Metric and $\omega^*$-Differentiability of Pointwise Lipschitz Mappings

UDC 517.5 Let ( 𝕏 , ‖ . ‖ 𝕏 ) denote a complex Banach space and L ( 𝕏 ) = B C ( ℝ → 𝕏 ) be the set of all 𝕏 -valued bounded continuous functions f : ℝ → 𝕏 . For f ∈ L ( 𝕏 ) we define ‖ f ‖ L ( 𝕏 ) = sup { ‖ f ( x ) ‖ 𝕏 : x ∈ ℝ } . Then ( L ( 𝕏 ) , ‖ . ‖ L ( 𝕏 ) ) itself is a Banach space. The Beurling spectrum S p e c ( f ) of a function f ∈ L ( 𝕏 ) is defined by S p e c ( f ) = { ζ ∈ ℝ : ∀ ϵ > 0 ∃ φ ∈ 𝒮 ( ℝ ) : supp φ ^ ⊂ ( ζ - ϵ , ζ + ϵ ) , φ * f ≢ 0 } . } . We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces: Let f ∈ L ( 𝕏 ) and K be an arbitrary compact set in ℝ . Then Spec ( f ) ⊂ K if and only if for any τ > 0 there exists a constant C τ < ∞ such that < b r > ‖ P ( D ) f ‖ L ( 𝕏 ) ≤ C τ < b r > ‖ f ‖ L ( 𝕏 ) sup x ∈ K ( τ ) | P ( x ) | < b r > for all polynomials with complex coefficients P ( x ) , where the differential operator P ( D ) is obtained from P ( x ) by substituting x → - i d ⅆ x , d ⅆ x is the usual derivative in L ( 𝕏 ) and K ( τ ) is the τ -neighborhood in ℂ of K . Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts K are also given.

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones.Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.The well-known Kolmogorov theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition adjoins the completeness problem (see the formulation of this theorem below).The purpose of this paper is the following problem: is there an analogue of Kolmogorov’s theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition for functional systems of polynomial functions and functional systems of rational functions?It turns out that the problem posed (an analogue of Kolmogorov’s theorem) has a negative answer for functional systems of polynomial functions with natural and integer coefficients, and for functional systems of polynomial functions with rational and real coefficients and for functional systems of rational functions with rational and real coefficients - the answer is positive. These theorems are the main results of this paper.

Using traditional methods it is possible to prove the Ito formula in a Hilbert space and some Banach spaces with special geometrical properties. The class of such Banach spaces is very narrow-they are subclass of reflexive Banach spaces. Using the definition of a generalized stochastic integral, early we proved the Ito formula in an arbitrary Banach space for the case, when as initial Ito process was the Wiener process. For an arbitrary Banach space and an arbitrary Ito process it is impossible to find the sequence of corresponding step functions with the desired convergence. We consider the space of generalized random processes, introduce general Ito process there and prove in it the Ito formula. Afterward, from the main Ito process in a Banach space we receive the generalized Ito process in the space of generalized random processes and we get the Ito formula in this space. Then we check decompasibilility of the members of the received equality and as they turn out Banach space valued, we get the Ito formula in an arbitrary Banach space. We implemented this approach when the stochastic integral in the Ito process was taken from a Banach space valued non-anticipating random process by the one dimensional Wiener process. In this paper we consider the case, when the stochastic integral is taken from an operator- valued non-anticipating random process by the Wiener process with values in a Banach space.

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.

In 1957, Kolmogorov and Arnold gave a solution to the 13th problem which had been formulated by Hilbert in 1900. Actually, it is known that there exist many open problems which can be derived from the original problem. From the function-theoretic point of view, Hilbert's 13th problem can be exactly characterized as the superposition representability problem for continuous functions of several variables. In this paper, the solution to the superposition representability problem for infinitely differentiable functions of several variables is given.

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.

Here we research the univariate multi-composite sigmoid activated quantitative approximation of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued multi-composite sigmoid activated neural network operators. We perform also the related Banach space valued multi-composite sigmoid activated fractional approximation. These multi-composite sigmoid acti-vated approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative or fractional derivatives. Our operators are deÖned by using a multi-composite density function induced by a gen- eral multi-composite sigmoid function. The approximations are pointwise and with respect to the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. We Önish with a convergence analysis.

We prove that functions with bounded n-variation and n-absolutely continuous functions of n-variables in the sense introduced previously by the author [Fund. Math. 173 (2002) 175–189] are stable under quasiconformal mappings. The class of quasiconformal mappings is the best possible since every homeomorphism which induces a bounded operator between BVn spaces is a quasiconformal mapping.

4. A. Garsia, S. Orey and E. Rodemich, Asymptotic behavior of successive coefficients of some power series, Illinois J. Math. 6 (1962), 620-629. 5. S. Karlin and J. McGregor, Random walks, Illinois J. Math. 3 (1959), 66-81. 6. D. G. Kendall, Unitary dilations of Markov transition operators, and the corresponding integral representation for transition-probability matrices, Harold Cram6r Volume (U. Grenander, Ed.), Almquist and Wiksell, Stockholm, 1960; pp. 139-161. 7. S. Orey, Strong ratio limit property, Bull. Amer. Math. Soc. 67 (1961), 571-574. 8. D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford Ser. (2) 13 (1962), 7-28.

CONTENTS § 1. Introduction § 2. What is entropy and -entropy? § 3. Which are more numerous-differentiable functions or analytic functions? § 4. The number of variables and the massiveness of function spaces § 5. Can one get along without continuous functions of several variables? § 6. A lecture by Kolmogorov

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

On the Structure of Continuous Functions of Several Variables

On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables

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

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.