Pointwise estimates for monotone polynomial approximation

We prove that iff is increasing on [?1,1], then for eachn=1,2,... there is an increasing algebraic polynomialP _n of degreen such that |f(x)?P _n (x)|≤cω ₂(f,√1?x ²/n), whereω ₂ is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.     

Uniform estimates for monotonic polynomial approximation

The uniform estimate is established for a monotone polynomial approximation of functions whose smoothness decreases at the ends of a segment.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 38–43, January, 1993.     

Uniform estimates for polynomial approximation in domains with corners

Let be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γ_j, such that {z_j}Γ_j-1∩Γ_j≠, j=1,…,l, where Γ₀Γ_l. Denote by α_jπ, 0<α_j2, the angles at z_j's between the curves Γ_j-1 and Γ_j, exterior with respect to G. Let Φ be a conformal mapping of the exterior of onto the exterior of the unit disk, normed by Φ^′(∞)>0. We assume that there is a neighborhood U of , such that , where

z≠z_j if α_j1. Set g_Gsup{|g(z)|:zG}. Then we prove Theorem. Let and 0βr. If a function f is analytic in G and f^(r)β_G<+∞, then for each nlr there is an algebraic polynomial P_n of degree <n, such that

     

Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials

Let Δ ^q be the set of functionsf for which theqth difference, is nonnegative on the interval [? 1,1],P _n is the set of algebraic polynomials of degree not exceedingn, τ _k (f, δ) _p is the averaged Sendov-Popov modulus of smoothness in theL _p [?1,1] metric for 1≦p≦∞, ω _k (f, δ) and $\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } ,$ , are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionf∈C[?1,1]?Δ² we construct a polynomialp _n ∈P _n ?Δ² such that $$\begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in [ - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered}$$ As a consequence, for a functionf∈C ²[?1,1]?Δ³ a polynomialp _n ^* ∈P _n ?Δ³ exists such that $$\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ),$$ wheren≥2 andC is an absolute constant.     

Pointwise error estimates for interpolation

Pointwise error estimates are obtained for polynomial interpolants in the roots and extrema of the Chebyshev polynomials of the first kind. These estimates are analogous to those derived by Henrici [2] for trigonometric polynomial interpolants.     

Pointwise gradient estimates

We survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear, possibly degenerate, elliptic and parabolic equations.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

Pointwise simultaneous approximation for Baskakov quasi-interpolants

Recently P. Mache and M. W. Müller introduced the Baskakov quasi-interpolants and obtained an approximation equivalence theorem. In this paper we consider simultaneous approximation equivalence theorem for Baskakov quasi-interpolants.     

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.     

Pointwise estimates for B-spline gram matrix inverses

We present a new method to prove a certain geometric-decay inequality for entries of inverses of B-spline Gram matrices, which is given in [1].     

Pointwise estimates for marginals of convex bodies

We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, log-concave probability measure on Rn. For a typical subspace E⊂Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c>0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.     

Pointwise estimates for some kernels on compact Lie groups

Si ottengono delle stime per il nucleo di Gauss-Weierstrass su gruppi di Lie compatti e se ne indicano alcune applicazioni.     

Pointwise estimates of potentials for higher-order capacities

In a domain D=Ω\E∈ R ⁿ , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W _p ^m (D)?W _q ¹ (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)?kf(x)∈W _p ^m (D)?W _q ¹ (D), where k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.     

Pointwise estimates for the maximal multilinear singular integral operators

Two pointwise estimates relating the maximal multilinear singular integral operators and some classical maximal operators are established. These pointwise estimates imply the rearrangement estimate and the BLO(Rn) estimate for the maximal multilinear singular integral operators.     

Pointwise estimation of comonotone approximation

We prove that, for a continuous functionf(x) defined on the interval [–1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomialsP _n (x) with the same local properties of monotonicity as the functionf(x) and such that ¦f(x)–P _n (x) ¦C₂(f;n^–2+n ^–11–x ²), whereC is a constant that depends on the length of the smallest interval.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1467–1472, November, 1994.The author is grateful to Prof. I. A. Shevchuk for his permanent attention to the work.     

Pointwise error estimates for trigonometric interpolation and conjugation

An estimate due to Gaier [2] for the error committed in replacing a periodic function f by an interpolating trigonometric polynomial is sharpened in such a way that the estimate makes evident the interpolating property of the polynomial. A similar improvement is given for Gaier's estimate of the difference between the conjugate of f and the conjugate trigonometric polynomial.