Multivariate Quasi-interpolation Research Articles

Approximation properties of periodic multivariate quasi-interpolation operators

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions φ˜j and trigonometric polynomials φj. The class of such operators includes classical interpolation polynomials (φ˜j is the Dirac delta function), Kantorovich-type operators (φ˜j is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on φ˜j and φj, we obtain upper and lower bound estimates for the Lp-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, K-functionals, and other terms.

Multivariate error function based neural network approximations

Here we present multivariate quantitative approximations of real and complex valued continuous multivariate functions on a box or \(\mathbb{R}^{N},\) \(N\in \mathbb{N}\), by the multivariate quasi-interpolation, Baskakov type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last three types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the Gaussian error special function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Multivariate neural network operators with sigmoidal activation functions

In this paper, we study pointwise and uniform convergence, as well as order of approximation, of a family of linear positive multivariate neural network (NN) operators with sigmoidal activation functions. The order of approximation is studied for functions belonging to suitable Lipschitz classes and using a moment-type approach. The special cases of NN operators, activated by logistic, hyperbolic tangent, and ramp sigmoidal functions are considered. Multivariate NNs approximation finds applications, typically, in neurocomputing processes. Our approach to NN operators allows us to extend previous convergence results and, in some cases, to improve the order of approximation. The case of multivariate quasi-interpolation operators constructed with sigmoidal functions is also considered.

Multivariate sigmoidal neural network approximation

Here we study the multivariate quantitative constructive approximation of real and complex valued continuous multivariate functions on a box or RN, N∈N, by the multivariate quasi-interpolation sigmoidal neural network operators. The "right" operators for our goal are fully and precisely described. This approximation is derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the logarithmic sigmoidal function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Multivariate hyperbolic tangent neural network approximation

Here we study the multivariate quantitative approximation of real and complex valued continuous multivariate functions on a box or RN, N∈N, by the multivariate quasi-interpolation hyperbolic tangent neural network operators. This approximation is derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order partial derivatives. Our multivariate operators are defined by using a multidimensional density function induced by the hyperbolic tangent function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.

Multivariate quasi-interpolation in L p (ℝ d ) with radial basis functions for scattered data

In this paper, quasi-interpolation for scattered data was studied. On the basis of generalized quasi-interpolation for scattered data proposed in [Z.M. Wu and J.P. Liu, Generalized strang-fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), pp. 201–214.], we have developed a new method to construct the kernel in the scheme by the linear combination of the scales, instead of the gridded shifts of the radial basis function. Compared with the kernel proposed in [Z.M. Wu and J.P. Liu, Generalized strang-fix condition for scattered data quasi-interpolation, Adv. Comput. Math. 23 (2005), pp. 201–214.], the new kernel, which is still a radial function, possesses the feature of polynomial reproducing property. This opens a possibility for us to propose a different technique by obtaining a higher approximation order of the convergence.

The Error in Linear Interpolation at the Vertices of a Simplex

A formula for the error in multivariate quasi-interpolation which reproduces the linear polynomials is given. From it sharp pointwise $L_\infty$-bounds for the error in linear interpolation (interpolation by linear polynomials) to function values at the vertices of a simplex are obtained. The corresponding 'envelope theorem' giving the optimal recovery of functions is discussed.

A characterization of multivariate quasi-interpolation formulas and its applications

Let ? be a compactly supported function on ? s andS (?) the linear space withgenerator ?; that is,S (?) is the linear span of the multiinteger translates of ?. It is well known that corresponding to a generator ? there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator ? at the multi-integers ? s facilitates the above study. An algorithm to yield this information for box splines is discussed.