Jitendriya Swain, Assistant Professor, Department of Mathematics, Indian Institute of Technology Guwahati for the award of the degree of Doctor of Philosophy and this work has not been submitted elsewhere for a degree. I would like to thank all the staff of the Mathematics and Administration Department for their necessary assistance when needed.
Frames in Hilbert Spaces
But unlike orthonormal bases, the coefficients in the frame expansions (1.2) are generally not unique. Whether the coefficients in the frame expansion (1.2) are uniquely determined under an additional assumption on the frame.
Gabor Frames
Necessary and Sufficient Conditions
The algebraic lattice structure Λ ={(αk, βn) : k, n∈Zd} was used to derive the necessary condition that the Gabor system G(g, α, β) is a complete, frame, or exact frame in the sense of the product αβ. For the network Λ ⊂ R2d, they proved the incompleteness of Gabor systems that are uniformly discrete (i.e. there is a minimum distance δ between the elements of Λ) in the sense of Beurling.
Representations of Gabor Frame Operators
Before examining the Gabor frame operator in detail, we define synthesis and analysis operators. The Gabor frame operator Sg,γ can be written using the correlation functions Gn of the pair (g, γ), where.
Wiener Amalgam Spaces
The dual and K¨othe dual of amalgam spaces are given in the following lemma. For a detailed study of classic amalgam spaces, we refer to the article by Fournier and Stewart [70].
The Schwartz Space and the Fourier Transform
Modulation Spaces
The Feichtinger algebra has many interesting properties: it forms a Banach algebra with respect to dot products and convolution, and is invariant with respect to the Fourier transform. Then the expansions of the Gabor framework given in (1.13) converge not only in L2 but in all modulation spaces as follows (see [77] for proofs).
Wilson Bases
Moreover, a Wilson basis will be an unconditional basis not only for L2(R) but also for all the modulation spaces if the original window g has sufficient collection concentration in the time-frequency plane. Tensor products of Wilson bases are also unconditional bases for the higher dimensional modulation spaces.
Literature Review and Overview of Main Results
- Gabor Frame Operators
- Generalization of Frames
- Identities and Inequalities for Frames
- Feichtinger’s Problem
- The Balian-Low Theorem
In Chapter 3, we investigate the above theorems when the window function is in the vector-valued Wiener space. Calculations of the inverse frame operator for HS frames in infinite dimensional Hilbert space are also very difficult.
Preliminaries
In this chapter, we show that Gabor expansions converge to the identity operator in the operator norm as well as in the weak* sense on amalgam spaces as the sampling density tends to infinity. We also obtain an analogue of the Janssen representation and the Wexler-Raz biorthogonality condition for Gabor framework operators on amalgam spaces.
Gabor Frame Operators on Amalgam Spaces
To prove the strong and weak* convergence, we make use of the following two lemmas. Next, we state and prove the convergence of the Gabor frame operator onW(Lp, Lq).
The Structure of Gabor Systems
As in the case of Gabor frames for L2(Rd), assuming that G(g, α, β) is a frame for L2(Rd,H), we list the basic properties of H-valued Gabor frames for L2(Rd, H) in the following sentence. The analysis operator Cg,hf = (hf, MβnTαkgih)k,n. ii) The synthesis operator Rg,hd=P. The series defining Rg,hd converges unconditionally in L2 for every ∈ℓ2. iii) Rg,h =Cg,h∗, and the frame operator Sg =Rg,hCg,h:L2(Rd,H) →L2(Rd,H) is strictly positive, the invertible operator satisfies.
Weight Functions
Then the following statements are equivalent. ii) Lpv(Rd,H) is translation-invariant (ie, for every x ∈ Rd, Tx is a continuous map of Lpv(Rd,H) onto itself). iii). Throughout this chapter, a submultiplicative weight function is denoted by w and a moderated w-function is denoted by v.
Weighted Amalgam Spaces
The K¨othe dual ofWH(Lp, Lqv) is the space of all measurable functionsg:Rd→ Hsodatg·WH(Lp, Lqv)⊆L1(Rd). It is equal to WH(Lp. For detailed study on K¨othe) dual for scalar valued amalgam spaces we refer to the text of Bennett and Sharpley [21].
H -Valued Gabor Expansions in Weighted Amalgam Spaces
The synthesis operator is defined by Rg,hd(x) = From the above observations we obtain the analogy of the Walnut representation for the H-value Gabor frames in the following statement. Now in the next note we will present some important results on H-valued Gabor frames.
The Algebra of H-Valued L ∞ -Weighted Shifts
H-Valued L ∞ -Weighted Shifts
The identification of Aw with the subclass of bounded operators on L2(Rd,H) is as follows: Given (mx)x. Note that the identification of families in Aw and operators on B(Lp(Rd,H)) is one-to-one. The continuity properties of operators defined by the family inAw are established by the following theorem.
Spectral invariance
Within the class of ρ-almost periodic operators, consider APwp(ρ), the subclass of the operators for which the Fourier series in (3.28) can be summed, where w is an allowed weight. More precisely, a ρ-almost periodic operator M belongs to APwp(ρ) if its Fourier coefficients with respect to ρ satisfy. For p ∈ [1,∞), the class Aw ⊂ B(Lp(Rd,H)) coincides with APwp(ρ), the class of ρ-almost periodic elements that have w-summable Fourier coefficients.
Invertibility of the H-Valued Gabor Frame Operator
We present some characterizations of HS frames and show that HS frames share many important properties with frames. We then show that the inverse of the HS frame operator can be approximated using finite-dimensional methods. We also present a classical perturbation result and prove that HS frames are stable under small perturbations.
Preliminaries
Let L(H) be the C∗-algebra of all bounded linear operators on a complex separable Hilbert space H. For 1 ≤ p < ∞, the von Neumann−Estimate p-class Cp is defined as the set of all compact drives T for which. We recall that C2 is a Banach space with respect to tok.k2, and also a Hilbert space with the inner product defined by.
Characterization of HS-Frames
A sequence {Gj :j ∈J} ⊆ L(H, C2) is an HS-Bessel sequence for Hmet bound B if and only if the synthesis operator T is a well-defined bounded operator with kTk. A row{Gj :j∈J} ⊆ L(H, C2) is an HS frame for Hif if and only if the synthesis operator T is a well-defined, bounded and surjective operator. A sequence {Gj :j∈J} ⊆ L(H, C2) is a HS-Riesz basis for Hwith boundaries A and B if and only if the synthesis operator T is a linear homeomorphism such that.
Approximation of the Inverse HS-Frame Operator
The result stated below can be found in ([33], Theorem 4.5) for Hilbert space frames. We are now in a position to prove the generalization of Theorem 4 in [26] by setting Hilbert space frames to HS frames. C2 is the orthogonal sum of the range of T∗ and the kernel of T, we can write any {Aj}j∈J ∈ L.
Stability of HS-Frames
A forg-frames counterexample can be found in [128], and an example can be similarly constructed for HS-frames. Motivated by these results, we next discuss some interesting results of concerns about HS frameworks. Since g-frames can be considered as a class of HS-frames, the previous results on g-frames can be taken as a special case of the results we established for HS-frames.
Identities and Inequalities for HS-Frames
The following inequality for a Parseval HS frame, which appears in Corollary 4.5.3, is a simple consequence of Theorem 1.7.2. Note that operators SK and SKc are self-adjoint and therefore SK∗ = SK, SK∗c = SKc. Next, we give a simplified presentation of Theorem 1.7.6 for HS frames, which generalizes Theorem 4.5.7 to a more general form that does not involve the real parts of the complex numbers.
Preliminaries
Integral operators
Also ifk lies in the two-dimensional version of the Feichtinger algebra, i.e., k∈M1(R2), then T is an operator of trace class. This result was proved in [92] and is a special case of the more general theorems proved in [80]. Since every operatorwm⊗wn belongs to I1 and the scalars hk, Wmni are summable, the series (5.8) converges absolutely to I1, and thereforeT ∈ I1.
Type A and Type B Operators
The Feichtinger Problem
We define an infinite block-diagonal matrix T by T = T1⊕T2⊕..⊕Tn⊕..Then T is a positive semi-definite trace class operator of Type B but not of Type A with respect to the orthonormal basis { wℓ}. Then T is a positive semi-definite trace class operator of Type B but not of Type A with respect to the orthonormal basis {wℓ}ℓ≥1. In this chapter we prove the Balian-Low type theorem (BLT) on L2(C) using the operators Z and ¯Z, that is, we prove that kZgk2 and kZg¯ k2 cannot both be finite at the same time if the rotated Gabor frame is not generated. by g ∈ L2(C) forms an orthonormal basis or an exact frame for L2(C).
Preliminaries
- Heisenberg Group and the Weyl Transform
- Hermite Operators
- Twisted Gabor Frames
- Twisted Zak Transform
The inversion formula for Weyl transform isf(z) =τ(π(z)∗W(f)),whereπ(z)∗ is the adjoint ofπ(z) andτ is the usual trace onB. To define this operator L, we must define the operators Z and ¯Z as follows: Z= dzd +12z¯and ¯Z = d¯dz−12z. The functions φm,n are eigenfunctions of the special Hermite operator. Since we are interested in obtaining BLT for convoluted Gabor frames, we define the convoluted Zak transform with a slight modification as follows.
The Amalgam BLT
Assume that Ztf(z, w)6= 0 for all (z, w)∈C2. Since Ztf is continuous on a simply connected domain C2, there is a continuous function ϕ(z, w) such that. Now we investigate the relations between the operators Z,Z¯ and the continuity of the rotated Zak transformation. Given that g is continuous and therefore the fundamental theorem of calculus for complex variables and ML inequality can be applied.
Non-Distributional Calculations and the BLT
We use the following notation to estimate the upper bound for the fluctuation of the convoluted Zak transform on small cubes. Using the Cauchy–Schwartz inequality on the left-hand side of (6.11) and (6.12), the proof immediately follows where . Then the inequality (6.15) can be obtained by (6.14) and using the Cauchy-Schwartz inequality in the last term of the above calculation.
Uncertainty Principle Approach to BLT
The Weak BLT
If we consider the special Hermite operatorL onC, then the inequality onL2(C) has the form k|z|fk2 kL12fk2≥. In [111] the generalized Fourier transform on L2(R) is defined in relation to the differential-difference operator. One can examine Balian-Lae type theorems (as discussed in Chapter 6) with respect to the generalized Fourier transform.
The Differential-Difference Operator
In this chapter we describe some issues related to the results discussed in this thesis.
The Generalized Fourier Transform
Balan, Density and redundancy of incoherent Weyl-Heisenberg superframes, in Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, TX, 1999), Contemp. Christensen, Frames containing the Riesz basis and approximation of frame coefficients using finite-dimensional methods, J. Feichtinger, Amalgam spaces and generalized harmonic analysis, in Proceedings of the Norbert Wiener Centenary Congress (East Lansing, MI, 1994), Proc .
