On hyperbolic summation and hyperbolic moduli of smoothness

The purpose of this series of papers is to present a general theory of Banach modules and to give some applications of it. The applications of the theory arise from observations that certain important notions of functional analysis are very closely related to certain Banach algebras and thereby to a module structure. This holds in particular for Banach lattices (which are C(X)-modules), interpolation spaces, tensor products and operator ideals.

C. Le Merdy in (Proc. Amer. Math. Soc. 126 (1998), 715- 719) proved the following complex interpolation formula for injective tensor products: ('2 ˜ '1,'2 ˜ '1) 1 = S4. We investigate whether related formulas hold when considering arbitrary 0 <

This work introduces and analyzes B-spline approximation spaces defined on general geo- metric domains obtained through a mapping from a parameter domain. These spaces are constructed as sparse-grid tensor products of univariate spaces in the parameter domain and are mapped to the physical domain via a geometric parametrization. Both the univariate approximation spaces and the geometric mapping are built using maximally smooth B-splines. We construct two such spaces, em- ploying either the sparse-grid combination technique or the hierarchical subspace decomposition of sparse-grid tensor products, and we prove their mathematical equivalence. Furthermore, we derive ap- proximation error estimates and inverse inequalities that highlight the advantages of sparse-grid tensor products. Specifically, under suitable regularity assumptions on the solution, these spaces achieve the same approximation order as standard tensor product spaces while using significantly fewer degrees of freedom. Additionally, our estimates indicate that, in the case of non-tensor-product domains, stronger regularity assumptions on the solution—particularly concerning isotropic (non-mixed) derivatives—are required to achieve optimal convergence rates compared to sparse-grid methods defined on tensor- product domains.

That the classical interpolation theorems of Riesz-Thorin and Marcinkiewicz are of great importance in harmonic analysis has long since been realized. The functional analytic generalization of these interpolation theorems — abstract interpolation space theory — also is now proving to be of importance in harmonic analysis (cf., for instance, [1]). In this paper we reverse the flow in the sense that a circle of ideas from Lp-multiplier theory and tensor products of Banach spaces is used to provide answers to a number of questions that arise naturally in the general theory of abstract interpolation spaces. While some of these questions have been answered before (though not always published), the unified approach used in this paper introduces techniques which may well be of wider utility in interpolation space theory. We shall be interested in both the real interpolation spaces (X o, X 1)θq;J, (X o, X 1)θ, q;k of Peetre and the complex interpolation spaces (X o,X 1)θ of Lions and Calderon. For all unexplained notation and terminology see [2], [5], [6] or [11].

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in L^p-setting, 1 le p le +infty , have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order r in mathbb {N}^+. Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the L^p modulus of smoothness of order r have been established. As a consequence, the qualitative order of approximation is also derived assuming f in suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in L^p(mathbb {R}), 1 le p le +infty , for the sampling Kantorovich operators.

We construct the bivariate form of Bernstein–Schurer operators based on parameter α. We establish the Voronovskaja-type theorem and give an estimate of the order of approximation with the help of Peetre’s K-functional of our newly defined operators. Moreover, we define the associated generalized Boolean sum (shortly, GBS) operators and estimate the rate of convergence by means of mixed modulus of smoothness. Finally, the order of approximation for Bögel differentiable function of our GBS operators is presented.

There have been various studies on approximation ability of feedforward neural networks. The existing studies are, however, only concerned with the density or upper bound estimation on how a multivariate function can be approximated by the networks, and consequently, the essential approximation ability of networks cannot be revealed. In this paper, by establishing both upper and lower bound estimations on approximation order, the essential approximation ability of a class of feedforward neural networks with trigonometric hidden layer units is clarified in terms of the second order modulus of smoothness of approximated function.KeywordsNeural NetworkHide LayerApproximation OrderFeedforward Neural NetworkHide UnitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order mathscr {O}(h^{2M}), M=1,2,3,4, are provided.

Fast formation of isogeometric Galerkin matrices by weighted quadrature

In this paper we solve electromagnetic scattering problems by approximating Maxwell's equations in the time‐domain with a high‐order quadrilateral discontinuous spectral element method (DSEM). The method is a collocation form of the discontinuous Galerkin method for hyperbolic systems where the solution is approximated by a tensor product Legendre expansion and inner products are replaced with Gauss–Legendre quadratures. To increase flexibility of the method, we use a mortar element method to couple element faces. Mortars provide a means for coupling element faces along which the polynomial orders differ, which allows the flexibility to choose the approximation order within an element by considering only local resolution requirements. Mortars also permit local subdivision of a mesh by connecting element faces that do not share a full side. We present evidence showing that the convergence of the non‐conforming approximations is spectral along with examples of their use. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Generalized Szász-Mirakjan type operators via q-calculus and approximation properties

On Hyperbolic Summation and Hyperbolic Moduli of Smoothness