notion of soft discrete frame in soft Hilbert spaces. We study the pre-frame operator and the frame operator of a soft discrete frame. In addition, we establish the most important result, called the decomposition theorem of soft frames.
Definition 24. Let be a sequence of soft vectors of a soft Hilbert space having a finite set of parameters. We say that is a soft frame on if there exist soft real numbers such that the following holds: The soft real numbers
and
are called bounds of the soft frame. These are not unique, as the optimal bounds are the largest possible value of
and the smallest possible value of
that satisfy (
1). In the case that
, the soft frame is called tight.
Remark 1. The relationship between the previous definition and the discrete frames in usual Hilbert spaces is that every discrete frame on a Hilbert space induces a soft discrete frame on a soft Hilbert space, with respect to any finite set of parameters. The proof of this claim is as follows: Let be any Hilbert space and let be a discrete frame on H with bounds . Then, if A is a finite set of parameters, we know that, by virtue of Theorem 17, is a soft Hilbert space, where for all and all , whereby, if for each we define by for all , we can affirm that is a soft frame on with bounds defined by , and . Indeed, for each and each , we have the following:where we have used that is a frame on H. On the other hand, we have the following: Thus, we have proved the following:that is,Therefore, is a soft frame on . Definition 25. Given a soft frame on a soft Hilbert space having a finite set of parameters, we define the pre-frame operator by the following: Proposition 5. The pre-frame operator associated to is well defined and bounded.
Proof. Observe that
. Then, we have the following:
Therefore,
□
Proposition 6. The adjoint soft operator of T is given by
Proof. Let
and
then
with
Additionally, for all
, the following holds:
Therefore,
□
Definition 26. We define the associated frame operator to the soft frame by Proposition 7. The associated frame operator to the soft frame with bounds is bounded and self-adjoint.
Proof. Note the following:
Hence,
S is bounded. Additionally, the following holds:
Thus,
S is self-adjoint. □
Proposition 8. Let be the associated frame operator to the soft frame . Then Proof. Let
; then,
□
Remark 2. Note that for all we have the following:Hence, the frame condition can be written in the form Lemma 6. If is a soft frame on a soft Hilbert space with bounds then is complete on
Proof. Let
be such that
for all
. Then, by the frame condition, we have
and so
□
Proposition 9. Given a sequence in the soft Hilbert space the following statements are equivalent:
- 1.
is a soft frame with bounds
- 2.
is a positive and bounded soft linear operator of to , which satisfies
Proof. (i)⇒(ii) Suppose that
is a soft frame with bounds
then
is bounded by Proposition 7 and since
it follows by Remark 2 that
Thus,
and hence,
(ii)⇒(i) Suppose that
is true, that is,
for all
. Then
is a soft frame with bounds
□
Lemma 7. Let be a soft Hilbert space and be a positive soft linear operator. Then for any the mappingdefined by for all , is a soft inner product. Proof. Indeed, let and be soft scalars. Then, the following hold:
because
. Additionally,
□
The following result is a version of the Cauchy–Schwarz inequality for positive soft linear operators. We will see its usefulness later.
Proposition 10. Let be a soft Hilbert space. If is a positive soft linear operator; then, Proof. By Lemma 7, it is satisfied that for all
defines a soft inner product on
Thus, by the Cauchy–Schwarz inequality, we have the following:
for all
and all
; this is,
Thus, making
, we obtain the following
□
Theorem 18. If is a soft frame with bounds for then the following statements are satisfied:
- (i)
The soft linear operator is invertible andwhere for all - (ii)
is a soft frame for with bounds called the dual soft frame of
Proof. Since
then
and also,
Therefore, by Corollary 1, it follows that
S is invertible. In addition, observe that by Remark 2 and Cauchy–Schwarz inequality, we have the following:
Hence,
Thus, the following holds:
Therefore,
On the other hand, by Proposition 10 and frame condition, the following is satisfied:
Thus, and hence In summary, we have proved that
Since
is positive and self-adjoint, we have the following:
and as
, by Proposition 9, we obtain that
is a soft frame for
with bounds
□
Theorem 19 (Decomposition Theorem for soft frames).
Let be a soft frame with bounds for and S be the associated frame operator. Then, for all , we have the following: Proof. If
, then we have the following:
and also,
□
Note that Theorem 19 tells us that there is no restriction to write every element of a soft Hilbert space as a linear combination of the soft frame and the associated inverse soft frame operator, which differs from Theorem 16-(3).