arXiv subjects
Publications and source records attributed to Vugar Ismailov.
In this paper, we show that the pair of classes of locally Hölder continuous functions (considered on $\mathbb{R}$ and $\mathbb{R}^{2}$, respectively) has the double difference property.
In this paper we prove that if a multivariate function of a certain smoothness class is represented by a sum of $k$ arbitrarily behaved ridge functions, then it can be represented by a sum of $k$ ridge functions of the same smoothness class and a polynomial of degree at most $k-1$. This solves the problem posed by A. Pinkus in his monograph "Ridge Functions" up to a multivariate polynomial.
We consider the approximation of a continuous function, defined on a compact set of the $d$-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best approximation. The result resembles the classical Chebyshev equioscillation theorem for polynomial approximation.
We consider the problem of representation of a bivariate function by sums of ridge functions. We show that if a function of a certain smoothness class is represented by a sum of finitely many, arbitrarily behaved ridge functions, then it can also be represented by a sum of ridge functions of the same smoothness class. As an example, this result is applied to a homogeneous constant coefficient partial differential equation.
In a number of papers, Y. Sternfeld investigated the problems of representation of continuous and bounded functions by linear superpositions. In particular, he proved that if such representation holds for continuous functions, then it holds for bounded functions. We consider the same problem without involving any topology and establish a rather practical necessary and sufficient condition for representability of an arbitrary function by linear superpositions. In particular, we show that if some representation by linear superpositions holds for continuous functions, then it holds for all functions. This will lead us to the analogue of the well-known Kolmogorov superposition theorem for multivariate functions on the $d$-dimensional unit cube.
We consider the problem of the representation of real continuous functions by linear superpositions $\sum_{i=1}^{k}g_{i}\circ p_{i}$ with continuous $g_{i}$ and $p_{i}$. This problem was considered by many authors. But complete, and at the same time explicit and practical solutions to the problem was given only for the case $k=2$. For $k>2$, a rather practical sufficient condition for the representation can be found in Sternfeld [17] and Sproston, Strauss [16]. In this short note, we give a necessary condition of such kind for the representability of continuous functions.
Let $X_{1},...,X_{n}$ be compact spaces and $X=X_{1}\times ... \times X_{n}.$ Consider the approximation of a function $f\in C(X)$ by sums $g_{1}(x_{1})+... g_{n}(x_{n}),$ where $g_{i}\in C(X_{i}),$ $i=1,...,n.$ In [8], M.Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of $X$, called "projection cycles". However, his proof had a gap, which was pointed out by Marshall and O'Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb's formula holds in a stronger form.
Using two results of Garkavi, Medvedev and Khavinson, we give sufficient conditions for proximinality of sums of two ridge functions with bounded and continuous summands in the spaces of bounded and continuous multivariate functions respectively. In the first case, we give an example which shows that the corresponding sufficient condition cannot be made weaker for some subsets of $\mathbb{R}^{n}$. In the second case, we obtain also a necessary condition for proximinality. All the results are furnished with plenty of examples. The results, examples and following discussions naturally lead us to a conjecture on the proximinality of the considered class of ridge functions. The main purpose of the paper is to draw readers' attention to this conjecture.
We characterize the best $L_{2}$ approximation to a multivariate function by linear combinations of ridge functions multiplied by some fixed weight functions. In the special case when the weight functions are constants, we propose explicit formulas for both the best approximation and approximation error.