Continuous Functions Of Several Variables Research Articles

Divergence everywhere of the Fourier series of continuous functions of several variables

The Fourier series of a function of real variables is said to be -convergent at a point for if there exists the limit over all indices such that for all and . An example of a continuous function of variables with modulus of continuity is constructed such that the Fourier series of with respect to the trigonometric system -diverges everywhere for an arbitrary fixed .

Fast sigmoidal networks via spiking neurons.

We show that networks of relatively realistic mathematical models for biological neurons in principle can simulate arbitrary feedforward sigmoidal neural nets in a way that has previously not been considered. This new approach is based on temporal coding by single spikes (respectively by the timing of synchronous firing in pools of neurons) rather than on the traditional interpretation of analog variables in terms of firing rates. The resulting new simulation is substantially faster and hence more consistent with experimental results about the maximal speed of information processing in cortical neural systems. As a consequence we can show that networks of noisy spiking neurons are "universal approximators" in the sense that they can approximate with regard to temporal coding any given continuous function of several variables. This result holds for a fairly large class of schemes for coding analog variables by firing times of spiking neurons. This new proposal for the possible organization of computations in networks of spiking neurons systems has some interesting consequences for the type of learning rules that would be needed to explain the self-organization of such networks. Finally, the fast and noise-robust implementation of sigmoidal neural nets by temporal coding points to possible new ways of implementing feedforward and recurrent sigmoidal neural nets with pulse stream VLSI.

On a Class of Lipschitz Continuous Functions of Several Variables

On a Class of Lipschitz Continuous Functions of Several Variables

On a class of Lipschitz continuous functions of several variables

We establish an estimate via initial values for functions in a class of Lipschitz continuous functions of several variables. This estimate can be used to investigate the uniqueness of quasi-classical solutions of Cauchy problems for first-order nonlinear partial differential equations (PDEs). Particularly, we give an answer to an open problem posed by S. N. Kružkov.

On the radon-nikodym derivatives of nonfredholm linear transformations in the space of continuous functions of several variables

In the space of continuous functions of m variables we study the transformation of a generalized Wiener measure under a linear change of variable described by an integral equation that is transgressive in the sense of Romanovskii.

Approximation from the topological viewpoint

Let X be a metric space. A family H of continuous functions of several variables of X with values in X is said to be generating if, whenever A ⊂ C( K, X) separates points and H operates on A, then A is dense in C( K, X). (For example, the family H = { x + y, xy, constants} in C( R 2, R) is generating (for R) by the Stone-Weierstrass theorem.) We identify metric spaces which admit generating families (not all do), and among those, we search for spaces X that admit generating families in C( X 2, X)—such as R. (This may be considered a topological version of Hilbert's 13th problem.) Once we know this, we try to identify some (small) generating families in C( X 2, X). (This is done in particular when X = R.) As a fringe benefit we obtain a “topological” proof of the Stone-Weierstrass theorem.

On the unicity of bestL 1-approximation by polynomials of several variables

In the present paper we shall consider certain problems of best mean approximation for functions of several variables. It is wellknown that there is an essential difficulty in the extension of the theory of best Chebyshev approximation to the multivariate case. The basic problem here is the absence of Haar subspaces of functions of more than one variable. However, this fact seems not to be an insuperable obstacle in developing the theory of best Lt-approximation of functions of several variables. Some first steps in this direction have been made in [1]. In particular, it was shown in [1] that the L~-approximation of continuous functions of two variables by the tensor product of a two-dimensional Haar subspaces and an arbitrary Haar subspace is unique. This result gives some hope that the general problem of unicity of best Ll-approximation of continuous functions of several variables by tensor products ofHaar subspaces can be solved positively. Nevertheless, this problem is still unsolved even in the case of ordinary polynomials. In the present paper we continue the study of uniqueness of multivariate L~-approximation presenting another approach to this problem. We shall consider approximation by arbitrary multivariate polynomials but restrict ourselves only to approximating rational functions. The main advantage of this approach is that it includes the study of the unicity of polynomials of several variables with fixed leading coefficients which have minimal Ll-deviation from zero. In case of Chebyshev approximation this problem was considered in [2], [6] and [7]. The results of our paper imply the unicity of the polynomials of least Ll-deviation having fixed leading coefficients. Thus we show that the solution of the Korkin--Zolotarjov type extremal problem is unique in the multivariate case, too. For rectangular regions this unique solution is given by the product of Chebyshev polynomials of second kind.

Der Kolmogoroff'sche Darstellungsatz bei halbstetigen Funktionen

In the present article the following problem is studied: Can you transfer the theorem of KOLMOGOROFF about the representation of continuous functions of several variables to semicontinuous functions? It is proved that this is not possible.

Some results on best L1-approximation of continuous functions

The chief purpose of this paper is to study the question of uniqueness of best L1-approximations for weighted approximation of continuous vector-valued functions and continuous functions of several variables by finite dimensional subspaces G. A uniqueness result for special subspaces G of the type G=G1, ×…×Gm is given. Moreover It is studied whether a sufficient condition for uniqueness introduced by DeVore and Strauss and in a more general context by Kroo is also necessary or not and a class of finite dimensional subspaces G for which a positive answer can be given is presented. Finally it is shown that subspaces of linear spline functions of two variables fail to have that sufficient property.

Transformation of repeated Wiener integrals in a space of continuous functions of several variables under a nonlinear change of variables

Transformation of repeated Wiener integrals in a space of continuous functions of several variables under a nonlinear change of variables

Representations of Continuous Functions of Several Variables

Representations of Continuous Functions of Several Variables

The area and variation of linearly continuous functions

area is equal to the Hausdorff 2-dimensional measure of the part of the graph. A similar result is valid for linearly continuous functions of several variables (3.6). In this paper, a co-area formula is established (3.7) which relates the integral of the Hausdorff n — 1 dimensional measure of the slices by hyperplanes of the essential portion of the graph to the total variation of the gradient measure. In other situations, this has been shown to be a very useful tool, [F3], [GE], [ZI], [Z2]. Our results have also been established independently by Herbert Federer by different methods. His results will be incorporated in his forthcoming book Geometric measure theory. 2. Notation and preliminaries. Throughout, x=(xi, • • • , x„) will be an arbitrary point of Euclidean «-space En. Ln will denote «-dimensional Lebesgue measure and for k^n, 77* and Ik will stand for kdimensional Hausdorff measure and k-dimensional integralgeometric measure respectively. In particular, H°iA) is the number of points in the set A. Recall that 77* and 7* agree on the (77*, k) rectifiable sets of finite 77* measure, cf. [F2, Theorem 9.7]. Gn is the orthogonal group on En and „((?„) = 1. With RCGn and wCEn-i associate the line \(R, w) given by

On the Structure of Representations of Continuous Functions of Several Variables as Finite Sums of Continuous Functions of One Variable

On the Structure of Representations of Continuous Functions of Several Variables as Finite Sums of Continuous Functions of One Variable

On the structure of representations of continuous functions of several variables as finite sums of continuous functions of one variable

On the structure of representations of continuous functions of several variables as finite sums of continuous functions of one variable

A Representation Theorem for Continuous Functions of Several Variables

A Representation Theorem for Continuous Functions of Several Variables

On the Structure of Continuous Functions of Several Variables

On the Structure of Continuous Functions of Several Variables

A representation theorem for continuous functions of several variables

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On the structure of continuous functions of several variables

On the structure of continuous functions of several variables