arXiv subjects

Liming Yang

Publications and source records attributed to Liming Yang.

34 records · Page 2Linked to original sources

arXiv2023

Invertibility in Weak-Star Closed Algebras of Analytic Functions

For $K\subset \mathbb C$ a compact subset and $μ$ a positive finite Bore1 measure supported on $K,$ let $R^\infty (K,μ)$ be the weak-star closure in $L^\infty (μ)$ of rational functions with poles off $K.$ We show that if $R^\infty (K,μ)$ has no non-trivial $L^\infty$ summands and $f\in R^\infty (K,μ),$ then $f$ is invertible in $R^\infty (K,μ)$ if and only if Chaumat's map for $K$ and $μ$ applied to $f$ is bounded away from zero on the envelope with respect to $K$ and $μ.$ The result proves the conjecture $\diamond$ posed by J. Dudziak in 1984.

math.FA↗

arXiv2023

A Note on Spectral Mapping Theorems for Subnormal Operators

For a compact subset $K\subset \mathbb C$ and a positive finite Borel measure $μ$ supported on $K,$ let $\text{Rat}(K)$ denote the space of rational functions with poles off $K,$ let $R^\infty (K,μ)$ be the weak-star closure of $\text{Rat}(K)$ in $L^\infty (μ),$ and let $R^2 (K,μ)$ be the closure of $\text{Rat}(K)$ in $L^2(μ).$ We show that there exists a compact subset $K\subset \mathbb C,$ a positive finite Borel measure $μ$ supported on $K,$ and a function $f\in R^\infty (K,μ)$ such that $R^\infty (K,μ)$ has no non-trivial direct $L^\infty$ summands, $f$ is invertible in $R^2 (K,μ)\cap L^\infty(μ),$ and $f$ is not invertible in $R^\infty (K,μ).$ The result answers an open question concerning spectral mapping theorems for subnormal operators raised by J. Dudziak in 1984.

math.FA↗

arXiv2023

The Commutant of Multiplication by z on the Closure of Rational Functions in $L^t(μ)$

For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $μ$ on $K,$ and $1 \le t < ı,$ let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, μ)$ be the closure of $\text{Rat}(K)$ in $L^t(μ).$ For a bounded Borel subset $\mathcal D\subset \mathbb C,$ let $\area_{\mathcal D}$ denote the area (Lebesgue) measure restricted to $\mathcal D$ and let $H^ı(\mathcal D)$ be the weak-star closed sub-algebra of $L^ı(\area_{\mathcal D})$ spanned by $f,$ bounded and analytic on $\mathbb C\setminus E_f$ for some compact subset $E_f \subset \mathbb C\setminus \mathcal D.$ We show that if $R^t(K, μ)$ contains no non-trivial direct $L^t$ summands, then there exists a Borel subset $\mathcal R \subset K$ whose closure contains the support of $μ$ and there exists an isometric isomorphism and a weak-star homeomorphism $ρ$ from $R^t(K, μ) \cap L^\infty(μ)$ onto $H^\infty(\mathcal R)$ such that $ρ(r) = r$ for all $r\in\text{Rat}(K).$ Consequently, we obtain some structural decomposition theorems for $\rtkmu$.

math.FA↗

arXiv2023

Mean Rational Approximation for Compact Subsets with Thin Boundaries

In 1991, J. Thomson obtained a celebrated decomposition theorem for $P^t(μ),$ the closed subspace of $L^t(μ)$ spanned by the analytic polynomials, when $1 \le t < ı.$ In 2008, J. Brennan \cite{b08} generalized Thomson's theorem to $R^t(K, μ),$ the closed subspace of $L^t(μ)$ spanned by the rational functions with poles off a compact subset $K$ containing the support of $μ,$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. We extend the above decomposition theorems for $R^t(K, μ)$ when the boundary of $K$ is not too wild.

math.FA↗

arXiv2022

Mean Rational Approximation for Some Compact Planar Subsets

In 1991, J. Thomson obtained celebrated structural results for $P^t(μ).$ Later, J. Brennan (2008) generalized Thomson's theorem to $R^t(K,μ)$ when the diameters of the components of $\mathbb C\setminus K$ are bounded below. The results indicate that if $R^t(K,μ)$ is pure, then $R^t(K,μ) \cap L^\infty (μ)$ is the "same as" the algebra of bounded analytic functions on $\mbox{abpe}(R^t(K, μ)),$ the set of analytic bounded point evaluations. We show that if the diameters of the components of $\mathbb C\setminus K$ are allowed to tend to zero, then even though $\text{int}(K) = \mbox{abpe}(R^t(K, μ))$ and $K =\overline {\text{int}(K)},$ the algebra $R^t(K,μ) \cap L^\infty (μ)$ may "be equal to" a proper sub-algebra of bounded analytic functions on $\text{int}(K),$ where functions in the sub-algebra are "continuous" on certain portions of the inner boundary of $K.$

math.FA↗

arXiv2020

Approximation in the mean by rational functions

For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite positive measure $μ$ supported on $K$, $R^t(K, μ)$ denotes the closure in $L^t(μ)$ of rational functions with poles off $K$. Let $\text{abpe}(R^t(K, μ))$ denote the set of analytic bounded point evaluations. The objective of this paper is to describe the structure of $R^t(K, μ)$. In the work of Thomson on describing the closure in $L^t(μ)$ of analytic polynomials, $P^t(μ)$, the existence of analytic bounded point evaluations plays critical roles, while $\text{abpe}(R^t(K, μ))$ may be empty. We introduce the concept of non-removable boundary $\mathcal F$ such that the removable set $\mathcal R = K\setminus \mathcal F$ contains $\text{abpe}(R^t(K, μ))$. Recent remarkable developments in analytic capacity and Cauchy transform provide us the necessary tools to describe $\mathcal F$ and obtain structural results for $R^t(K, μ)$. Assume that $R^t(K, μ)$ does not have a direct $L^t$ summand. Let $H^\infty_{\mathcal R}(\mathcal L^2_{\mathcal R})$ be the weak$^*$ closure in $L^\infty (\mathcal L^2_{\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\mathcal F$, where $\mathcal L^2_{\mathcal R}$ denotes the planar Lebesgue measure restricted to $\mathcal R$. We prove that the identity map ($r\rightarrow r$, $r$ is a rational function with poles off $K$) extends an isometric isomorphism and a weak$^*$ homeomorphism from $R^t(K, μ)\cap L^\infty(μ)$ onto $H^\infty_{\mathcal R}(\mathcal L^2_{\mathcal R })$. Consequently, we show that a decomposition theorem (Main Theorem II) of $R^t(K, μ)$ holds for an arbitrary compact subset $K$ and a finite positive measure $μ$ supported on $K$, which extends the central results regarding $P^t(μ)$.

math.FA↗

arXiv2020

Kernel based regression with robust loss function via iteratively reweighted least squares

Least squares kernel based methods have been widely used in regression problems due to the simple implementation and good generalization performance. Among them, least squares support vector regression (LS-SVR) and extreme learning machine (ELM) are popular techniques. However, the noise sensitivity is a major bottleneck. To address this issue, a generalized loss function, called $\ell_s$-loss, is proposed in this paper. With the support of novel loss function, two kernel based regressors are constructed by replacing the $\ell_2$-loss in LS-SVR and ELM with the proposed $\ell_s$-loss for better noise robustness. Important properties of $\ell_s$-loss, including robustness, asymmetry and asymptotic approximation behaviors, are verified theoretically. Moreover, iteratively reweighted least squares (IRLS) is utilized to optimize and interpret the proposed methods from a weighted viewpoint. The convergence of the proposal are proved, and detailed analyses of robustness are given. Experiments on both artificial and benchmark datasets confirm the validity of the proposed methods.

cs.LG↗

arXiv2019

Approximation in the mean by rational functions II

For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite positive measure $μ$ supported on $K$, $R^t(K, μ)$ denotes the closure in $L^t(μ)$ of rational functions with poles off $K$. Conway and Yang (2019) introduced the concept of non-removable boundary $\mathcal F$ and removable set $\mathcal R = K\setminus \mathcal F$ for $R^t(K, μ)$. We continue the previous work and obtain structural results for $R^t(K, μ)$. Assume that $S_μ$, the multiplication by $z$ on $R^t(K, μ)$, is pure ($R^t(K, μ)$ does not have $L^t$ summand). Let $H^\infty_{\mathcal R}(A_{\mathcal R})$ be the weak$^*$ closure in $L^\infty (A_{\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\mathcal F$, where $A_{\mathcal R}$ denotes the area measure restricted to $\mathcal R$. $\mathcal R$ is $γ$-connected ($γ$ denotes analytic capacity) if for any two disjoint open set $G_1$ and $G_2$ with $\mathcal R \subset G_1 \cup G_2 ~γ-a.a.$, then $\mathcal R \subset G_1 ~γ-a.a.$ or $\mathcal R \subset G_2 ~γ-a.a.$. We prove: (1) $R^t(K, μ)$ contains no non-trivial characterization functions if and only if the removable set $\mathcal R$ is $γ$-connected. (2) There is an isometric isomorphism and a weak$^*$ homeomorphism from $R^t(K, μ)\cap L^\infty(μ)$ onto $H^\infty_{\mathcal R}(A_{\mathcal R })$.

math.FA↗

arXiv2019

Reproducing kernel of the space $R^t(K,μ)$

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $μ$ supported on $K,$ $R^t(K, μ)$ denotes the closure in $L^t (μ)$ of rational functions with poles off $K$. Let $Ω$ be a connected component of the set of analytic bounded point evaluations for $R^t(K, μ)$. In this paper, we examine the behavior of the reproducing kernel of $R^t(K, μ)$ near the boundary $\partial Ω\cap \mathbb T$, assuming that $μ(\partial Ω\cap \mathbb T ) > 0$, where $\mathbb T$ is the unit circle.

math.FA↗

arXiv2018

On Nontangential Limits and Shift Invariant Subspaces

In 1998, John B. Conway and Liming Yang wrote a paper in which they posed a number of open questions regarding the shift on $P^t(μ)$ spaces. A few of these have been completely resolved, while at least one remains wide open. In this paper, we review some of the solutions, mention some alternate approaches and discuss further the problem that remains unsolved.

math.FA↗

arXiv2018

Spectral Picture For Rationally Multicyclic Subnormal Operators

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal H,$ J. B. Conway and N. Elias (Analytic bounded point evaluations for spaces of rational functions, J. Functional Analysis, 117:1{24, 1993) showed that $clos(σ(S) \setminus σ_e (S)) = clos(Int (σ(S))).$ This paper examines the property for rationally multicyclic (N-cyclic) subnormal operators. We show: (1) There exists a 2-cyclic irreducible subnormal operator $S$ with $clos(σ(S) \setminus σ_e (S)) \neq clos(Int (σ(S))).$ (2) For a pure rationally $N-$cyclic subnormal operator $S$ on $\mathcal H$ with the minimal normal extension $M$ on $\mathcal K \supset \mathcal H,$ let $\mathcal K_m = clos (span\{(M^*)^kx: ~x\in\mathcal H,~0\le k \le m\}.$ Suppose $M |_{\mathcal K_{N-1}}$ is pure, then $clos(σ(S) \setminus σ_e (S)) = clos(Int (σ(S))).$

math.FA↗

arXiv2018

Aleman-Richter-Sundberg's Theorem On $P^t(μ)$-Spaces

Let $ν$ be a finite complex measure with support in $\bar {\mathbb D}$ and let $\mathcal Cν$ denote the Cauchy transform of $ν.$ Suppose that $ν$ annihilates polynomials in complex variable $z$ and $ν|_{\partial \mathbb D} = hm,$ where $m$ is the normalized Lebesgue measure on $\partial {\mathbb D}$. We show that, for $ε_0 > 0,$ $m$-almost all $e^{iθ}\in \partial {\mathbb D},$ and $a > 0,$ when $r$ tends to 1, there exists $E_r \subset B(re^{iθ}, \frac{1-r}{4})$ with analytic capacity $γ(E_r) < ε_0 \frac{1-r}{4}$ such that $|\mathcal Cν(λ) - e^{-iθ}h(e^{iθ}) | \le a$ area-almost all $λ\in B (re^{iθ}, \frac{1-r}{4} ) \setminus E_r .$ Using this result, we provide an alternative proof of Aleman-Richter-Sundberg's Theorem on nontangential limits in $P^t(μ)$-Spaces and the index of invariant subspaces.

math.FA↗

arXiv2017

Boundary values in $R^t(K,μ)$-spaces and invariant subspaces

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $μ$ supported on $K,$ $R^t(K, μ)$ denotes the closure in $L^t (μ)$ of rational functions with poles off $K.$ The paper examines the boundary values of functions in $R^t(K, μ)$ for certain compact subset $K$ and extends the work of Aleman, Richter, and Sundberg on nontangential limits for the closure in $L^t (μ)$ of analytic polynomials (Theorem A and Theorem C in \cite{ars}). We show that the Cauchy transform of an annihilating measure has some continuity properties in the sense of capacitary density. This allows us to extend Aleman, Richter, and Sundberg's results for $R^t(K, μ)$ and provide alternative short proofs of their theorems as special cases.

math.FA↗

arXiv2017

Bounded Point Evaluations For Certain Polynomial And Rational Modules

Let $K$ be a compact subset of the complex plane $\mathbb C.$ Let $P(K)$ and $R(K)$ be the closures in $C(K)$ of analytic polynomials and rational functions with poles off $K,$ respectively. Let $A(K) \subset C(K)$ be the algebra of functions that are analytic in the interior of $K$. For $1\le t <\infty,$ let $P^t(1, ϕ_1,...,ϕ_N,K)$ be the closure of $P(K)+P(K)ϕ_1+...+P(K)ϕ_N$ in $L^t(dA|_K),$ where $dA|_K$ is the area measure restricted to $K$ and $ϕ_1,...,ϕ_N\in L^t(dA|_K).$ Let $HP(ϕ_1,...,ϕ_N,K)$ be the closure of $P(K)ϕ_1+...+P(K)ϕ_N +R(K)$ in $C(K),$ where $ϕ_1,...,ϕ_N\in C(K).$ In this paper, we prove if $R(K)\ne C(K),$ then there exists an analytic bounded point evaluation for both $P^t(1, ϕ_1,...,ϕ_N,K)$ and $HP(ϕ_1,...,ϕ_N,K)$ for certain smooth functions $ϕ_1,...,ϕ_N,$ in particular, for $\bar z,\bar z^2,...,\bar z^N.$ We show that $A(K)\subset HP(\bar z,\bar z^2,...,\bar z^N,K)$ if and only if $R(K) = A(K).$ In particular, $C(K) \ne HP(\bar z,\bar z^2,...,\bar z^N,K)$ unless $R(K) = C(K).$ We also give an example of $K$ showing the results are not valid if we replace $\bar z^n$ by certain $ϕ_n,$ that is, there exist $K$ and a function $ϕ\in A(K)$ such that $R(K) \ne A(K),$ but $A(K) = HP (ϕ,K).$

math.FA↗

arXiv2017

Bounded Point Evaluations For Rationally Multicyclic Subnormal Operators

Let $S$ be a pure bounded rationally multicyclic subnormal operator on a separable complex Hilbert space $\mathcal H$ and let $M_z$ be the minimal normal extension on a separable complex Hilbert space $\mathcal K$ containing $\mathcal H.$ Let $bpe(S)$ be the set of bounded point evaluations and let $abpe(S)$ be the set of analytic bounded point evaluations. We show $abpe(S) = bpe(S) \cap Int(σ(S)).$ The result affirmatively answers a question asked by J. B. Conway concerning the equality of the interior of $bpe(S)$ and $abpe(S)$ for a rationally multicyclic subnormal operator $S.$ As a result, if $λ_0\in Int(σ(S))$ and $dim(ker(S-λ_0)^*) = N,$ where $N$ is the minimal number of cyclic vectors for $S,$ then the range of $S-λ_0$ is closed, hence, $λ_0 \in σ(S) \setminus σ_e (S).$

math.FA↗

arXiv2002

Extended Quark Potential Model from Random Phase Approximation

The quark potential model is extended to include the sea quark excitation using the random phase approximation (RPA). The effective quark interaction preserves the important Quantum Chromodynamics (QCD) properties -- chiral symmetry and confinement simultaneously. A primary qualitive analysis shows that the $π$ meson as a well-known typical Goldstone boson and the other mesons made up of valence $q\bar{q}$ quark pair such as the $ρ$ meson can also be described in this extended quark potential model.

hep-ph↗