Polynomial approximation on infinite intervals with weights having inner zeros

We generalize Laguerre weights on R⁺ by multiplying them by translations of finitely many Freud type weights which have singularities, and prove polynomial approximation theorems in the corresponding weighted spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Shape preserving approximation on the real line with exponential weights

The well-known “Bernstein’s weighted problem” deals with the possibility of weighted approximation on the whole real line. In this paper, we show the possibility of k-monotone approximation on the real line with Freud’s weight .     

Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight $u(x)= x^{\gamma}e^{-x^{-\alpha}-x^{\beta}}$ , α>0, β>1 and γ≧0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation.     

Inequalities with exponential weights

Let R=(-∞,∞)$\mathbb{R}=(-\infty ,\infty )$ and let Q∈C²:R→R⁺=[0,∞)$Q\in C^2:\mathbb{R}\to {\mathbb{R}}^{+}=[0,\infty )$ be an even function. Then in this paper we consider the infinite–finite range inequality, an estimate for the Christoffel function, and the Markov–Bernstein inequality with the exponential weights _wρ(x)=|x|^ρe^-Q(x),x∈R$w_{\rho }(x)=|x{|}^{\rho }{\mathrm{e}}^{-Q(x)},x\in \mathbb{R}$.     

Bergman spaces with exponential weights

In this paper we introduce a kind of Bergman space $A_{\phi }^p$ on the unit disk D with exponential weights, which cover those defined by Borichev et al. (2007) [6]. We obtain upper and lower bound estimates on the Bergman kernel. As an application, we discuss the Bergman projection and duality.     

Singular integrals with exponential weights

We study the operators

where is the Hardy-Littlewood maximal function, the Hilbert transform or Carleson operator.

Under suitable conditions on the weight of exponential type, we prove boundedness of from spaces, defined on with respect to the measure to with the same density measure. These operators, that arise in questions of harmonic analysis on noncompact symmetric spaces, are bounded from to if and only if .

     

Sharp Nikolskii inequalities with exponential weights

w _a(x)=exp(–x^a), xR, a0. , N _n (a,p,q) — (2), _n P _nw_ap, CN_n(a,p, q)P_nw_aq. , — , {P _n}, .

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This material is based upon research supported by the National Science Foundation under Grant No. DMS-84-19525, by the United States Information Agency under Senior Research Fulbright Grant No. 85-41612, and by the Hungarian Ministry of Education (first author). The work was started while the second author visited The Ohio State University between 1983 and 1985, and it was completed during the first author's visit to Hungary in 1985.     

Weighted polynomial approximation with Freud weights

We show that ifw(x)=exp(–|x|), then in the case =1 for every continuousf that vanishes outside the support of the corresponding extremal measure there are polynomialsP _n of degree at mostn such thatw ⁿ P _n uniformly tends tof, and this is not true when <1. these=" are=" the=" missing=" cases=" concerning=" approximation=" by=" weighted=" polynomials=" of=" the=">w ⁿ P _n wherew is a Freud weight. Our second theorem shows that even if we are only interested in approximation off on the extremal support, the functionf must still vanish at the endpoints, and we actually determine the (sequence of) largest possible intervals where approximation is possible. We also briefly discuss approximation by weighted polynomials of the formW(a_nx)P _n (x).Communicated by Edward B. Saff.     

Rational approximation with varying weights I

We investigate two problems concerning uniform approximation by weighted rationals {w ⁿr_n～ _∞ ⁿ⁼¹ }, wherer _n=p_n Namely, forw(x):=e ^x we prove that uniform convergence to 1 ofw ⁿr_n is not possible on any interval [0,a] witha>2π. Forw(x):=x ^?, ?>1, we show that uniform convergence to 1 ofw ⁿr_n is not possible on any interval [b, 1] withb<tan ⁴(π(??1)/4?). (The latter result can be interpreted as a rational analogue of results concerning “incomplete polynomials.”) More generally, for α≥0, β≥0, α+β>0, we investigate forw(x)=e ^x andw(x)=x ^?, the possibility of approximation byw ⁿ p_n/q_n～ _∞ ⁿ⁼¹ , where depp _n≤αn, degq _n≤βn. The analysis utilizes potential theoretic methods. These are essentially sharp results though this will not be established in this paper.     

Form-preserving exponential approximation

In this paper we continue our research in constructing linear methods for approximating functions by parabolic splines with equidistant knots, inheriting such properties of approximated functions as monotony and convexity. The obtained results are generalized to the case of nonequidistant knots and to some kinds of exponential splines of the third order.     

Rational exponential approximation with real poles

This paper shows that, in the set of rational functions with real poles there exists a best minimax approximation to the exponential function over the non-negative real axis. This minimax approximation has an equal-ripple property similar to the classical Chebyshev approximation and, under certain conditions, it has a form that could be gainfully exploited in the numerical solutions of heat-conduction type problems.     

On weighted polynomial approximation with monotone weights

We construct an even weight monotone on the right half line such that the logarithmic integral of the largest -convex minorant of converges and the polynomials are dense in .     

Weighted exponential polynomial approximation

A necessary and sufficient condition for completeness of systems of exponentials with a weight in Lp is established and a quantitative relation between the weight and the system of exponential in Lp is obtained by using a generalization of Malliavin's uniqueness theorem about Watson's problem.