Classical Bernstein Theorem Research Articles

Polynomial approximation on a compact subset of the real line

We prove an analogue of the classical Bernstein theorem concerning the rate of polynomial approximation of piecewise analytic functions on a compact subset of the real line.

Bernstein-type theorems for spacelike stationary graphs in Minkowski spaces

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean space and Calabi's theorem for spacelike maximal surfaces in 3-dimensional Minkowski space.

A Bernstein problem in warped products

Uniqueness and non-existence of entire solutions to the minimal surface equation in warped products R 2 ×f R are provided. As a consequence of our results, the classical Bernstein's Theorem is extended.

An extension to Bernstein theorem on minimal surface equation

By using the property of Monge-Ampere equations, we give an asymptotic behavior at infinity of solutions of the minimal surface equations outside a bounded plane set, which extends the classical Bernstein theorem.

Interpolation and approximation in Taylor spaces

The univariate Taylor formula without remainder allows to reproduce a function completely from certain derivative values. Thus one can look for Hilbert spaces in which the Taylor formula acts as a reproduction formula. It turns out that there are many Hilbert spaces which allow this, and they should be called Taylor spaces. They have certain reproducing kernels which are either polynomials or power series with nonnegative coefficients. Consequently, Taylor spaces can be spanned by translates of various classical special functions such as exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions. Since the theory of kernel-based interpolation and approximation is well-established, this leads to a variety of results. In particular, interpolation by shifted exponentials, rationals, hyperbolic cosines, logarithms, and Bessel functions provides exponentially convergent approximations to analytic functions, generalizing the classical Bernstein theorem for polynomial approximation to analytic functions. Finally, we prove sampling inequalities in Taylor spaces that allow to derive similar convergence rates for non-interpolatory approximations.

On Approximation of Continuous Functions by Entire Functions on Subsets of the Real Line

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. Approximation of continuous bounded functions by entire functions of exponential type on unbounded closed proper subsets of the real line is studied.

On Approximation by Entire Functions on an Unbounded Quasi-Smooth Curve

We generalize a classical Bernstein Theorem about approximation of functions on the real line by entire functions of the exponential type, i.e. for any function f continuous and bounded on an unbounded quasi-smooth (in the sense of Lavrentiev) curve L in the complex plane we construct entire functions of exponential type σ > 0 which in some sense converge optimally to f on L as σ → ∞.

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

We discuss Totik’s extension of the classical Bernstein theorem on polynomial approximation of piecewise analytic functions on a closed interval. The error of the best uniform approximation of such functions on a compact subset of the real line is studied.

Power Series Kernels

We introduce a class of analytic positive definite multivariate kernels which includes infinite dot product kernels as sometimes used in machine learning, certain new nonlinearly factorizable kernels, and a kernel which is closely related to the Gaussian. Each such kernel reproduces in a certain “native” Hilbert space of multivariate analytic functions. If functions from this space are interpolated in scattered locations by translates of the kernel, we prove spectral convergence rates of the interpolants and all derivatives. By truncation of the power series of the kernel-based interpolants, we constructively generalize the classical Bernstein theorem concerning polynomial approximation of analytic functions to the multivariate case. An application to machine learning algorithms is presented.

A Bernstein-type theorem for Riemannian manifolds with a Killing field

The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space \(\mathbb{R}^3\) that can be written as the graph of a function on \(\mathbb{R}^2\) must be a plane. In this paper, we extend Bernstein’s result to complete minimal surfaces in (may be non-complete) ambient spaces of non-negative Ricci curvature carrying a Killing field. This is done under the assumption that the sign of the angle function between a global Gauss map and the Killing field remains unchanged along the surface. In fact, our main result only requires the presence of a homothetic Killing field.

Amoebas, Monge-Ampàre measures, and triangulations of the Newton polytope

The amoeba of a holomorphic function f is, by definition, the image in R n of the zero locus of f under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function N f with the property that its Monge-Ampére mass is concentrated to the amoeba of f We obtain results of two kinds; by approximating N f with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of f; by computing the Monge-Ampére measure, we find sharp bounds for the area of amoebas in ℝ . We also consider systems of functions f 1 ,…, f n and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.

A Bernstein type theorem for minimal volume preserving maps

We show that any minimal volume preserving map from the Euclidean plane into itself is a linear diffeomorphism. We derive this from a similar result on minimal diffeomorphisms. We also show that the classical Bernstein theorem on minimal graphs is a corollary of our result.

A duality result between the minimal surface equation and the maximal surface equation

In this note we show how classical Bernstein's theorem on minimal surfaces in the Euclidean space can be seen as a consequence of Calabi-Bernstein's theorem on maximal surfaces in the Lorentz-Minkowski space (and viceversa). This follows from a simple but nice duality between solutions to their corresponding differential equations.

The value distribution of the hyperbolic Gauss map

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in H3(−1), and prove that it cannot omit more than four points unless the surface is a horosphere. 0. Introduction In minimal surface theory, the value distribution of the Gauss map has been studied for a long time. An early result is the classical Bernstein Theorem: Any complete minimal graph in R is a plane. Replacing the graph by some geometric conditions, R. Osserman [3] showed that if the complete minimal surface is nonflat, then the Gauss map cannot omit a set of positive logarithmic capacity, which answered a conjecture of Nirenberg. Afterwards, it was generalized by F. Xavier [8] that the Gauss map of such a surface can omit at most six points. In 1989, H. Fujimoto [2] proved the best result finally that the Gauss map of the nonflat complete minimal surface in R can omit at most four points. It is natural to consider the similar properties of minimal surfaces in H. An interesting feature is that there exists a family of absolutely area-minimizing hypersurfaces in H and only one of them is totally geodesic (see [7]). The question seems to be how to raise an adequate Bernstein problem in hyperbolic space. Professor Y. L. Xin pointed out to me that the striking work done by R. Bryant [1] supplies a framework to solve the Bernstein problem in hyperbolic space. In this paper, we shall be concerned with the surfaces in hyperbolic space of constant mean curvature one. We abbreviate “constant mean curvature one” by CMC-1. These surfaces share many properties with minimal surfaces in R. They possess the “Weierstrass representation” in terms of holomorphic data. This formula was discovered by R. Bryant [1]. Many other properties may be found in papers by M. Umhara and K. Yamada ([4], [5]). Here we try to investigate the hyperbolic analogue of the Gauss map. It is a natural question how the values of the hyperbolic Gauss map distribute. Using Bryant’s representation formula we are able to answer this question as follows. Theorem. The hyperbolic Gauss map of nonflat complete CMC-1 surfaces in H3(−1) can omit at most four points. Received by the editors November 1, 1995 and, in revised form, April 2, 1996. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42.

On Bernstein type theorems concerning the growth of derivatives of entire functions

A subspace X of Ll,,(R) which is invariant under all left translation operators Tt, t E R, is called admissible if X is a Banach space satisfying the following properties: (i) If IIfnlIx 0, then there exists a subsequence (nk) such that fink (s) 0 almost everywhere. (ii) The group TX := {Ttlx : t E R} is a bounded strongly continuous group. In this case, let Cx := sup{HITtIx : t E RI. Typical admissible spaces are Co(R), BUC(R) and all spaces LP(R) for 1 0, e-6Itg(t) E L1 (R). With these definitions our main result goes as follows: Theorem 1. If g is an entire function of exponential type r such that its restriction to the real axis, denoted by gR, is subexponential and belongs to some admissible space X, then the derivative g' is also in X. Moreover, Ia + 9' IIX < (Oa2 + r2)1/2* CX 1lg1IIx for each real oz. This result yields as consequences and in a systematic way many new and old Bernstein type inequalities. 1. THE MAIN THEOREMS Recall that an entire function f is called to be of exponential type r if log M(r) lim Sup =7 r-woo r where M(r) := max{lf(z)I: IzI < r} is the maximum modulus of f. The classical Bernstein theorem [Bo, p.206, Theorem 11.1.2] states that if f is an entire function of exponential type r and is bounded on the real axis by M, then If'(t)I < TFM for all t c IR, or equivalently, IIf'IIIo < TIIflIIo for each entire function f of exponential type r. The latter is referred as to Bernstein's inequality. Many extensions are given, see [Bo, Chap. 11], [D-S], [Le, Chap. IV] and [K-S-T]. It is the purpose of Received by the editors August 16, 1995. 1991 Mathematics Subject Classification. Primary 30D20; Secondary 47D03, 47A10.

Bernstein's theorem for compact groups

In this paper we generalize the classical Bernstein theorem concerning the absolute convergence of the Fourier series of Lipschitz functions. More precisely, we consider a group G which is finite dimensional, compact, and separable and has an infinite, closed, totally disconnected, normal subgroup D, such that G D is a Lie group. Using this structure, we define in a natural way the notion of Lipschitz condition, and then prove that a function which satisfies a Lipschitz condition of order greater than ( dim G + 1) 2 belongs to the Fourier algebra of G.