Classes Of Convolutions Research Articles

Widths and best approximations for classes of convolutions of periodic functions

We establish exact lower bounds for the Kolmogorov widths in the metrics ofC andL for classes of functions with high smoothness; the elements of these classes are sourcewise-representable as convolutions with generating kernels that can increase oscillations. We calculate the exact values of the best approximations of such classes by trigonometric polynomials.

Convolution-Generated Motion as a Link between Cellular Automata and Continuum Pattern Dynamics

Cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical, biological, and ecological systems. However, for patterns in which sharp interfaces form and propagate, automata simulations can exhibit undesirable properties, including spurious anisotropy and poor representation of interface curvature effects. These simulations are also prohibitively slow when high accuracy is required, even in two dimensions. Also, the highly discrete nature of automata models makes theoretical analysis difficult. In this paper, we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata, but based on continuous convolutions rather than discrete sums. These convolution-generated motions naturally achieve the fine-grid limit of the corresponding automata, and they are also well suited to numerical and theoretical analysis. Because of this, the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast Fourier transform techniques, and for a large class of convolutions the limiting interface motion laws can be derived analytically. Thus convolution-generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics. This is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics. In this latter role, it also has a number of benefits over the traditional reaction–diffusion/Ginzburg–Landau continuum PDE models of pattern formation, which yield true moving interfaces only as singular limits. We illustrate the power of this approach with convolution-generated motion models for pattern dynamics in developmental biology and excitable media.

Several statements for convex functions

For the setM of convex-downward functions Ψ (•) vanishing at infinity, we present its decomposition into subsets with respect to the behavior of special characteristics η (Ψ;•) and μ(Ψ;•) of these functions. We study geometric and analytic properties of the elements of the subsets obtained, which are necessary for the investigation of problems of the theory of approximation for classes of convolutions.

Construction of correlation functions in two and three dimensions

AbstractThis article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data‐assimilation applications. A self‐contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical‐analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross‐covariance functions on three‐dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross‐covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross‐covariance functions on S2, referred to as being space‐limited, cannot be obtained using truncated spectral expansions. Compactly supported and space‐limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data‐analysis algorithms.Convolution integrals leading to practical examples of compactly supported covariance and cross‐covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O < di,dj ≦, where |. | denotes Euclidean distance in R3. the parameters di and dj are ‘cut‐off’ distances. Closed‐form expressions are determined for classes of convolution cross‐covariance functions Cij(x,y) := (gi * gj)(x‐y), i ≠ j, and convolution covariance functions Cii(x,y) := (gi * gi)(x‐y), vanishing for |x ‐ y| > di + dj and |x ‐ y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second‐ and third‐order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x ‐ y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th‐order piecewise rational functions, are also constructed. These functions are used to develop space‐limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

On n-widths of some periodic convolution classes in Orlicz spaces

In this paper we extend the some accurate results on n-width of Sobolev class W∞t in Lp spaces to Orlicz spaces, and establish the corresponding results for dual cases.

Approximation of $$\bar \psi $$ -integrals of periodic functions by fourier sums (small smoothness). II

We investigate the rate of convergence of Fourier series on the classes $$L^{\bar \psi } $$ N in the uniform and integral metrics. The results obtained are extended to the case where the classes $$L^{\bar \psi } $$ N are the classes of convolutions of functions from ℜ with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets $$L^{\bar \psi } $$ N, which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.

Approximation of $$\bar \psi - integrals$$ of periodic functions by Fourier sums (small smoothness). Iof periodic functions by Fourier sums (small smoothness). I

We investigate the rate of convergence of Fourier series on the classes \(L^{\bar \psi } \)N in the uniform and integral metrics. The results obtained are extended to the case where the classes \(L^{\bar \psi } \)N are the classes of convolutions of functions from N with kernels with slowly decreasing coefficients. In particular, we obtain asymptotic equalities for the upper bounds of deviations of the Fourier sums on the sets \(L^{\bar \psi } \)N which are solutions of the Kolmogorov-Nikol’skii problem. In addition, we establish an analog of the well-known Lebesgue inequality.

Estimates of the Kolmogorov widths for classes of infinitely differentiable periodic functions

Lower estimates of the Kolmogorov widths are obtained for certain classes of infinitely differentiable periodic functions in the metrics of C and L. For many important cases, these estimates coincide with the values of the best approximations of convolution classes by trigonometric polynomials calculated by Nagy, and, hence, they are exact.

Approximation of classes of convolutions by linear methods of summation of Fourier series

We consider a family of special linear methods of summation of Fourier series and establish exact equalities for the approximation of classes of convolutions with even and odd kernels by polynomials generated by these methods.

The identical equation in 𝜓-products

In Bull. Amer. Math. Soc. 36 (1930), 762–772, R. Vaidyanatha- swamy established a remarkable identity valid for any multiplicative arithmetic function and involving Dirichlet convolution. D. H. Lehmer (Trans. Amer. Math. Soc. 33 (1931), 945–952) introduced a very general class of arithmetical convolutions, called ψ \psi -products, which include the well-known Dirichlet products, Eckford Cohen’s unitary convolutions, and in fact Narkiewicz’s so-called regular A A -convolutions. In this paper, we establish an identical equation valid for multiplicative arithmetic functions and Lehmer’s ψ \psi -convolutions which yields, as special cases, all known identical equations valid for the Dirichlet and unitary convolutions, besides establishing identical equations for several new convolutions.

On fractional integration of generalized functions on a half-line

A new approach to fractional integrals of distributions on a half-line is suggested. The results admit an extension to a large class of Mellin convolutions.

On the best approximation of classes of convolutions of periodic functions by trigonometric polynomials

We present sufficient conditions for kernels to belong to the classN n * . In certain cases, this enables us to find exact values of the best approximations of classes of convolutions by trigonometric polynomials in the metrics ofC andL.

Lower bounds for widths of classes of convolutions of periodic functions in the metrics ofC andL

We give new sufficient conditions for kernels to belong to the set C y,2n introduced by Kushpel'. These conditions give the possibility of extending the set of kernels belonging to C y,2n . On the basis of these results, we obtain lower bounds for the Kolmogorov widths of classes of convolutions with these kernels. We show that these estimates are exact in certain important cases.

LOWER ESTIMATES OF THE WIDTHS OF CLASSES OF FUNCTIONS DEFINED BY A MODULUS OF CONTINUITY

For kernels satisfying the condition introduced by the author, lower bounds are found for the Kolmogorov widths of classes of convolutions , convex, in the uniform metric. In a number of cases these bounds are sharp.

Best one-sided approximation of convolution classes by cardinal splines

Let G be a Polya frequency density and $$\tilde G$$ its periodization. We get the exact value of upper approximation of S1 (G), which is a convolution class with kernel G, by cardinal splines corresponding to G. Similar result has been obtained for periodic convolution class with kernel $$\tilde G$$ as well.

On the cardinal spline interpolation corresponding to infinite order differential operators

This paper discusses some problems on the cardinal spline interpolation corresponding to infinite order differential operators. The remainder formulas and a dual theorem are established for some convolution classes, where the kernels arePF densities. Moreover, the exact error of approximation of a convolution class with interpolation cardinal splines is determined. The exact values of averagen-Kolmogorov widths are obtained for the convolution class.

On approximation of convolution classes

Asymptotic equalities are found for the least upper bounds of the best approximations of some convolution classes with even kernel in the metric of a spaceLp.

On the optimal quadrature least L∞-norm of monosplines with free knots on the real axis

In this paper, the exact solutions of the L ∝ norm on some classes of monosplines with free knots on the real axis are obtained, and the optimal quadrature formula and optimal error on the convolution class taking a PF density as its kernal are given.

Widths of classes of convolutions with Poisson kernel

Widths of classes of convolutions with Poisson kernel

Extremal relations for classes of convolutions

Classes of periodic functions which can be represented as convolutions are found whose approximations by linear operators defined by convolutions in metrics of dual spaces are equal, independent of the choice of approximating operator.