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We consider generalized Riccati operator equations and study their solutions in terms of Berezin transforms. We also investigate invariant subspaces of some Toeplitz operators on the Bergman space L a 2 over the unit disc D of the complex plane C. Some other related questions are also discussed. [ABSTRACT FROM AUTHOR]

In this paper,the linear space $\mathcal F$ of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in $\mathcal F$ is established from a continuous function on I. We show that, for a finite set of linearly independent continuous functions on I , we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) $\mathcal F_{\mathcal B}\subset\mathcal F$ , and the reproducing kernel $\mathbf k$ for $\mathcal F_{\mathcal B}$ is defined by a basis of $\mathcal F_{\mathcal B}$. For a given data set $\mathcal D=\{(t_k, y_k) : k=0,1,\ldots,N\}$ , we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form $f_{\mathcal V}+f_{\mathcal B}$ , where $f_{\mathcal V}\in C_{\mathcal V}$ and $f_{\mathcal B}\in \mathcal F_{\mathcal B}$. Here $C_{\mathcal V}$ is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel $\mathbf k^*$. We show that the solution function can be written in the form $f_{\mathcal V}+f_{\mathcal B}=\sum_{m=0}^N\gamma_m\mathbf k^*_{t_m} +\sum_{j=0}^N \alpha_j\mathbf k_{t_j}$ , where ${\mathbf k}_{t_m}^\ast(\cdot)={\mathbf k}^\ast(\cdot,t_m)$ and $\mathbf k_{t_j}(\cdot)=\mathbf k(\cdot,t_j)$ , and the coefficients γm and αj can be solved by a system of linear equations. [ABSTRACT FROM AUTHOR]

AbstractMany methods have been developed to analyze complex data, such as non-Euclidean shape, network, and manifold data. However, there is a lack of methods for studying interactions among complex data. In this article, we first propose a novel kernel function for a metric space and construct its associated reproducing kernel Hilbert space. The new nonstationary kernel function provides a flexible and powerful tool for learning complex structures in non-Euclidean data. We then construct an analysis of variance (ANOVA) decomposition of the nonparametric regression function defined on metric space, which provides a hierarchical structure for investigating the main effects and interactions. We develop estimation and computational methods for a semi-parametric model with a multivariate function on a product of metric spaces modeled by the ANOVA decomposition. We establish the convergence rates of parameter and nonparametric function estimates. The application of the proposed methods to the Alzheimer’s Disease Neuroimaging Initiative hippocampus shape data confirms some existing and suggests some new interactions among hippocampal regions. Simulations indicate that the proposed methods work well. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the sufficient dimension reduction problem in regression analysis with real‐valued response and functional or longitudinal covariates. We propose a new method based on gradients of the conditional distribution function to estimate the sufficient dimension reduction subspace. While existing inverse‐regression‐type methods relies on a linearity condition, our method is based on the gradient of conditional distribution function and its validity only requires smoothness conditions on the population parameters. Practically, the proposed estimator can be obtained by standard algorithm of functional principal component analysis. The proposed method is demonstrated through extensive simulations and two empirical examples. [ABSTRACT FROM AUTHOR]

For a bounded linear operator T on a Hilbert space H , the Berezin symbol (or Berezin transform) of T is the bounded function T ~ on a set Θ defined by T ~ τ = T k ^ τ z , k ^ τ z for all τ ∈ Θ , where k ^ τ = k τ k τ is the normalized reproducing kernel of the space H . The Berezin norm and the Berezin number (radius) of an operator T are defined, respectively, by T ber : = sup τ , μ ∈ Θ | ⟨ T k ^ τ , k ^ μ ⟩ | and ber T = sup τ ∈ Θ T ~ τ. In this paper, we obtain some Berezin radius inequalities for certain operator matrices. Among other inequalities, it is shown that if A and B are bounded linear operators on a complex Hilbert space H , then ber 0 A B 0 ≤ 1 2 sup θ ∈ R ‖ e i θ A + e - i θ B ∗ ‖ ber ≤ 1 2 (‖ A ‖ ber + ‖ B ‖ ber). [ABSTRACT FROM AUTHOR]

In this article we study Banach spaces of solenoidal solutions, that is, vector fields of zero divergence that satisfy the vector Helmholtz equation in R 3 . As it is known, these solutions are related to the time-harmonic Maxwell equations or Maxwell equations in a simple medium which makes this study important. We define an integral operator given by the Fourier transform of the tangential fields which are square integrable on the unit sphere S 2 that will generate pairs of vector fields satisfying the time-harmonic Maxwell equations, which we will call electromagnetic Herglotz pairs. By last, in the context of Hilbert spaces, the space of solutions is studied, seen as a reproducing kernel Hilbert space and its reproducing kernel is calculated. [ABSTRACT FROM AUTHOR]

A numerical approach is proposed for space fractional partial differential equations by the reproducing kernel approach. Some procedures are presented for improving the existing methods. The presented method is easy to accomplish. Approximate solutions and their partial derivatives are shown to converge to exact solutions, respectively. Experiments show that the presented technique is efficient, and that high-precision global approximate solutions can be obtained.

A numerical approach is proposed for space fractional partial differential equations by the reproducing kernel approach. Some procedures are presented for improving the existing methods. The presented method is easy to accomplish. Approximate solutions and their partial derivatives are shown to converge to exact solutions, respectively. Experiments show that the presented technique is efficient, and that high-precision global approximate solutions can be obtained. [ABSTRACT FROM AUTHOR]

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers H t , t ∈ R ∗ , of which the H - 1 = H is the space of quaternions and H 1 is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on H t . Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function. [ABSTRACT FROM AUTHOR]

AbstractWe consider the issue of screening uninformative variables associate with survival outcome in ultra high-dimensional situation. specifically, we focus on Hilbert-Schmits Independent criteria (HSIC) statistic defined in the Reproducing kernel Hilbert Space(RKHS) and use the HSIC as a feature screening approach to avoid from difficulty occurred from the ultra high dimensionality. Generally, it is known that the HSIC can account for the flexible relationship between the outcome and predictor variables. Compared with the typical approaches such as regularization and model free screening approaches, our approach does not require the complex non parametric computation or the complicated numerical optimization, Thereby, our proposed approach effectively and flexibly filters out the uninformative predictor. We will show the advantages of our method in numerical simulation studies and illustrate the usefulness of it by applying to diffuse large-B-cell lymphoma data. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a marginal screening method for variable screening in high-dimensional binary classification based on the Wasserstein distance accounting for the distributional difference. Many existing screening methods, such as the two-sample t-test and Kolmogorov test, have been developed under the parametric/nonparametric modeling assumptions to reduce the dimension of the predictors. However, such modeling specifications or nonparametric approaches are associated with the probability measure induced by the predictor in a Euclidean space. While many machine learning methods have successfully found the nonlinear decision boundary in the transformed space, called the reproducing kernel Hilbert space (RKHS), we consider the Wasserstein filter's capacity to detect the distributional difference between two probability measures induced by the nonlinear function of the predictor in the RKHS. Thereby, we can flexibly filter out the non-informative predictors associated with the binary classification, as well as escape the modeling assumptions required in a Euclidean space. We prove that the Wasserstein filter satisfies the sure screening property under some mild conditions. We also demonstrate the advantages of our proposed approach by comparing the finite sample performance of it with those of the existing choices through simulation studies, as well as through application to lung cancer data. [ABSTRACT FROM AUTHOR]

Let n , m ≥ 1 and α > 0 . We denote by F α , m the m-analytic Bargmann–Segal–Fock space, i.e., the Hilbert space of all m-analytic functions defined on C n and square integrables with respect to the Gaussian weight exp (- α | z | 2) . We study the von Neumann algebra A of bounded linear operators acting in F α , m and commuting with all "horizontal" Weyl translations, i.e., Weyl unitary operators associated to the elements of R n . The reproducing kernel of F 1 , m was computed by Youssfi [Polyanalytic reproducing kernels in C n , Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of F α , m by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel K is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of K in the "horizontal direction" and decompose it into the sum of d products of Hermite functions, with d = n + m - 1 n . Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that F α , m is isometrically isomorphic to the space of vector-functions L 2 (R n) d , and A is isometrically isomorphic to the algebra of matrix-functions L ∞ (R n) d × d . [ABSTRACT FROM AUTHOR]

In 2012, we obtained Oppenheim's inequality for positive semidefinite matrices and its equality condition by the reproducing kernel method. In this paper, as a continuation, we give a reproducing kernel proof of the block matrix version of the Oppenheim-Schur's inequality and its equality condition in the positive definite case. [ABSTRACT FROM AUTHOR]

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established.

A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set Θ ⊆ C such that the evaluation functionals φ τ (f) = f (τ) , τ ∈ Θ , are continuous on ℋ. The Berezin number of an operator X is defined by ber (X) = sup τ ∈ Θ | X ˜ (τ) | = sup τ ∈ Θ | 〈 X k ˆ τ , k ˆ τ 〉 | , where the operator X acts on the reproducing kernel Hilbert space H = H (Θ) over some (non-empty) set Θ . In this paper, we introduce a new family involving means ∥ ⋅ ∥ σ t between the Berezin radius and the Berezin norm. Among other results, it is shown that if X ∈ L (H) and f , g are two non-negative continuous functions defined on [ 0 , ∞) such that f (t) g (t) = t , (t ⩾ 0) , then ∥ X ∥ σ 2 ⩽ ber (1 4 (f 4 (| X |) + g 4 (| X ∗ |)) + 1 2 | X | 2) and ∥ X ∥ σ 2 ⩽ 1 2 ber (f 4 (| X |) + g 2 (| X | 2)) ber (f 2 (| X | 2) + g 4 (| X ∗ |)) , where σ is a mean dominated by the arithmetic mean ∇ . [ABSTRACT FROM AUTHOR]

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established. [ABSTRACT FROM AUTHOR]

Studying convection–diffusion problems of delayed type in physics helps us to understand transport phenomena and has practical applications in various fields. The mathematical analysis of this model has practical applications in various fields, such as flow dynamics, material science, and environmental modeling. In this paper, the theory of reproducing kernel spaces (RKS) is utilized to solve singularly perturbed 2D parabolic convection–diffusion problems of delayed type. To this end, a series form for the solution is first constructed in reproducing kernel Hilbert space, and then, the approximate solution is given as an N -term summation. The main contribution of the present research is that, for the first time, a novel formula is found for the homogenization of 2D initial-boundary-value problems. Furthermore, a semi-analytical RKS method is employed without employing the Gram–Schmidt orthogonalization algorithm. We derive theorems to reveal stability and convergence properties which are examined by numerical experiments. The technique is especially suited for problems having boundary-layer behavior. Numerical results are provided to demonstrate the efficiency, stability, and superiority of the proposed technique. [ABSTRACT FROM AUTHOR]

This study aims to improve the predictive performance for the event time through the machine learning model and find informative variables in the time-to-event data, simultaneously. To address this issue, after regarding the time-to-event data as the dichotomized counting processes data for predicting survival time, we consider the time-dependent support vector machine (SVM) framework for the dichotomized counting process data, where the decision function in this framework consists of the time-independent risk score and time-dependent intercept. Also, we consider the empirical partial derivative of the risk score function with respect to each marginal predictor as the indicator for the important predictor. Through this approach, it is possible to predict survival time and find variables that affect on the survival time at the same time. Simulation studies were conducted to confirm the performance of the model, and real data analysis was conducted by predicting the survival time of the lung cancer after the diagnosis and selecting genes associate with lung cancer through human gene data. [ABSTRACT FROM AUTHOR]

As an advanced detection technique, electrical resistive tomography (ERT) has been applied to detect the solid–liquid two-phase flow velocity based on available ERT measurements. The flow velocity computation by ERT must depend on the relative algorithms, including both the cross-correlation (CC) principle and convolutional neural networks (CNNs). However, these two types of algorithms have poor accuracy and generalization under complex measuring conditions and various flow patterns. To address this issue, in this paper, a hybrid network is proposed that combines a CNN with a reproducing kernel-based support vector machine (RKSVM) technique. The features hidden in ERT measurements are extracted using the CNN, and then the flow velocity is computed by the RKSVM in a high-dimensional feature space. According to the ERT measurements in an actual experimental platform, the results show that the hybrid network has higher accuracy and generalization ability for flow velocity computation compared with the existing CC, RKSVM, and CNN methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study the linear canonical Fourier–Bessel wavelet transform. We prove an orthogonality relation, inversion formula and some inequality for linear canonical Fourier–Bessel wavelet transform. We first present a direct relationship between the linear canonical Bessel wavelet transform and ordinary Bessel wavelet transform. Based on this relation, we provide an alternative proof of the orthogonality relation for the linear canonical Bessel wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical Bessel wavelet transform. [ABSTRACT FROM AUTHOR]

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on $ {\mathbb C}^n $ C n [Klembeck P. A complete Kähler metric of positive curvature on $ {\mathbb C}^n $ C n . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature. [ABSTRACT FROM AUTHOR]

The semi-Lagrangian Reproducing Kernel (SL RK) has been successfully applied to simulations of extreme deformation problems thanks to intensive advancements in meshfree methods in recent decades. The SL RK shape function is constructed centered at a material point with a fixed spatial radius, and therefore it is a time-dependent function. When the SL RK is introduced in the weak form of the equation of motion, the time derivatives of SL RK lead to convective terms in the semi-discrete equation, significantly reducing the computational efficiency and accuracy. This paper presents a novel approximation scheme in which displacement, velocity, and acceleration are approximated by the same SL RK at a time instant. Consequently, the proposed concurrent SL RK formulation avoids the need to compute the convective terms, offering an effective numerical solution for high-strain rate dynamic problems. Furthermore, an eigenvalue analysis is performed on the semi-discrete equation of motion to investigate the temporal stability of the concurrent SL RK formulation. A comprehensive study of the eigenvalue and critical time step is conducted to identify the effective parameters for the stability of the system. Several manufactured dynamic problems are studied to validate the analytical estimate. The estimated analytical eigenvalue and the critical time step are consistent with the numerical result. [ABSTRACT FROM AUTHOR]

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on $ {\mathbb C}^n $ C n [Klembeck P. A complete Kähler metric of positive curvature on $ {\mathbb C}^n $ C n . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature. [ABSTRACT FROM AUTHOR]

In this note, we shall consider extension formulas and norm inequalities for some typical Sobolev Hilbert spaces. We see many related open problems. [ABSTRACT FROM AUTHOR]

We intend to introduce and investigate a new functional space on the quaternionic unit ball of slice hyper-meromorphic functions with unique pole at the origin. Mainly, we provide a concrete characterization of their elements and give the closed explicit expression of the associated reproducing kernel function. Moreover, we show that they are isometrically isomorphic to the configuration space on the positive real half line by means of an integral transform of Bargmann type. The closed formulas for this transform are given in two special cases. [ABSTRACT FROM AUTHOR]

The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations J f (z) = f (z ¯) ¯ , J s f (z) = f (z ¯ + i s) ¯ , and J ∗ f (z) = 1 z α + 2 f (1 z ¯ ) ¯ . The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric. The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations , , and . The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric. [ABSTRACT FROM AUTHOR]

The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations $ \mathcal{J}f(z) = \overline{f(\bar{z})} $, $ \mathcal{J}_sf(z) = \overline{f(\bar{z}+is)} $, and $ \mathcal{J}_*f(z) = \frac{1}{z^{{\alpha}+2}}\overline{f(\frac{1}{\bar{z}})} $. The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric.

We studied Toeplitz operators acting on certain poly-Bergman-type spaces of the Siegel domain $ D_{2} \subset \mathbb{C}^{2} $. Using continuous nilpotent symbols, we described the $ C^* $-algebras generated by such Toeplitz operators. Bounded measurable functions of the form $ \tilde{c}(\zeta) = c(\text{Im}\, \zeta_{1}, \text{Im}\, \zeta_{2} - |\zeta_1|^{2}) $ are called nilpotent symbols. In this work, we considered symbols of the form $ \tilde{a}(\zeta) = a(\text{Im}\, \zeta_1) $ and $ \tilde{b}(\zeta) = b(\text{Im}\, \zeta_2 -|\zeta_1|^{2}) $, where both limits $ \lim\limits_{s\rightarrow 0^+} b(s) $ and $ \lim\limits_{s\rightarrow +\infty} b(s) $ exist, and $ a $ belongs to the set of piece-wise continuous functions on $ \overline{\mathbb{R}} = [-\infty, +\infty] $ and with one-sided limits at $ 0 $. We described certain $ C^* $-algebras generated by such Toeplitz operators that turned out to be isomorphic to subalgebras of $ M_n(\mathbb{C}) \otimes C(\overline{\Pi}) $, where $ \overline{\Pi} = \overline{ \mathbb{R}} \times \overline{ \mathbb{R}}_+ $ and $ \overline{\mathbb{R}}_+ = [0, +\infty] $.

Over the last decades, learning methods using kernels have become very popular. The main reason is that real data analysis often requires nonlinear methods to detect the dependencies that allow successful predictions of properties of interest. Gaussian kernels have been used in many studies such as learning algorithms and data analysis. Most of these studies have shown that the parameter chosen for a Gaussian kernel could have a huge impact on the desired results. Therefore, it is essential to understand this impact on a theoretical level. The main contribution of this paper is to study the effect of the Gaussian kernel bandwidth parameter on how well an empirical operator defined from data approximates its continuous counterpart. Some results in spectral approximations are provided as well as some examples.

We extend the assertion of Problem 12340 in Amer. Math. Monthly 129 (2022), 686, by deriving some additional asymptotic behaviour of that special kernel.

A novel method is presented for reproducing kernel of a 2D fractional diffusion equation. The exact solution is expressed as a series, which is then truncated to get an approximate solution. In addition, some techniques to improve existing methods are also proposed. The proposed approach is easy to implement. It is proved that both the approximate solution and its partial derivatives converge to their exact solutions. Numerical results demonstrate that the proposed approach is effective and can provide a high precision global approximate solution.

We consider a novel bounded integral transform with a kernel function being the n-th polyanalytic Intissar–Hermite polynomial. We provide a concrete description of its range, shown to be a reproducing kernel Hilbert space for which we provide an explicit closed formula of its reproducing kernel. The limit of these ranges leads, in a precise sense, to the so-called van Eijndhoven–Meyers Bargmann space, which also corresponds to the limit case. The null space of the considered transform is also characterized. It gives rise in particular to new integral representation of the true-polyanalytic Bargmann space by means of the polyanalytic Intissar–Hermite integral transform. [ABSTRACT FROM AUTHOR]

Let D = {z G C : |z| < 1} and L(La²(D)) be the space of all bounded linear operators from the Bergman space La²(D) into itself. In this paper we shall associate symbols to bounded linear operators in L(La²(D)) and analyse if a symbol calculus can be obtained. [ABSTRACT FROM AUTHOR]

For a scalar inner function θ, the model space of Sz.-Nagy and Foias is the subspace Kθ = H²⊖θH² of the classical Hardy space H² = H²(D) over the unit disc D = {z ∈ C: |z| < 1} . For a bounded linear operator A on the model space Kθ, its Berezin symbol is the function AKθ defined on D by AKθ(λ)... is the normalized reproducing kernel of the subspace Kθ. We shall consider the following question: Let A: Kθ → Kθ be an operator for which there exists a constant δ > 0 such that | AKθ (λ)| ≥ δ > 0, for all λ ∈ D. Under which additional conditions is A invertible? In this article we investigate this question in the case when θ is an interpolation Blaschke product. In particular, the invertibility property of functions of model operators is investigated. Some other problems are also discussed. [ABSTRACT FROM AUTHOR]

In [4] a model is presented of a finite-dimensional Pontryagin space with a symmetric operator without eigenvalues. In this note we show that this model is unique up to an equivalence relation. [ABSTRACT FROM AUTHOR]

In this paper, we introduced a reproducing kernel space which is a particular class of Hilbert space. We discuss various properties of the reproducing kernel. In particular, our aim is to construct kernel in reproducing kernel Hilbert space of the specific function space (Sobolev space) with the improved inner product and norm. Also, we derive the reproducing kernel for periodic boundary conditions. [ABSTRACT FROM AUTHOR]

In the paper, 2-complex symmetric weighted composition operators induced by three type symbols on the weighted Bergman spaces of the right half-plane with three conjugations J f (z) = f (z ¯) ¯ , J s f (z) = f (z ¯ + i s) ¯ and J ∗ f (z) = 1 z α + 2 f (1 z ¯ ) ¯ are characterized by building the relations between some parameters. Some examples of such operators are also given. [ABSTRACT FROM AUTHOR]

D. A. Hejhal showed the positive definiteness of the matrix H = (h jk) appearing in Schiffer’s identity K B = 4 π K S 2 + ∑ j , k h jk v j v k ¯ , where K B and K S are, respectively, the Bergman and the Szegö kernel on a planar regular region. We give several conditions equivalent to Hejhal’s theorem by means of an integral transform whose kernel is the product of two Szegö kernels. In particular, we show that the positive definiteness of the matrix H follows from the contractivity of the above integral transform together with the equality condition fornorms. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study for the Dunkl–Stockwell transform some problems of the reproducing kernels theory. In particular, we investigate the Saitoh approach to give some applications of the Tikhonov regularization for some generalized Sobolev spaces associated with the Dunkl–Stockwell transform and we study also some time-frequency concentration problems. [ABSTRACT FROM AUTHOR]

Measuring and testing the dependency between multiple random functions is often an important task in functional data analysis. In the literature, a model-based method relies on a model which is subject to the risk of model misspecification, while a model-free method only provides a correlation measure which is inadequate to test independence. In this paper, we adopt the Hilbert–Schmidt Independence Criterion (HSIC) to measure the dependency between two random functions. We develop a two-step procedure by first pre-smoothing each function based on its discrete and noisy measurements and then applying the HSIC to recovered functions. To ensure the compatibility between the two steps such that the effect of the pre-smoothing error on the subsequent HSIC is asymptotically negligible when the data are densely measured, we propose a new wavelet thresholding method for pre-smoothing and to use Besov-norm-induced kernels for HSIC. We also provide the corresponding asymptotic analysis. The superior numerical performance of the proposed method over existing ones is demonstrated in a simulation study. Moreover, in a magnetoencephalography (MEG) data application, the functional connectivity patterns identified by the proposed method are more anatomically interpretable than those by existing methods. [ABSTRACT FROM AUTHOR]

For any real $ \beta $ let $ H^2_\beta $ be the Hardy-Sobolev space on the unit disc $ {\mathbb D} $. $ H^2_\beta $ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $ \beta > 1/2 $. In this paper, we prove that $ C_{\varphi } $ has dense range in $ H_{\beta }^{2} $ if and only if the polynomials are dense in a certain Dirichlet space of the domain $ \varphi({\mathbb D}) $ for $ 1/2 < \beta < 1 $. It follows that if the range of $ C_{\varphi } $ is dense in $ H_{\beta }^{2} $, then $ \varphi $ is a weak-star generator of $ H^{\infty} $, although the conclusion is false for the classical Dirichlet space $ \mathfrak{D} $. Moreover, we study the relation between the density of the range of $ C_{\varphi } $ and the cyclic vector of the multiplier $ M_{\varphi}^{\beta}. $

For any real \beta let H^2_\beta be the Hardy-Sobolev space on the unit ball \mathbb {B}_{n}, n\geq 1. H^2_\beta is a reproducing kernel Hilbert space and its reproducing kernel is bounded when \beta >n/2. In this paper, we characterize when the composition operator C_{\varphi } on H^{2}_{\beta } is Fredholm for a non-constant analytic map \varphi :\mathbb {B}_{n}\to \mathbb {B}_{n}. [ABSTRACT FROM AUTHOR]