Continuous frame in Hilbert space and its applications

In this paper, we study continuous frames in Hilbert spaces using a family of linearly independent vectors called coherent state (CS) and applying it in any physical space. To accomplish this goal, the standard theory of frames in Hilbert spaces, using discrete bases, is generalized to one where the basis vectors may be labeled using discrete, continuous or a mixture of the two types of indices. A comprehensive analysis of such frames is presented and illustrated by the examples drawn from a toy example Sea Star and the affine group.


INTRODUCTION
The Hilbert space is the natural framework for the mathematical description of many areas of Physics and Mathematics.The most economical solution, is of course to use an orthonormal basis,     n n ,  which gives in addition the uniqueness of decomposition: .
The uniqueness of the decomposition and the orthogonality of the basis vectors, while maintaining its others useful properties: fast convergence, numerical stability of the reconstruction       n , etc.The resulting object is called a discrete frame, a concept introduced by Duffin and Schaeffer (1952) in the context of non-harmonic Fourier analysis.Latter the concept of generalization of frames was proposed by Daubechies et al. (1986) and then independently by Ali et al., (1993)   .Combining the two allows for the recovery of any element  from its frame components: In mathematical physics, the coherent states, actually yield the following continuous resolution of the identity on the group, G with the (left) Haar measure where U being a strongly continuous unitary representation of G on the H , and  a fixed, suitable vector in H . Rewriting Equation 5, we obtain: Analogy with a tight frame is clear and it seems natural to call the set of vectors   One should mention furthermore that the theory of frames has been elaborated by Ali et al. (1993Ali et al. ( , 2000) ) and Daubechies et al. (1986) and Daubechies (1990).Moreover, the interested reader can refer to Antoine and Grossmann (1976), Duffin and Schaeffer (1952), Gazeau (2016), andChristensen (2003).In this paper, continuous frames in Hilbert space were studied and applied to any physical space.

MATERIALS AND METHODS
Here, mathematical formulation of frames in H was discussed by taking the famous articles by Ali et al., (1993) and Rahimi et al., (2017).

Mathematical formulation of frames
A frame H is the union of a choice of linearly independent sets of vectors, satisfying a specific completeness-or rather over completeness-condition.Each set of vectors could be labeled by a set of discrete, continuous, or mixed indices.Let  be a locally compact space (which could be partly, or completely discrete) and let  be a regular Borel measure on  with support equal to  .
A set of vectors the integral converging weakly.
To be more explicit, we shall denote the frame so defined by  


. Note that if J X  is a discrete set and  is a counting measure, then the condition of Equation 7 yields the following equation.
It is in this form that the definition of a frame is conventionally given (Duffin and Schaeffer, 1952;Fornasier and Rauhut, 2005) 7), implies the usual frame condition: In other words,

 
Satisfying the frame condition


. The width or snugness of the frame  have the same width and the frame is self-dual iff  are said to be unitarily equivalent frames.In this case, of course, that Equation 18is not the only way to obtain a self-dual tight frame from  


. Indeed, if U is any unitary operator on H , then since we can always write , we see that with: There is a sense in which any two frames


, related by Equation 19, are equivalent and we proceed to study this point a little more closely.If we introduce the positive operator: For each X x , Equation 7 assumes the form: Of course, for each x there is more than one choice of linearly independent vectors  i x for which Equation 21 is satisfied.The arbitrariness is quantified by Ali et al., (1993) and Friedberg et al., (2003).
if there exists an n n

Proof
It is clear from the unitary of Equation 24, then Equation 23 holds.On the other hand, assume that i x  are linearly independent and satisfy Equation 23.Then, for all ,

Example 1: A toy example -Sea star
Consider the Euclidean plane with Dirac notations and the resolution of the identity comes through the sum of their corresponding orthogonal projections, Again the resolution of the identity for Sea Star is as follows: distance on X to be defined: Similarly, any regular N-fold polygon in the plane have satisfied the resolution of unity by the following way: Finally, if we consider the continuous case, then we have,

Example 2: The affine group
In order to obtain a concrete situation where the more general considerations of the results do indeed apply, let us construct a rather unorthodox family of CS for the affine group A G .The connected affine group consists of transformation of  of the following type: We have the multiplication law,  .
reproduces this composition rule.The inverse is given by the matrix x , then trivially the action of the matrix Equation 30 on this vector is given by so that coherent states may be constructed 14 for a suitable choice is easily computed to be a multiplication operator on In order for  to be bounded and invertible with , 1  A we must, therefore, impose on These conditions, together with the fact that Let

H
be an abstract, separable Hilbert space (over the complexes then the frame is called tight.Writing

*
Corresponding author.E-mail: hkdas_math@du.ac.bd.Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Here, S is a positive bounded operator with bounded inverse to be continuous (in the Equations 51 norm).However, the reproducing kernel, level theory of continuous frames in Hilbert space was studied, focusing primarily on the analysis and ending with its applications to possible physical space.The mathematical construction of frames was illustrated by the examples drawn from a toy example Sea Star and the affine group.
Euclidean plane, denotes the unit vector with polar angle a