Showing papers on "Shift operator published in 2019"

Consistent systems of linear differential and difference equations

Abstract: We consider systems of linear differential and difference equations ∂Y (x) = A(x)Y (x), σY (x) = B(x)Y (x) with ∂ = d/dx , σ a shift operator σ(x) = x + a, q-dilation operator σ(x) = qx or Mahler operator σ(x) = x p and systems of two linear difference equations σ Y (x) = A(x)Y (x), σ ' Y (x) = B(x)Y (x) with (σ , σ') a sufficiently independent pair of shift operators, pair of q-dilation operators or pair of Mahler operators. Here A(x) and B(x) are n × n matrices with rational function entries. Assuming a consistency hypothesis, we show that such system can be reduced to a system of a very simple form. Using this we characterize functions satisfying two linear scalar differential or difference equations with respect to these operators. We also indicate how these results have consequences both in the theory of automatic sets, leading to a new proof of Cobham's Theorem, and in the Galois theories of linear difference and differential equations, leading to hypertranscendence results.

Inner functions in reproducing kernel spaces

Abstract: In Beurling’s approach to inner functions for the shift operator S on the Hardy space H2, a function f is inner when f ⊥ Snf for all \(n \geqslant 1\). Inspired by this approach, this paper develops a notion of an inner vector x for any operator T on a Hilbert space, via the analogous condition x ⊥ Tnx for all \(n \geqslant 1\). We study these inner vectors in a variety of settings. Using Birkhoff–James orthogonality, we extend this notion of inner vector for an operator on a Banach space. We then apply this development of inner function to recast a theorem of Shapiro and Shields to discuss the zero sets for functions in Hilbert spaces, as well as obtain a corresponding result for zero sets for a wide class of Banach spaces.

Output feedback based event-triggered sliding mode control for delta operator systems

Abstract: In this paper, we present an output feedback based design of event-triggered sliding mode control for delta operator systems. For discrete-time systems, multi-rate output sampling based state estimation technique is very useful if the output information is available. But at high sampling rates, the discrete-time representation of the system using shift operator becomes numerically ill-conditioned and as a result, the observability matrix becomes singular as the sampling period tends to zero. Here, a new formulation of multi-rate state estimation (MRSE) for a small sampling period is presented. We first propose a new observability matrix and then discuss its relationship with the observability matrix defined in the conventional sense. For the delta operator system with matched uncertainty, we have presented the design of MRSE based sliding mode control (SMC). Additionally, to make the control efficient in terms of resource utilization, MRSE based event-triggered SMC is proposed. The absence of Zeno phenomenon is guaranteed as the control input is inherently discrete in nature. Finally, the effectiveness of the proposed method is illustrated through numerical simulations, considering a ball and beam system and a general linear system as a numerical example.

A Beurling Theorem for almost-invariant subspaces of the shift operator

Abstract: A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorems. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.

Some results on quantum Hahn integral inequalities

Abstract: In this paper the quantum Hahn difference operator and the quantum Hahn integral operator are defined via the quantum shift operator $_{\theta }\varPhi _{q}(t)=qt+(1-q)\theta $ , $t\in [a,b]$ , $\theta = \omega /(1-q)+a$ , $0< q<1$ , $\omega \ge 0$ . Some new fractional integral inequalities are established by using the quantum Hahn integral for one and two functions bounded by quantum integrable functions. The Hermite–Hadamard type of ordinary and fractional quantum Hahn integral inequalities as well as the Polya–Szego type fractional Hahn integral inequalities and the Gruss–Cebysev type fractional Hahn integral inequality are also presented.

Unitary Shift Operators on a Graph.

Abstract: A unitary shift operator (GSO) for signals on a graph is introduced, which exhibits the desired property of energy preservation over both backward and forward graph shifts. For rigour, the graph differential operator is also derived in an analytical form. The commutativity relation of the shift operator with the Fourier transform is next explored in conjunction with the proposed GSO to introduce a graph discrete Fourier transform (GDFT) which, unlike existing approaches, ensures the orthogonality of GDFT bases and admits a natural frequency-domain interpretation. The proposed GDFT is shown to allow for a coherent definition of the graph discrete Hilbert transform (GDHT) and the graph analytic signal. The advantages of the proposed GSO are demonstrated through illustrative examples.

Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets

Abstract: Abstract Taking two and three shared set problems into background, the uniqueness problem of a meromorphic function together with its shift operator have been studied. Our results will improve a number of recent results in the literature. Some examples have been provided in the last section to show that certain conditions used in the paper, is the best possible.

Generalized shift operator of certain encodings of real numbers.

Abstract: The present article is devoted to the investigation of some properties of the generalized shift operator of numbers represented in terms of numeral systems with a variable alphabet.

Strict Versions of Integrable Hierarchies in Pseudodifference Operators and the Related Cauchy Problems

Abstract: In the algebra PsΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator Λ0. The first deformation is by the group in PsΔ corresponding to the Lie subalgebra Ps<0 of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra PsΔ≤0 of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on Λ0 that respectively complements the Lie subalgebra PsΔ<0 or PsΔ≤0. This yields two integrable hierarchies associated with Λ0, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For Λ0 equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.

Norms of Truncated Toeplitz Operators and Numerical Radii of Restricted Shifts

Abstract: This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved.

One-dimensional Hadamard Quantum Walk on a Cycle with Rotational Implementation

Abstract: Quantum walks have been extensively studied recently, mainly due to their vast difference in behavior to classical random walks. This paper is concerned with discrete time and space quantum walks of particles that propagate through a one-dimensional line. This line can be either a lattice or a graph or any other form of mathematical structure that can be viewed as a one-dimensional line. First is defined a concrete way to describe the unitary evolution of a quantum walk through a balanced coin operator and a shift operator. Then follows the implementation of the quantum walk on an $8$-cycle, i.e a cycle graph with $8$ nodes, which is then run locally as a simulation and on IBM's quantum computer. The paper explores two implementations of the quantum walk as a quantum circuit: the first one consists of generalised controlled inversions, as introduced in \cite{EffWalk}, whereas the second one tries to replace them with rotation operators around the basis states. The main aim is to find a way around the caveat resulting from the large amount of ancilla qubits required to carry out the computation in the case where only generalised inverters are used. Next, another three experiments are computed, involving cycles with a larger state space, more specifically $16$, $32$ and $64$ possible positions. In order to measure the magnitude of the error of the circuit we use the cross entropy benchmarking method, calculated through the Hellinger distance. Finally, a derivation of the variance of the quantum walk is provided along with a calculation of the variance for our experiment.

Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting

Abstract: It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ which is invariant under the backward shift operator can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the inner Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. The main focus of this paper is to present the extension of this structure to a more general setting. The Hardy space is replaced by the full weighted Bergman-Fock space of formal power series in $d$ freely noncommutative indeterminates, where the shift is replaced by the right shift tuple, where the conservative/dissipative discrete-time linear system becomes a certain type of conservative/dissipative multidimensional linear system with time-varying weights and with evolution along a rooted tree with each node having $d$ forward branches, where a backward shift-invariant subspace ${\mathcal N}$ is the range of the observability operator for such a weighted-Bergman multidimensional linear system, and where the transfer function of this system is the Beurling-Lax representer for the forward shift-invariant subspace ${\mathcal M} = {\mathcal N}^{[\perp]}$, and where this transfer function also serves as the characteristic function for the operator tuple having hypercontractive-operator-tuple adjoint equal to the restriction of the backward-shift tuple to $\mathcal N$.

Algebras of frequently hypercyclic vectors

Abstract: We show that the multiples of the backward shift operator on the spaces $\ell_{p}$, $1\leq p<\infty$, or $c_{0}$, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of $\mathcal{A}$-hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator $T$ on any Banach space, $FHC(T)$ is algebrable for a suitable product, and in some cases it is even strongly algebrable.

On Invariant Subspaces for the Shift Operator

Abstract: In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 ≤ p < ∞ ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( ∂ D ) is reducing under the shift operator M e i θ on L 2 ( ∂ D ) if and only if it is hyperinvariant under M e i θ . At the same time, we show that there are two large classes of invariant subspaces for M e i θ that are not hyperinvariant subspaces for M e i θ and are also not reducing subspaces for M e i θ . Moreover, we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.

Hypertranscendence and linear difference equations

Abstract: After Holder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator $x\mapsto x+h$ ($h\in\mathbb{C}^*$), the $q$-difference operator $x\mapsto qx$ ($q\in\mathbb{C}^*$ not a root of unity), and the Mahler operator $x\mapsto x^p$ ($p\geq 2$ integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable $x$ with complex coefficients (or in the variable $1/x$ in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

Abstract: We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients urn:x wiley:mma:media:mma5716:mma5716-math-0001 verifying urn:x-wiley:mma:media:mma5716:mma5716-math-0002, ie, associated with measures verifying Szego condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D=SR+K2 with K2, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that urn:x-wiley:mma:media:mma5716:mma5716-math-0003 with K2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if urn:x-wiley:mma:media:mma5716:mma5716-math-0004, then D=SR+Kp with Kp an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle.

An analytic model for left invertible weighted translation semigroups

Abstract: M. Embry and A. Lambert initiated the study of a semigroup of operators $\{S_t\}$ indexed by a non-negative real number $t$ and termed it as weighted translation semigroup. The operators $S_t$ are defined on $L^2(\mathbb R_+)$ by using a weight function. The operator $S_t$ can be thought of as a continuous analogue of a weighted shift operator. In this paper, we show that every left invertible operator $S_t$ can be modeled as a multiplication by $z$ on a reproducing kernel Hilbert space $\cal H$ of vector-valued analytic functions on a certain disc centered at the origin and the reproducing kernel associated with $\cal H$ is a diagonal operator. As it turns out that every hyperexpansive weighted translation semigroup is left invertile, the model applies to these semigroups. We also describe the spectral picture for the left invertible weighted translation semigroup. In the process, we point out the similarities and differences between a weighted shift operator and an operator $S_t.$

A Sharp Jackson Inequality in Lp(ℝd) with Dunkl Weight

Abstract: A sharp Jackson inequality in the space Lp(ℝd), 1 ≤ p < 2, with Dunkl weight is proved. The best approximation is realized by entire functions of exponential spherical type. The modulus of continuity is defined by means of a generalized shift operator bounded on Lp, which was constructed earlier by the authors. In the case of the unit weight, this operator coincides with the mean-value operator on the sphere.

Some questions of harmonic analysis in weighted Morrey type spaces

Abstract: Weighted Morrey-type classes of functions that are harmonic in the unit disk and in the upper half plane are defined in this work. Under some conditions on the weight function, we study some properties of functions belonging to these classes. Estimation of the maximum values of harmonic functions for a nontangential angle through the Hardy–Littlewood maximal function are generalized to more general case, and then the boundedness of Hardy–Littlewood operator is applied in the Morrey-type spaces. Weighted Morrey–Lebesgue type space is defined, where the shift operator is continuous with respect to shift, and its invariance with regard to the singular operator is proved. The validity of Minkowski inequality in Morrey–Lebesgue type spaces is also proved. An approximation properties of the Poisson kernel are studied.

Beurling Type Theorem on the Hilbert Space Generated by a Positive Sequence

Abstract: Let H2(γ) be the Hilbert space over the bidisk $$\mathbb{D}^2$$ generated by a positive sequence γ = {γnm}n,m≥0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H2(γ) with γ = {γnm}n,m≥0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk $$\mathbb{D}^2$$ .

On the ellipticity of operators associated with Morse-Smale diffeomorphisms

Abstract: We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm property of a given operator depends on the Sobolev smoothness exponent $s$.

$N$-hypercontractivity and similarity of Cowen-Douglas operators

Abstract: When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0} ^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an $n$-hypercontraction, we prove that the weights must satisfy the inequality $$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ As an application of this result, it is shown that such an operator cannot be subnormal. We also give an example to illustrate the important role that the $n$-hypercontractivity assumption plays in determining the similarity of Cowen-Douglas operators in terms of the curvatures of their eigenvector bundles.

Kernels of Toeplitz Operators and Rational Interpolation

Abstract: The kernel of a Toeplitz operator on the Hardy class H2 in the unit disk is a nearly invariantsubspace of the backward shift operator, and, by D. Hitt’s result, it has the form g · Kω where ω is an inner function, Kω = H2 ⊝ ωH2, and g is an isometric multiplier on Kω. We describe the functions ω and g for the kernel of the Toeplitz operator with symbol .$$ \overline{\theta}\varDelta $$ where θ is an inner function and Δ is a finite Blaschke product.

Chaotic backward shift operator on Chebyshev polynomials

Abstract: We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

On the monotone complexity of the shift operator.

Abstract: We show that the complexity of minimal monotone circuits implementing a monotone version of the permutation operator on $n$ boolean vectors of length $q$ is $\Theta(qn\log n)$. In particular, we obtain an alternative way to prove the known complexity bound $\Theta(n\log n)$ for the monotone shift operator on $n$ boolean inputs.