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Abstract In this paper, we study and analyze the regularized least squares for functional linear regression model. The approach is to use the reproducing kernel Hilbert space framework and the integral operators. We show with a more general and realistic assumption on the reproducing kernel and input data statistics that the rate of excess prediction risk by the regularized least squares is minimax optimal. [ABSTRACT FROM AUTHOR]

A number of recent papers have established connections between reproducing kernel Hilbert spaces H of entire functions, de Branges spaces, sampling formulas and a class of symmetric operators with deficiency indices ( 1 , 1 ) . In this paper analogous connections between reproducing kernel Hilbert spaces of entire vector valued functions, de Branges spaces of entire vector valued functions, sampling formulas and symmetric operators with deficiency indices ( p , p ) are obtained. Enroute, an analog of L. de Branges' characterization of the reproducing kernel Hilbert spaces of entire functions that are now called de Branges spaces is obtained for the p × 1 vector valued case. A special class of these de Branges spaces of p × 1 vector valued entire functions is identified as a functional model for M. G. Krein's class of entire operators with deficiency indices ( p , p ) . [ABSTRACT FROM AUTHOR]

This paper is concerned with spherical harmonics, and two refinements thereof: complex harmonics and symplectic harmonics. The reproducing kernels of the spherical and complex harmonics are explicitly given in terms of Gegenbauer or Jacobi polynomials. In the first part of the paper we determine the reproducing kernel for the space of symplectic harmonics, which is again expressible as a Jacobi polynomial of a suitable argument. In the second part we find plane wave formulas for the reproducing kernels of the three types of harmonics, expressing them as suitable integrals over Stiefel manifolds. This is achieved using Pizzetti formulas that express the integrals in terms of differential operators. [ABSTRACT FROM AUTHOR]

We study two geometric properties of reproducing kernels in model spaces K θ where θ is an inner function: overcompleteness and existence of uniformly minimal systems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern–Clark point. It is shown that “uniformly minimal non-Riesz” sequences of reproducing kernels exist near each Ahern–Clark point which is not an analyticity point for θ , while overcompleteness may occur only near the Ahern–Clark points of infinite order and is equivalent to a “zero localization property”. In this context the notion of quasi-analyticity appears naturally, and as a by-product of our results we give conditions in the spirit of Ahern–Clark for the restriction of a model space to a radius to be a class of quasi-analyticity. [ABSTRACT FROM AUTHOR]

We present a formulation of smoothed particle hydrodynamics (SPH) that utilizes a first-order consistent reproducing kernel, a smoothing function that exactly interpolates linear fields with particle tracers. Previous formulations using reproducing kernel (RK) interpolation have had difficulties maintaining conservation of momentum due to the fact the RK kernels are not, in general, spatially symmetric. Here, we utilize a reformulation of the fluid equations such that mass, linear momentum, and energy are all rigorously conserved without any assumption about kernel symmetries, while additionally maintaining approximate angular momentum conservation. Our approach starts from a rigorously consistent interpolation theory, where we derive the evolution equations to enforce the appropriate conservation properties, at the sacrifice of full consistency in the momentum equation. Additionally, by exploiting the increased accuracy of the RK method's gradient, we formulate a simple limiter for the artificial viscosity that reduces the excess diffusion normally incurred by the ordinary SPH artificial viscosity. Collectively, we call our suite of modifications to the traditional SPH scheme Conservative Reproducing Kernel SPH, or CRKSPH. CRKSPH retains many benefits of traditional SPH methods (such as preserving Galilean invariance and manifest conservation of mass, momentum, and energy) while improving on many of the shortcomings of SPH, particularly the overly aggressive artificial viscosity and zeroth-order inaccuracy. We compare CRKSPH to two different modern SPH formulations (pressure based SPH and compatibly differenced SPH), demonstrating the advantages of our new formulation when modeling fluid mixing, strong shock, and adiabatic phenomena. [ABSTRACT FROM AUTHOR]

In general, the Beurling theorem does not hold for an invariant subspace in the Hardy space over the bidisk. In 1991, Nakazi posed a conjecture that the Beurling theorem holds for a singly generated invariant subspace. In this paper, a relation between a singly generated invariant subspace and a weighted Hardy space over the bidisk is studied. It is showed that there exists a weighted Hardy space over the bidisk which has a non-cyclic reproducing kernel. Also a counterexample for Nakazi's conjecture is given. [ABSTRACT FROM AUTHOR]

This paper mainly studies the time sampling and reconstruction of signals in weighted reproducing kernel subspaces of L ν p . Firstly, a class of examples for weighted reproducing kernel subspaces are given. Then, using samples produced by a crossing time encoding machine or an Integrate-and-Fire time encoding machine satisfying some density constraints, iterative approximation algorithms are established for perfect recovery of signals in weighted reproducing kernel subspaces. Finally, an Integrate-and-Fire sampling scheme is discussed and an approximate reconstruction procedure is proposed for signals in weighted reproducing kernel subspaces. Moreover, we give detailed analysis for the approximate reconstruction method in the setting of shift-invariant subspaces. [ABSTRACT FROM AUTHOR]

This paper constitutes an extension of the work of Mendez et al. (2014) [36] , for three-dimensional simulations of deformable membranes under flow. An immersed thick boundary method is used, combining the immersed boundary method with a three-dimensional modeling of the structural part. The immersed boundary method is adapted to unstructured grids for the fluid resolution, using the reproducing kernel particle method. An unstructured finite-volume flow solver for the incompressible Navier–Stokes equations is coupled with a finite-element solver for the structure. The validation process relying on a number of test cases proves the efficiency of the method, and its robustness is illustrated when computing the dynamics of a tri-leaflet aortic valve. The proposed immersed thick boundary method is able to tackle applications involving both thin and thick membranes/closed and open membranes, in significantly high Reynolds number flows and highly complex geometries. [ABSTRACT FROM AUTHOR]

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators { A ( t ) | t ∈ R } that converges in norm to asymptotes A ± at ±∞. Then under certain conditions [22] that include the assumption that the operators { D ( t ) = D + A ( t ) , t ∈ R } all have discrete spectrum the spectral flow along the path { D ( t ) } can be shown to be equal to the index of ∂ t + D ( t ) when the latter is an unbounded Fredholm operator on L 2 ( R , H ) . In [16] an investigation of the index = spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In [11] the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index. In this paper we take the investigation begun in [11] much further. We show how to generalize a formula known from the setting of the L 2 index theorem to the non-Fredholm setting. Restricting back to the case of selfadjoint Fredholm operators our formula extends the result of [22] in the sense of relaxing the discrete spectrum condition. It also generalizes some other Fredholm operator results of [21,16,11] that permit essential spectrum for the operators in the path. Our result may also apply however when the operators { D ( t ) } have essential spectrum equal to the whole real line. Our main theorem gives a trace formula relating the homological index of [7] to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalization of spectral flow to the non-Fredholm setting. [ABSTRACT FROM AUTHOR]

We obtain by elementary methods necessary and sufficient conditions for a k -dimensional cubature formula to hold for all polynomials of degree up to 2 m − 1 when the nodes of the formula have Lagrange polynomials of degree at most m . The main condition is that the Lagrange polynomial at each node is a scalar multiple of the reproducing kernel of degree m − 1 evaluated at the node plus an orthogonal polynomial of degree m . Stronger conditions are given for the case where the cubature formula holds for all polynomials of degree up to 2 m . This result is applied in one dimension to obtain a quadrature formula where the nodes are the roots of a quasi-orthogonal polynomial of order 2. In two dimensions the result is applied to obtain constructive proofs of cubature formulas of degree 2 m − 1 for the Geronimus and the Morrow–Patterson classes of nodes. A cubature formula of degree 2 m is obtained for a subclass of Morrow–Patterson nodes. Our discussion gives new proofs of previous theorems for the Chebyshev points and the Padua points, which are special cases. [ABSTRACT FROM AUTHOR]

Abstract: In the previous works, the authors presented the reproducing kernel method (RKM) for solving various differential equations. However, to the best of our knowledge, there exist no results for functional differential equations. The aim of this paper is to extend the application of reproducing kernel theory to nonlocal functional differential equations with delayed or advanced arguments, and give the error estimation for the present method. Some numerical examples are provided to show the validity of the present method. [Copyright &y& Elsevier]

Abstract: The theory of symmetric operators has several deep applications to the function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions (model subspaces of Hardy spaces, de Branges–Rovnyak spaces, Herglotz spaces), Sturm–Liouville and Schrodinger differential operators, Toeplitz operators, and infinite Jacobi matrices. In this paper we develop a general representation theory of simple symmetric operators with equal deficiency indices, and obtain a collection of results which refine and extend classical works of Krein and Livsic. In particular, we provide an alternative proof of a theorem of Livsic which characterizes when two simple symmetric operators with equal deficiency indices are unitarily equivalent. Moreover, we provide a new, more easily computable formula for the Livsic characteristic function of a simple symmetric operator. [Copyright &y& Elsevier]