The classical theorem of K. Weierstrass about polynomial approximation says that any function g continuous on a compact interval I, and so bounded and uniformly continuous on I, can be approximated arbitrarily closely on I by polyomials. However, for any given non-constant function g that is continuous and bounded on the real axis, and a positive number ε, we cannot find a polynomial p such that |g(x)−p(x)| < ε for all real x. We will see that entire functions of exponential type bounded on the real axis constitute the smallest class of entire functions which, for any g bounded and uniformly continuous on R and any ε > 0, contains a function f such that max{|g(x)− f(x)| : −∞ < x < ∞} < ε. Problems of best approximation by polynomials whose degree does not exceed a fixed given integer n were concidered by P. L. Chebyshev much before Weierstrass proved his theorem. We shall discuss analogous questions for best approximation by entire functions of exponential type. In particular, we shall prove the existence of an entire function f∗ of exponential type τ minimizing the quantity sup{|g(x) − f(x)| : −∞ < x < ∞} as f varies in the class of all entire functions of exponential type τ . There is no loss of generality in supposing that the interval I in the theorem of Weierstrass is the unit interval [−1 , 1]. A theorem of L. Féjer says that if H2n−1 is the polynomial of degree 2n − 1, which

The paper contains results on best approximation by logarithmically concave classes of functions. For example, we prove the following: Let P-c denote the class of real polynomials, having -1 and 1 as consecutive zeros, and whose zeros z(k) = x(k)+iy(k), i(2) = -1, satisfy the inequality vertical bar y(k)vertical bar <= vertical bar x(k)vertical bar-1. Let i(x) = 1, x is an element of [-1, 1] be the unit function on the interval [-1, 1] and 1 <= p < infinity. Then, there exists a unique constant c(p) such that inf(q is an element of Pc) integral(1)(-1) vertical bar i(x) - q(x)vertical bar(p) dx = integral(1)(-1) vertical bar i(x) - c(p)(1 - x(2))vertical bar p dx. The exact values of the best approximation are found in the particular cases p = 1 and p = 2.

In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1approximants are characterized as transfinite interpolants on canonical sets. The notion of a HaarChebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.

In mathematics, the term approximation usually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are called canonical sets of best approximation. The present paper summarizes results on canonical sets of best L1-approximation with emphasis on multivariate interpolation and best L1-approximation by blending functions. The best L1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariate L1-approximation by sums of univariate functions. Explicit constructions of best one-sided L1-approximants give rise to well-known and new inequalities.

We discuss error representations for Hermite-Lagrange trigonometric interpolation introduced in Dryanov and Petrov (Interpolation and L 1-approximation by trigonometric polynomials and blending functions, J. Approx. Theory 164, 1049–1064 (2012)) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite-Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with prescribed order.

We present results on interpolation and L-1-approximation of periodic functions by trigonometric polynomials and trigonometric blending functions. In Section 1, we obtain an error-representation formula for Hermite-Lagrange interpolation by trigonometric polynomials in terms of the differential operator D(2n+1) := D Pi(n)(K=1) (D-2 + k(2)). In Sections 2 and 3, we establish canonical set characterization of the best and best one-sided trigonometric L-1-approximants under some restrictions. In Section 4, we obtain an error-representation formula for multivariate Hermite-Lagrange transfinite interpolation by trigonometric blending functions that form the kernel of the differential operator D-theta((2m+1)) D-eta((2n+1)). In Section 5, we give explicit constructions of the best trigonometric blending L-1-approximants to multivariate periodic functions in terms of Hermite Lagrange transfinite interpolation on canonical sets. Our results on best and best one-sided L-1-approximation reveal the close relationship between interpolation and best L-1-approximation (see e.g. Pinkus (1989) [15]). The non-linear problem of best Li-approximation becomes a linear interpolation problem on certain convexity functional cones. The interpolation point set of the interpolants that are best L-1-approximants does not depend on the function to be approximated. For that reason, such an interpolation set is called canonical set of best L-1-approximation. In Section 6, we construct one-sided transfinite trigonometric blending interpolants to multivariate periodic functions. Then, we show that the best one-sided trigonometric blending L-1-approximants to multivariate periodic functions are not transfinite trigonometric blending interpolants on interpolation sets consisting of vertical and horizontal line segments. (C) 2012 Elsevier Inc. All rights reserved.

Let P n denote the linear space of polynomials p ( z :=Σ k =0 n a k ( p ) z k of degree ≦ n with complex coefficients and let | p | [−1,1] : = max x ∈[−1,1] | p ( x )| be the uniform norm of a polynomial p over the unit interval [−1, 1]. Let t n ∈ P n be the n th Chebyshev polynomial. The inequality | p |_[ - 1,1]/| a_n (p)|≧| t_n |_[ - 1,1]/| a_n (t_n )|,p ∈ P_n due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t n in P n . The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.

In Section 1 we present results on interpolating recovery of entire functions of exponential type having a polynomial asymptotic on the real line. The interpolating formulas obtained extend the Shannon-Kotelnikov sampling formula for bandlimited signals to digitizing of signals having a polynomial time asymptotic and bounded highest frequency. The recovery formulas can be useful to accelerate the convergence of sampling series even in the Shannon-Kotelnikov classical case of time-bounded signals. By blending interpolants we obtain an interpolating formula for recovery of entire functions of exponential type without the restriction of a polynomial asymptotic behavior on a line, i.e., beyond the Paley-Wiener-Schwartz space. In Section 2, by using the interpolating formulas from Section 1, we construct oscillating Chebyshev entire functions of exponential type. They serve as a testing function to study the distribution of the zeros of real entire functions of exponential type. The results on zero distribution of an entire function of exponential type are sharp with respect to the prescribed asymptotic and generalize a theorem by Duffin and Schaeffer. In Section 3 we apply the results on zero distribution of entire functions from Section 2 to conclude one-sided local geometric behavior for real entire functions of exponential type. We give a precise version and extend a result by Hörmander on local geometric behavior of entire functions of exponential type.

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L p -metric is ε −1/2, ε→0+.

Let D denote the unit disc of the complex plane and P-n the class of polynomials of degree at most n with complex coefficients. We prove that[GRAPHICS]where p(0) := p belongs to P-n and fork >= 0, p(k+1)(z) := zp(k)'(z). We also obtain a new proof of a well-known inequality of Duffin and Schaeffer and sharpenings of some other classical inequalities.

The following Bernstein inequality[GRAPHICS]valid for all complex polynomials p of degree n, has been extended by Ruscheweyh to[GRAPHICS]We prove in this note that two other Bernstein, inequalities, i.e., or[GRAPHICS]where t(theta) is a complex trigonometric polynomial of degree n do not admit similar extensions. In addition we obtain a new proof of Marcel Riesz interpolation formula.

Let D be the unit disk in the complex plane C. We prove that for any polynomial p of degree at most nmaxz∈∂Dp(z)-p(z¯)z-z¯⩽nmax0⩽j⩽npeijπ/n+pe-ijπ/n2,where ∂D denotes the boundary of D. We show how this result is related to classical inequalities of Bernstein and Markov and to more recent results due to Duffin and Schaeffer.

In this paper we study the local behaviour of a trigonometric polynomial t(theta) := Sigma(nu=-n)(n) alpha(nu) e(inutheta) around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in [0, 2pi). The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form Sigma(nu=0)(n) alpha(nu) x(nu) are also proved.

Let D be the unit disk in the complex plane ℂ and ‖ p‖:=max_z ∈∂ D| p(z)| , where p(z)=∑_k=0^na_k(p)z^k is a polynomial of degree at most n and a k ( p ) ∈ ℂ;. The following sharpening of Bernstein’s inequality ‖ p^'‖+2n n+2| a_0(p)|≤ n ‖ p‖ has been proved by Ruscheweyh. Our main contribution concerns the case of equality which has remained unsolved since 1982. We prove another inequality of Bernstein type that leads to an improvement of the upper bound for ‖ p^'‖ under some additional condition.

The Bernstein inequality max |z|≤1 |p′(z)| ≤ nmax |z|≤1 |p(z)|, valid for all polynomials p with complex coefficients, has been extended by Ruscheweyh to max |z|≤1 |p′(z)| ≤ nmax |z|≤1 |p(z)| − 2n n+ 2 |p(0)|, n ≥ 2. We prove in this note that other Bernstein inequalities, i.e., max −1≤x≤1 ∣∣∣√1− x2p′(x)∣∣∣ ≤ n max −1≤x≤1 |p(x)|. or max 0≤θ≤2π |t′(θ)| ≤ n max 0≤θ≤2π |t(θ)|, where t(θ) is an arbitrary trigonometric polynomial with complex coefficients, do not admit similar extensions. We also discuss other improvements of known Bernstein-type inequalities. 2000 Mathematical Subject Classification. Primary: 41A17 Introduction Let Pn denote the set of polynomials p(z) := ∑n k=0 ak(p)z k with complex coefficients endowed with the norm |p|D := sup z∈∂D |p(z)| where D := {z | |z| < 1} is the unit disc of the complex plane. Let Tn denote the set of trigonometric polynomials t(θ) := ∑n k=−n ak(t)e ikθ with complex coefficients endowed with the norm |t|R := sup θ∈R |t(θ)|. We shall also consider the class P̃n of polynomials

Let f epsilon C-2,C-2([-1, 1](2)) be a real function satisfying partial derivative(4)f/partial derivativex(2) partial derivativey(2) greater than or equal to 0 on [-1, 1](2). We study the problem of best one-sided L-1-approximation to f from the linear space {h epsilon C-2,C-2([-1. 1](2)) :partial derivative(4) h/partial derivativex(2) partial derivativey(2) = 0} of all blending functions of order (2, 2). The unique best one-sided L-1-approximant to f from above is characterized by transfinite Hermite interpolation on the canonical grid {(x, y) epsilon [-1, 1](2): \x\ = \y\}. For f even with respect to one of its variables we characterize the unique best one-sided L-1-approximant to f from below by transfinite Hermite interpolation on the canonical grid {(x. y) epsilon [-1, 1](2) : \x\ = \y\}. There is no canonical grid for the entire cone class of functions f with partial derivative(4)f/partial derivativex(2) partial derivativey(2) greater than or equal to 0 on [-1, 1](2) when we approximate from below. The best one-sided L-1-approximant from above has the smoothness of f. The best one-sided L-1-approximant to f from below is a blending-spline function with two line segment knots {(x, 0) : -1 less than or equal to x less than or equal to 1} and {(0, y): -1 less than or equal to y less than or equal to 1}; i.e., the best one-sided approximation to f from below possesses a saturation effect with respect to the smoothness off. (C) 2002 Elsevier Science (USA).

Let f epsilon C-2,C-1 ([-1, 1](2)) be a real function satisfying partial derivative(3) f/partial derivativex(2) partial derivativey greater than or equal to 0 on [-1, 1](2). We prove existence and uniqueness of the best one-sided L-1-approximant h* to f from above (resp. h(*) from below) with respect to the infinite dimensional linear space of all B-2,B-1-blending functions:B-2,B-1 := {g epsilon C-2,C-1 ([-1, 1](2)) : partial derivative(3) g/partial derivativex(2) partial derivativey = 0}.The unique best one-sided approximants h* (resp. h(*)) are characterized by Hermite type interpolation conditions with respect to canonical point sets V* := {(x, y) epsilon [-1, 1](2) : y = 2\x\ - 1} resp. Lambda(*) := {(x, y) epsilon [-1, 1](2) : y = -2\x\ + 1}.

In the present paper we provide a multivariate generalization of the Euler-Maclaurin formula which is based on an appropriate multivariate extension of the Bernoulli functions.