This paper deals with an alternative proof of Beurling-Lax theorem by adopting a constructive approach instead of the isomorphism technique which was used in the original proof.
It is known that every C·_(0)-contraction has a dilation to a Hardy shift.This leads to an elegant analytic functional model for C·_(0)-contractions,and has motivated lots of further works on the model theor...It is known that every C·_(0)-contraction has a dilation to a Hardy shift.This leads to an elegant analytic functional model for C·_(0)-contractions,and has motivated lots of further works on the model theory and generalizations to commuting tuples of C·_(0)-contractions.In this paper,we focus on doubly commuting sequences of C·_(0)-contractions,and establish the dilation theory and the analytic model theory for these sequences of operators.These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivariable operator theory to the operator theory in the infinite-variable setting.展开更多
Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and ...Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in approximation.During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations.The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies.The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate functions.This article mainly serves as a survey.It also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f.Such function F has the form F=f+iHf,where H stands for the Hilbert transformation of the context.We develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,a.e.The original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called mono-components.If f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of f.The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion.Mono-components are crucial to understand the concept instantaneous frequency.We will present several most important mono-component function classes.Decompositions of signals into mono-components are called adaptive Fourier decompositions(AFDs).Wc note that some scopes of the studies on the ID mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds.We finally provide an account of related studies in pure and applied mathematics.展开更多
基金supported by the Multi-Year Research Grant(No.MYRG115(Y1-L4)-FST13-QT)the Multi-Year Research Grant(No.MYRG116(Y1-L3)-FST13-QT)+1 种基金Macao Government(No.FDCT098/2012/A3)the Natural Science Foundation of Guangdong Province(No.S2011010004986)
文摘This paper deals with an alternative proof of Beurling-Lax theorem by adopting a constructive approach instead of the isomorphism technique which was used in the original proof.
基金supported by National Natural Science Foundation of China(Grant Nos.11871157 and 12101428)。
文摘It is known that every C·_(0)-contraction has a dilation to a Hardy shift.This leads to an elegant analytic functional model for C·_(0)-contractions,and has motivated lots of further works on the model theory and generalizations to commuting tuples of C·_(0)-contractions.In this paper,we focus on doubly commuting sequences of C·_(0)-contractions,and establish the dilation theory and the analytic model theory for these sequences of operators.These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivariable operator theory to the operator theory in the infinite-variable setting.
基金Macao University Multi-Year Research Grant(MYRG)MYRG2016-00053-FSTMacao Government Science and Technology Foundation FDCT 0123/2018/A3.
文摘Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in approximation.During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations.The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies.The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate functions.This article mainly serves as a survey.It also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f.Such function F has the form F=f+iHf,where H stands for the Hilbert transformation of the context.We develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,a.e.The original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called mono-components.If f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of f.The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion.Mono-components are crucial to understand the concept instantaneous frequency.We will present several most important mono-component function classes.Decompositions of signals into mono-components are called adaptive Fourier decompositions(AFDs).Wc note that some scopes of the studies on the ID mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds.We finally provide an account of related studies in pure and applied mathematics.
