A Version of Hilbert’s 13th Problem for Infinitely Differentiable Functions
نویسندگان
چکیده
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.
منابع مشابه
Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces
We study an interesting class of Banach function algebras of innitely dierentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of innitely dierentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)...
متن کاملA VARIATIONAL APPROACH TO THE EXISTENCE OF INFINITELY MANY SOLUTIONS FOR DIFFERENCE EQUATIONS
The existence of infinitely many solutions for an anisotropic discrete non-linear problem with variable exponent according to p(k)–Laplacian operator with Dirichlet boundary value condition, under appropriate behaviors of the non-linear term, is investigated. The technical approach is based on a local minimum theorem for differentiable functionals due to Ricceri. We point out a theorem as a spe...
متن کاملThe representability hierarchy and Hilbert’s 13th problem
We propose the representability hierarchy of algebraic functions over C, which relates classic questions like the unsolvability of the quintic by radicals to unresolved questions like Hilbert’s 13th problem in its original algebraic form. We construct the theory of algebraic functions via both superpositions and algebraic geometry and then justify the known positions of the universal algebraic ...
متن کاملRemarks on Lp-approximation of infinitely differentiable multivariate functions
We study the Lp-approximation problem (1 ≤ p < ∞) for infinitely differentiable d-variate functions with respect to the worst case error. In particular, we correct a mistake in the argumentation of Novak and Woźniakowski [2], who showed that the problem is intractable. The main ingredients are arguments from convex geometry, as well as a probabilistic calculation.
متن کاملgH-differentiable of the 2th-order functions interpolating
Fuzzy Hermite interpolation of 5th degree generalizes Lagrange interpolation by fitting a polynomial to a function f that not only interpolates f at each knot but also interpolates two number of consecutive Generalized Hukuhara derivatives of f at each knot. The provided solution for the 5th degree fuzzy Hermite interpolation problem in this paper is based on cardinal basis functions linear com...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010