Continuous frames in Krein spaces

In recently years, frames in Krein spaces had been considered. The paper presents a family of generators for a Krein space by their frames. These generators are dual frames and operator dual frames corresponding to a given frame in a Krein space. We characterize all generalized dual frames of a primary frame. Also, approximately dual frames in a Krein space are introduced and, we study the relation between approximately dual frames and operator duals in a Krein space. Finally, perturbation of frames in this space is considered.

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

In this present paper we introduce weaving Hilbert space frames in the continuous case, we propose new approaches for manufacturing pairs of woven continuous frames, and we obtain new properties in continuous weaving frame theory related to dual frames. Also, we provide some approaches for constructing continuous weaving frames by using small perturbations. These methods not only enhance the understanding of frame theory but also open avenues for practical applications in signal processing and data representation. Future research may further explore the implications of these findings in more complex systems and their potential interdisciplinary benefits.

A $J$-frame is a frame $\mathcal{F}$ for a Krein space $(\mathcal{H}, [\, , \,])$ which is compatible with the indefinite inner product $[\, , \, ]$ in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in $\mathcal{H}$. With every $J$-frame the so-called $J$-frame operator is associated, which is a self-adjoint operator in the Krein space $\mathcal{H}$. The $J$-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of $J$-frame operators in a Krein space by a $2\times 2$ block operator representation. The $J$-frame bounds of $\mathcal{F}$ are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the $2\times 2$ block representation. Moreover, this $2\times 2$ block representation is utilized to obtain enclosures for the spectrum of $J$-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all $J$-frames associated with a given $J$-frame operator.

Continuous framings for Banach spaces

Reproducing kernel almost Pontryagin spaces

A model for a class of cyclic analytic 2-isometries acting on Pontryagin spaces is given, generalizing a Hilbert space version given by S. Richter. Furthermore, an example is constructed showing that, unlike in the Hilbert space case, a cyclic analytic 2-isometry need not have a cyclic vector in the orthogonal complement of its range.

As a starting point we assume to have a continuous frame in a Hilbert space with respect to a measure space. This frame inherits a unitary structure from a unitary representation of a locally compact abelian group in the Hilbert space. In this setting we state a continuous sampling result for the range space of the associated analysis frame operator. The data sampling are functions also defined by using the underlying unitary structure. The result is illustrated by using continuous frames in Paley–Wiener and shift-invariant spaces generated by translates of fixed functions. A sampling strategy working only for discrete abelian groups is also discussed.

Some equalities and inequalities for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>g</mml:mi></mml:math>-Bessel sequences in Hilbert spaces

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

In this paper, we introduce the concept of orthogonal frames in Krein spaces, prove the independence of the choice of the fundamental symmetry, and from this, we obtain a number of interesting properties that they satisfy. We show that there is no distinction between orthogonal frames in a Krein space and orthogonal frames in its associated Hilbert. Furthermore, we characterize frames dual to a given frame, which is a useful tool for constructing examples.

This chapter demonstrates a Krein space based finite horizon \(H_\infty \)-filtering method for discrete-time nonlinear systems and, on this basis, handling \(H_\infty \) fault estimation issues for the INS/GPS integrated system by combing it with proportional and multi-integral (PMI) observer scheme. The core idea lies in converting the targeting problem into a minimum of an indefinite quadratic form such that a relationship between nonlinear filter in Hilbert space and nonlinear estimation in Krein space can be established. Then, by using first-order Taylor approximation and Krein space projection, a sufficient and necessary condition for the minimum is derived and a feasible solution of the nonlinear filter can be obtained by recursively computing Riccati recursions. Especially, for fault estimation purpose, a PMI based nonlinear fault estimator is constructed and designed in the framework of \(H_\infty \) filtering. Successful applications of the results in INS/GPS integrated system are demonstrated through experimental studies.

In this paper, a Krein space approach to finite horizon H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filtering is proposed for a class of affine nonlinear discrete-time systems. It is shown that the problem of H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> nonlinear filtering can be converted into a minimum of an indefinite quadratic form. Hence, a relationship between H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> nonlinear filter in Hilbert space and nonlinear estimation in Krein space is established. By using first-order Taylor approximation and Krein space projection, a sufficient and necessary condition for the minimum is derived. Moreover, a feasible solution of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> nonlinear filter can be obtained by recursively computing Riccati recursions. Finally, a numerical example and one kind of integration filter are used to demonstrate the effectiveness of the proposed method.

Index theory and second quantization of boundary value problems

Definitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces. A sufficient condition for definitizability of a selfadjoint operator A with a nonempty resolvent set ρ(A) in a Krein space (H,[·❘·]) is the finiteness of the number of negative squares of the form [Ax❘y] (see [10, p. 11]).KeywordsHilbert SpaceEssential SpectrumSelfadjoint OperatorKrein SpaceAlgebraic MultiplicityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.