Remez-type Inequalities Research Articles

A sharp Remez type inequalities for the functions with asymmetric restrictions on the oldest derivative

For odd $r\in \mathbb{N}$; $\alpha, \beta >0$; $p\in [1, \infty]$; $\delta \in (0, 2 \pi)$, any $2\pi$-periodic function $x\in L^r_{\infty}(I_{2\pi})$, $I_{2\pi}:=[0, 2\pi]$, and arbitrary measurable set $B \subset I_{2\pi},$ $\mu B \leqslant \delta/\lambda,$ where $\lambda=$ $\left({\left\|\varphi_{r}^{\alpha, \beta}\right\|_{\infty} \left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty}{E^{-1}_0(x)_\infty}\right)^{1/r}$, we obtain sharp Remez type inequality $$E_0(x)_\infty \leqslant \frac{\|\varphi_r^{\alpha, \beta}\|_\infty}{E_0(\varphi_r^{\alpha, \beta})^{\gamma}_{L_p(I_{2\pi} \setminus B_\delta)}} \left\|x \right\|^{\gamma}_{{L_p} \left(I_{2\pi} \setminus B \right)}\left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty^{1-\gamma},$$ where $\gamma=\frac{r}{r+1/p},$ $\varphi_r^{\alpha, \beta}$ is non-symmetric ideal Euler spline of order $r$, $B_\delta:= \left[M- \delta_2, M+ \delta_1 \right]$, $M$ is the point of local maximum of spline $\varphi_r^{\alpha, \beta}$ and $\delta_1 > 0$, $\delta_2 > 0$ are such that $\varphi_r^{\alpha, \beta}(M+ \delta_1) = \varphi_r^{\alpha, \beta}(M- \delta_2), \;\; \delta_1 + \delta_2 = \delta .$In particular, we prove the sharp inequality of Hörmander-Remez type for the norms of intermediate derivatives of the functions $x\in L^r_{\infty}(I_{2\pi})$.

Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.

“Smooth rigidity” and Remez-type inequalities

If a \((d+1)\)-smooth function f(x) on \([-1,1]\), with \(\mathrm{max\,}_{[-1,1]}|f(x)|\ge 1,\) has \(d+1\) or more distinct zeroes on \([-1,1]\), then \(\mathrm{max\,}_{[-1,1]}|f^{(d+1)}(x)|\ge 2^{-d-1}(d+1)!\). This follows from the polynomial interpolation of f at its zeroes, with Lagrange’s remainder formula. This is one of the simplest examples of what we call “smooth rigidity”: certain geometric properties of zero sets of smooth functions f imply explicit lower bounds on the high-order derivatives of f. In dimensions greater than one, the powerful one-dimension tools such as Lagrange’s remainder formula, and divided finite differences, are not directly applicable. Still, the result above implies, via line sections, rather strong restrictions on zeroes of smooth functions of several variables (Yomdin in Proc AMS 90(4):538–542, 1984). In the present paper we study the geometry of zero sets of smooth functions, and significantly extend the results of Yomdin (1984), including into consideration, in particular, finite zero sets (for which the line sections usually do not work). Our main goal is to develop a pure multi-dimensional approach to smooth rigidity, based on polynomial Remez-type inequalities (which compare the maxima of a polynomial on the unit ball, and on its subset). Very informally, one of our main results is that the smooth rigidity of a zeroes set Z is approximately the reciprocal Remez constant of Z.

Sharp Remez-Type Inequalities of Various Metrics with Asymmetric Restrictions Imposed on the Functions

For any p ∈ (0, ∞], ω > 0, β ∈ (0, 2ω), and any measurable set B ⊂ Id ≔ [0, d], μB ≤ β, we deduce a sharp Remez-type inequality $$ {\left\Vert {x}_{\pm}\right\Vert}_{\infty}\le \frac{{\left\Vert {\left(\varphi +c\right)}_{\pm}\right\Vert}_{\infty }}{{\left\Vert \varphi +c\right\Vert}_{L_p\left({I}_{2\upomega}\backslash {B}_y^c\right)}}{\left\Vert x\right\Vert}_{L_p\left({I}_d\backslash B\right)} $$ on the set S𝜑'(𝜔) of d-periodic functions x with zeros and a given sine-shaped 2𝜔-periodic comparison function 𝜔; here, c ∈ [−‖φ‖∞,‖φ‖∞] satisfies the condition $$ {\left\Vert {x}_{+}\right\Vert}_{\infty}\cdot {\left\Vert {x}_{-}\right\Vert}_{\infty}^{-1}={\left\Vert {\left(\varphi +c\right)}_{+}\right\Vert}_{\infty}\cdot {\left\Vert {\left(\varphi +c\right)}_{-}\right\Vert}_{\infty}^{-1}, $$ $$ {B}_y^c:= \left\{t\in \left[0,2\omega \right]:\kern0.5em \left|\varphi (t)+c\right|>y\right\}, $$ and y is such that $$ \mu {B}_y^c=\beta . $$ In particular, inequalities of this type are established on Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient ‖x+‖∞/‖x−‖∞.

Sharp Remez-type Inequalities of Various Metrics in the Classes of Functions with Given Comparison Function

For any p ϵ [1,∞], 𝜔 > 0, β ϵ (0, 2𝜔), and any measurable set B ⊂ I d := [0,d], μB ≤ β, we establish a sharp Remez-type inequality of various metrics $$ {E}_0{(x)}_{\infty}\le \frac{{\left\Vert \varphi \right\Vert}_{\infty }}{E_0{{{\left(\varphi \right)}_L}_p}_{\left({I}_{2\omega}\backslash {B}_1\right)}}{\left\Vert x\right\Vert}_{Lp\;\left({I}_d\backslash B\right)} $$ in the classes S𝜑(ω) of d-periodic (d ≥ 2ω) functions x with a given sine-shaped 2ω-periodic comparison function 𝜑, where B1 := [(ω − β)/2, (ω+β)/2] and E0(f)Lp(G) is the best approximation of the function f by constants in the metric of the space Lp(G). In particular, we prove sharp Remez-type inequalities of various metrics in the Sobolev spaces of differentiable periodic functions. We also obtain inequalities of this type in the spaces of trigonometric polynomials and splines.

Sharp Remez-Type Inequalities of Different Metrics for Differentiable Periodic Functions, Polynomials, and Splines

We prove a sharp Remez-type inequality of different metrics $$ {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left\{\frac{{\left\Vert x\right\Vert}_{L_p\left(\left[0,2\uppi \right]\backslash B\right)}}{{\left\Vert {\varphi}_r\right\Vert}_{L_p\left(\left[0,2\uppi \right]\backslash {B}_1\right)}}\right\}}^{\alpha }{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{1-\alpha },\kern0.5em q>p>0,\kern0.5em \alpha =\left(r+1/q\right)/\left(r+1/p\right), $$ for 2π -periodic functions x ∈ $$ {L}_{\infty}^r $$ satisfying the condition where $$ L{(x)}_p\coloneq \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:\left[a,b\right]\subset \left[0,2\uppi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\} $$ , B ⊂ [0, 2π], μB ≤ β/λ (λ is chosen so that $$ {\left\Vert x\right\Vert}_p={\left\Vert {\varphi}_{\uplambda, r}\right\Vert}_{L_p\left[0,2\uppi /\uplambda \right]} $$ ), φ r is the ideal Euler’s spline of order r, and $$ {B}_1\coloneq \left[\frac{-\uppi -\beta /2}{2},\frac{-\uppi +\beta /2}{2}\right]\cup \left[\frac{\uppi -\beta /2}{2},\frac{\uppi +\beta /2}{2}\right] $$ . As a special case, we establish sharp Remez-type inequalities of different metrics for trigonometric polynomials and polynomial splines satisfying the condition (*).

Sharp Remez-type inequalities of various metrics for the best approximations by a constant

We prove new sharp Remez-type inequalities of various metrics on the sets of functions with a given comparison function.

On sharp Remez-type inequalities of various metrics for periodic functions

We prove new sharp Remez-type inequalities of various metrics for periodic functions, polynomials and splines.

Sharp Remez-Type Inequalities for Differentiable Periodic Functions, Polynomials, and Splines

For any $$ \omega $$ > 0, β $$ \epsilon $$ (0, 2 $$ \omega $$ ), and any measurable set B $$ \epsilon $$ I d := [0,d], μB = β, we obtain the following sharp inequality of the Remez type: on the classes S φ (ω) of functions x with minimal period d(d ≥ 2ω) and a given sine-shaped 2 $$ \omega $$ -periodic comparison function '. In particular, we prove sharp Remez-type inequalities on the Sobolev classes of differentiable periodic functions. We also obtain inequalities of the indicated type in the spaces of trigonometric polynomials and polynomial splines.

NORMING SETS AND RELATED REMEZ-TYPE INEQUALITIES

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’,Comm. Inst. Sci. Kharkov13(1936), 9–95] bounds the maximum of the absolute value of a real polynomial$P$of degree$d$on$[-1,1]$through the maximum of its absolute value on any subset$Z\subset [-1,1]$of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of$n$variables’,Math. USSR Izv.37(1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’,Israel. J. Math.186(2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’,J. Approx. Theory162(2010), 72–93]). Still, given a subset$Z\subset [-1,1]^{n}\subset \mathbb{R}^{n}$, it is not easy to determine whether it is${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming (here${\mathcal{P}}_{d}(\mathbb{R}^{n})$is the space of real polynomials of degree at most$d$on$\mathbb{R}^{n}$), that is, satisfies a Remez-type inequality:$\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$for all$P\in {\mathcal{P}}_{d}(\mathbb{R}^{n})$with$C$independent of$P$. (Although${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming sets are precisely those not contained in any algebraic hypersurface of degree$d$in$\mathbb{R}^{n}$, there are many apparently unrelated reasons for$Z\subset [-1,1]^{n}$to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces$V$of continuous functions on$[-1,1]^{n}$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for$Z$to be$V$-norming, partly known, partly new, restricting ourselves to the simplest nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants$N_{V}(Z)$in the Remez-type inequalities for$V$, as the function of the set$Z$, showing that it is Lipschitz in the Hausdorff metric.

George Lorentz and inequalities in approximation

George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the Introduction. 0. Introduction 1. Bernstein-type inequalities for exponential sums. 2. Remez-type inequalities for exponential sums. 3. Lorentz degree of polynomials. 4. Markovand Bernstein-type inequalities for constrained polynomials. 5. Muntz-type theorems. 6. Remez-type inequalities and Newman’s product problem. 7. Multivariate approximation. 8. Newman’s inequality. 9. Littlewood polynomials. 10. Inequalities for generalized polynomials. 11. Markovand Bernstein-type inequalities for rational functions. 12. Nikolskii-type inequalities for shift-invariant function spaces. 13. Inverse Markovand Bernstein-type inequalities. 14. Ultraflat sequences of unimodular polynomials. 15. Zeros of polynomials with coefficient constraints. 1. Bernstein-type Inequalities for Exponential Sums The results in this section were, in large measure, motivated by the letter of Lorentz below. “Dear Tamas: Feb. 27, 1988 I know you are interested in Bernstein-type inequalities and I am also. In some non-linear cases one has ‖P ‖X ≤ Φ(n)‖P‖Y , where n is the dimension of the set of the P ’s, and the norms are taken in different Banach spaces X and Y . For instance, inequalities of Dolzhenko and Pekarskii for rational 2000 Mathematics Subject Classifications: Primary: 41A17 Typeset by AMS-TEX 1 functions are of this type. I have proved an inequality of this type for exponential functions ∑n 1 aje λjx or the extended exponential sums

Local Remez-type inequalities for exponentials of a potential on a piecewise analytic ARC

Based on techniques from potential theory and geometric function theory, we give a new proof of the pointwise Remez-type inequality for exponentials of logarithmic potentials on the unit interval [−1, 1]. This inequality was first proved by Erdelyi, Li and Saff [8, Theorem 2.2.]. We generalize our result to the case of a piecewise analytic arc.

A REMEZ-TYPE THEOREM FOR HOMOGENEOUS POLYNOMIALS

Remez-type inequalities provide upper bounds for the uniform norms of polynomials p on given compact sets K, provided that |p(x)| ⩽ 1 for every x ∈ K\E, where E is a subset of K of small measure. In this paper we prove sharp Remez-type inequalities for homogeneous polynomials on star-like surfaces in Rd. In particular, this covers the case of spherical polynomials (when d = 2 we deduce a result of Erdélyi for univariate trigonometric polynomials).

Remez-Type Inequalities in Terms of Linear Measure

We obtain sharp uniform bounds for an exponential Q of a logarithmic potential on a quasi-smooth curve (in the sense of Lavrentiev) in terms of the linear measure of the subset of that curve on which Q is bounded by 1.

On Remez-type inequalities for polynomials in Rm and Cm

Remez-type inequalities provide estimates for the size of polynomials on given sets K⊂R m (or C m ) when the magnitude of polynomials on “largeldquo” subsets of K is known. We shall study this question on “smooth” sets K in R m and C m and show how the smoothness of K effects the estimates.

Polynomial inequalities on measurable sets and their applications

We study Polya-and Remez-type inequalities for univariate and multivariate polynomials and discuss their applications to Nikolskii-type inequalities and upper estimates of trigonometric integrals.

Polynomial Inequalities on Measurable Sets and Their Applications II. Weighted Measures

We study Remez-type inequalities for univariate and multivariate polynomials on bounded sets in weighted spaces and discuss their applications to Nikolskii-type inequalities and local approximation.

Lower bounds for derivatives of polynomials and Remez type inequalities

P. Turán [Über die Ableitung von Polynomen, Comositio Math. 7 (1939), 89–95] proved that if all the zeros of a polynomial p p lie in the unit interval I = def [ − 1 , 1 ] I \overset {\text {def}}{=} [-1,1] , then ‖ p ′ ‖ L ∞ ( I ) ≥ deg ⁡ ( p ) / 6 ‖ p ‖ L ∞ ( I ) \|p’\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\; . Our goal is to study the feasibility of lim n → ∞ ‖ p n ′ ‖ X / ‖ p n ‖ Y = ∞ \lim _{{n\to \infty } }{\|p_{n}’\|_{X}} / {\|p_{n}\|_{Y}} =\infty for sequences of polynomials { p n } n ∈ N \{p_{n}\}_{n\in \mathbb N } whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.

Remez-type inequalities and their applications

The Remez inequality gives a sharp uniform bound on [−1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [−1, 1], where |; p|; is at most 1, is known. Remez-type inequalities give bounds for classes of functions on a line segment, on a curve or on a region of the complex plane, given that the modulus of the functions is bounded by 1 on some subset of prescribed measure. This paper offers a survey of the extensive recent research on Remez-type inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and Müntz polynomials. Remez-type inequalities play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remez-type inequalities by giving a number of applications.

Inequalities for generalized nonnegative polynomials

We define generalized polynomials as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. We prove Markov-, Bernstein-, and Remez-type inequalities inLp (0<p<∞) and Nikolskii-type inequalities for such generalized polynomials. Our results extend the corresponding inequalities for ordinary polynomials.