On a smoothness problem in ridge function representation

A representation problem for smooth sums of ridge functions

“…Kuleshov [21] generalized our result [3,Theorem 2.3] to all possible cases of s. That is, he proved that if a function f ∈ C s (R n ), where s ≥ 0, is of the form (1.1) and (k − 1)-tuple of the given set of k directions a i forms a linearly independent system, then there exist g i ∈ C s (R), i = 1, ..., k, such that (1.2) holds (see [21,Theorem 3]). In [2], we reproved this result using completely different ideas.…”

“…Since F ≡ 0 on [0, 1), F (1) = 0 and F ∈ C(1, 2), we obtain that F is continuous on (0, 2). Consider the interval [2,3). For any x ∈ [2, 3) we can write that…”

“…In [3], we gave a partial solution to the above representation problem. Our solution comprises the cases in which s ≥ 1 and k −1 directions of the given k directions are linearly independent.…”

On the numerical solution of identification hyperbolic-parabolic problems with the Neumann boundary condition

“…Partial differential equations with unknown source terms are widely used in mathematical modeling of reallife systems in many different fields of science and engineering. They have been studied extensively by many researchers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein).…”

A note on the equioscillation theorem for best ridge function approximation

“…This theorem characterizes the unique best uniform approximation to a continuous real valued function F (t) by polynomials P (t) of degree at most n, by the oscillating nature of the difference F (t) − P (t). The result says that if such polynomial has the property that for some particular n + 2 points t i in [0, 1] 1] |F (t) − P (t)| , i = 1, ..., n + 2, then P is the best approximation to F on [0, 1]. The monograph of Natanson [30] contains a very rich commentary on this theorem.…”