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Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with ℓ 1 norm, the p -norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that leads to new RKBSs based on Orlicz spaces and unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs.

This paper structures some new reproductive kernel spaces based on Legendre polynomials to solve time variable order fractional advection-reaction-diffusion equations. Some examples are given to show the effectiveness and reliability of the method.

We obtain new inequalities involving Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space H . Among many inequalities obtained here, it is shown that if A is a positive bounded linear operator on H , then ‖ A ‖ ber = ber ( A ) , where ‖ A ‖ ber and ber ( A ) are the Berezin norm and Berezin number of A , respectively. In contrast to the numerical radius, this equality does not hold for selfadjoint operators, which highlights the necessity of studying Berezin number inequalities independently.

Fractional-order calculus has become a useful mathematical framework to describe the complex super-diffusive process; however, numerical solutions of the two-sided space-fractional super-diffusive model with variable coefficients are difficult to obtain, and almost no method can obtain an analytical solution. In this paper, a class of new fractional dimensional reproducing kernel spaces (RKS) based on Caputo fractional derivatives is given, and we give analytical and numerical solutions of the two-sided space-fractional super-diffusive model based on the class of new RKS. The analytical solution is represented in the form of series in the reproducing kernel space. Numerical experiments indicate that the piecewise reproducing kernel method is more accurate than the traditional reproducing kernel method (RKM), and these new fractional reproducing kernel spaces are efficient for the two-sided space-fractional super-diffusive model.

The High order Gradient Reproducing Kernel in conjunction with the Collocation Method (HGRKCM) is introduced for solutions of 2nd- and 4th-order PDEs. All the derivative approximations appearing in PDEs are constructed using the gradient reproducing kernels. Consequently, the computational cost for construction of derivative approximations reduces tremendously, basis functions for derivative approximations are smooth, and the accumulated error arising from calculating derivative approximations are controlled in comparison to the direct derivative counterparts. Furthermore, it is theoretically estimated and numerically tested that the same number of collocation points as the source points can be used to obtain the optimal solution in the HGRKCM. Overall, the HGRKCM is roughly 10–25 times faster than the conventional reproducing kernel collocation method. The convergence of the present method is estimated using the least squares functional equivalence. Numerical results are verified and compared with other strong-form-based and Galerkin-based methods.

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.

By introducing the dimension splitting method into the reproducing kernel particle method (RKPM), a hybrid reproducing kernel particle method (HRKPM) for solving three-dimensional (3D) wave propagation problems is presented in this paper. Compared with the RKPM of 3D problems, the HRKPM needs only solving a set of two-dimensional (2D) problems in some subdomains, rather than solving a 3D problem in the 3D problem domain. The shape functions of 2D problems are much simpler than those of 3D problems, which results in that the HRKPM can save the CPU time greatly. Four numerical examples are selected to verify the validity and advantages of the proposed method. In addition, the error analysis and convergence of the proposed method are investigated. From the numerical results we can know that the HRKPM has higher computational efficiency than the RKPM and the element-free Galerkin method.

We give some applications of Berezin transforms and Engliś C ∗ -algebras methods, namely we investigate the solution of generalized Riccati operator equation of the form X A X + ∑ i = 1 N B i X C i - D = 0 , N ≥ 2 via Berezin transforms. Also, we study finite product of operators, including finite zero-product of Toeplitz operators on the Bergman Hilbert space L a 2 D in terms of Berezin transforms. The same method is used for characterization of compact truncated operators on the reproducing kernel Hilbert space H = H Ω .

An interface-modified reproducing kernel particle method (IM-RKPM) is introduced in this work to allow for a direct model construction from image pixels of heterogeneous polycrystalline Li-ion battery microstructures. The interface-modified reproducing kernel (IM-RK) approximation is constructed through scaling of a kernel function by a regularized distance function in conjunction with strategic placement of interface node locations. This leads to RK shape functions with either weak or strong discontinuities across material interfaces, suitable for modeling various interface mechanics. With the placement of a triple junction node and distance-based scaling of kernel functions, the resulting IM-RK shape function also possesses proper discontinuities at the triple junctions. This IM-RK approximation effectively remedies the well-known Gibb’s oscillation in the smooth approximation of discontinuities. Different from the conventional meshfree approaches for interface discontinuities, this IM-RK approach is done without additional degrees of freedom associated with the enrichment functions, and it is formulated with the standard procedures in the RK shape function construction. This work focuses on identifying the accuracy and convergence properties of IM-RKPM for modeling the coupled electro-chemo-mechanical system. A linear patch test is formulated and numerically tested for the electro-chemo-mechanical coupled problem with a Butler–Volmer boundary condition representing the physical conditions in Li-ion battery microstructures. This is followed by verification of the optimal rates of convergence of IM-RKPM for solving the coupled problem with higher order solutions. The image-based modeling of Li-ion battery microstructures in the numerical examples demonstrates the applicability of the proposed method to realistic Li-ion battery materials modeling.

We study some problems of operator theory by using Berezin symbols approach. Namely, we investigate in terms of Berezin symbols invariant subspaces of isometric composition operators on H Ω . We discuss operator corona problem, in particular, the Toeplitz corona problem. Further, we characterize unitary operators in terms of Berezin symbols. We show that the well known inequality w A ≥ 1 2 A for numerical radius is not true for the Berezin number of operators, which is defined by ber A : = sup λ ∈ Ω A ~ λ , where A ~ λ : = A k ^ λ , k ^ λ is the Berezin symbol of operator A : H Ω → H Ω . Finally, we provide a lower bound for ber A .