On Discretizing Uniform Norms of Exponential Sums

In this paper, we consider the uniform norm discretization problem for general real multivariate exponential sums p(w)=∑0≤j≤ncje⟨μj,w⟩,μj,w∈Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p({{\mathbf{w}}})=\sum _{0\le j\le n}c_je^{\langle \mu _j, {{\mathbf{w}}}\rangle }, \;\;\mu _j, {{\mathbf{w}}}\in \mathbb {R}^d$$\end{document}. Given arbitrary 0<τ≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\tau \le 1$$\end{document} this problem consists in finding discrete point sets wj∈K,1≤j≤N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {{\mathbf{w}}}_j\in K, 1\le j\le N$$\end{document} in the compact domain K⊂Rd,d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\subset \mathbb {R}^d, d\ge 1$$\end{document} so that for every p(w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p({{\mathbf{w}}})$$\end{document} as above we have maxw∈K|p(w)|≤(1+τ)max1≤j≤N|p(wj)|.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \max _{{{\mathbf{w}}}\in K}|p({{\mathbf{w}}})|\le (1+\tau )\max _{1\le j\le N}|p({{\mathbf{w}}}_j)|. \end{aligned}$$\end{document}Using certain new Bernstein–Markov type inequalities for exponential sums it will be verified that for convex polytopes and convex polyhedral cones K in Rd,d≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d, d\ge 1$$\end{document} there exist meshes w1,…,wN⊂K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf{w}}}_1,\ldots ,{{\mathbf{w}}}_N\subset K$$\end{document} of cardinality N≤cnτdlndμn∗δτ,μn∗:=max1≤j≤n|μj|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N\le c\left( \frac{n}{\sqrt{\tau }}\right) ^{d}\ln ^{d}\frac{\mu _n^*}{\delta \tau }, \;\;\;\mu _n^*:=\max _{1\le j\le n}|\mu _j| \end{aligned}$$\end{document}for which the above inequality holds for any multivariate exponential sum p with exponents satisfying the separation condition |μk-μj|≥δ,j≠k,δ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\mu _{k}-\mu _j|\ge \delta , j\ne k, \delta >0$$\end{document}. In addition, the optimality of the cardinality estimates will be also discussed.


Introduction
In the past 15-20 years the problem of discretization of uniform and L q norms in various finite dimensional spaces has been widely investigated. In case of L q norms for trigonometric polynomials this problem is usually referred to as the Marcinkiewicz-Zygmund type problem, on the other hand when uniform norm and algebraic polynomials are considered then the terms norming sets or optimal meshes are usually used in the literature.
Consider the space L q (K ), 1 ≤ q ≤ ∞ endowed with some probability measure on the compact set K ⊂ R d . Then, given an n +1-dimensional subspace U n ⊂ L q (K ) the Marcinkiewicz-Zygmund type problem for 1 ≤ q < ∞ consists in finding discrete point sets Y N = {x 1 , . . . , x N } ⊂ K such that for any u ∈ U n we have (1) with some constants c 1 , c 2 > 0 depending only on q, d and K .
Similarly if q = ∞ and U n ⊂ C(K ) the goal is to ensure that for every u ∈ U n Here and throughout this paper p K stands for the uniform norm of the function on the compact set K ⊂ R d . The crucial question in the above-mentioned problems is the cardinality N of the discrete point set. Naturally, the goal is to ensure relations (1) and (2) with possibly minimal number of points. Clearly, we must have N > n in order for (1) or (2) to hold, that is cardinality of the mesh cannot be less than the dimension of the space. So ideally one would like to achieve discretization with N ∼ n points.
Discretization of uniform and L q norms for various spaces of algebraic and trigonometric polynomials is widely used in the study of the convergence of Fourier series, Lagrange and Hermite interpolation, positive quadrature formulas, scattered data interpolation, construction of discrete extremal sets of Fekete and Leja type, etc. The first discretization result for the L q norm of univariate trigonometric polynomials goes back to Marcinkiewicz and Zygmund, see [1]. Various generalizations of Marcinkiewicz-Zygmund type inequalities for multivariate trigonometric and algebraic polynomials can be found in recent papers [2], [3] and [4]. In this paper, we will be concerned primarily with the uniform version (2) of the discretization problem. Some early results on discretization of uniform norm for multivariate algebraic polynomials can be found in [5] and [6] where the discrete sets satisfying (2) were studied. Let us denote by P d n the space of real algebraic polynomials in d variables of degree ≤ n. In [7] it was conjectured that any convex body K ⊂ R d possesses a discrete mesh Y N = {x 1 , . . . , x N } ⊂ K of cardinality N ∼ n d ∼dimP d n for which relation (2) holds. Subsequently, this conjecture was verified in [8] in case d = 2.
In this paper, we will consider problem (2) for general real multivariate exponential polynomials of the form with arbitrary given μ j ∈ R d , d ≥ 1. Throughout the paper the "degree" of these exponential polynomials is defined by μ * n := max 0≤ j≤n |μ j |.
We will be seeking the solution of the discretization problem in an exact form with constant c(d, K ) = 1 + τ, ∀τ > 0 in (2). Naturally in this case we aim for discrete meshes with possibly optimal cardinality with respect to both n and τ . Such exact meshes were introduced in [3], p. 5, see also [9]. One of the main results of this paper (Theorem 3) asserts that when d = 1 and K = [0, 1] then for arbitrary 0 < τ < 1 there exist meshes Y N = {x 1 , . . . , x N } ⊂ [0, 1] of cardinality so that for every exponential polynomial (3) with arbitrary μ j ∈ R, μ j+1 − μ j ≥ δ, 0 ≤ j ≤ n − 1. An important feature of this result is the fact that the number of points in the discrete mesh depends primarily on the number n of exponents μ j ∈ R, while the degree μ * n and separation parameter δ of the exponential polynomials appears only in the logarithmic term. So this bound is "almost" degree and exponent independent. In addition, above result provides universal discrete meshes which are suitable for all n term univariate exponential sums of given degree and separation. A similar result is shown to hold even on unbounded domain [0, ∞) provided that μ j < 0 (Theorem 5).
The upper bound (4) for the cardinality of the discrete mesh turns out to be near optimal in the sense that it is sharp in general up to the logarithmic term with respect to both n and τ (Theorem 4). Let us note that existence of near optimal discretization meshes for multivariate algebraic polynomials was verified in [10], p. 74 using Fekete points for general compact sets. The crucial property used in [10] was the fact that algebraic polynomials are closed relative to multiplication. This multiplicative property of course does not hold for general exponential polynomials (3) and hence the method of Fekete points is not applicable in our case. Moreover, in contrast with the trigonometric case considered, e.g., in [3], since the exponents μ j ∈ R d in the exponential sums (3) are arbitrary the basis functions e μ j ,w are not orthogonal in our setting and hence this crucial Fourier analytic tool is not available, too. Therefore, we needed to rely on a different method for treating general exponential sums. This method will be based on various new Bernstein and Markov type inequalities for univariate and multivariate exponential sums which are discussed in Sects. 1 and 2, respectively. Section 3 is dealing with the discretization of the uniform norm for univariate exponential sums. Subsequently, in Sect. 4 we will study the discretization problem for multivariate exponential sums (3).
A crucial feature of the results of the present paper is their universality in the sense that the upper bounds for their cardinality are independent of the exponents μ j of exponential sums (3), as long as they are separated by |μ k − μ j | ≥ δ, j = k, δ > 0 and an upper bound |μ j | ≤ M is imposed. Therefore, we introduce the following set of n term exponential sums of variable w ∈ R d with exponents separated by δ and bounded by M It should be noted that d (n, δ, M) is not a linear subspace.
The main result of Sect. 4 (Theorem 6) asserts that for convex polytopes K in R d and any so that for every p given by (3) we have Throughout this paper, we will always denote by c(...) positive constants depending only on the quantities specified in the brackets, while c will stand for absolute constants.
Finally, we would like to emphasize an important advantage of the Bernstein-Markov methods used in this paper: all the discrete meshes are constructed explicitly and the proofs of the main results essentially provide algorithms for their construction.

Bernstein and Markov Type Inequalities for Univariate Exponential Sums
In this section, we will use and extend various Bernstein and Markov type inequalities for univariate exponential sums given in the monograph [11]. These inequalities are related to the well-known Newman inequality, see [11], p. 276. First let us mention the following modification of the Newman inequality verified in [11], p. 287.
Given any λ j ∈ R + , 0 ≤ j ≤ n, λ 0 = 0, λ j+1 − λ j ≥ 1 and p(x) = 0≤ j≤n c j x λ j , c j ∈ R it follows that Next, we will need an extension of the Newman type inequality given in [11], p. 301, E.10 by allowing negative exponents and providing explicit constants depending on [a, b] ⊂ (0, ∞). Knowledge of the explicit constants will be crucial in the multivariate case.
Thus, we may assume that λ 0 < 0 < λ n . Choose any y ∈ (a, b]. Setting μ j := λ j − λ 0 and g(x) := 0≤ j≤n c j x μ j we have that μ j ≥ j, 0 ≤ j ≤ n and p(x) = x λ 0 g(x). Thus, applying (7) for g on the interval [a, y] yields Hence, using that λ 0 < 0 and thus g [a,y] ≤ y −λ 0 p [a,y] we obtain by the last estimate Hence, Finally, using (8) and (9) if y ∈ [(a + b)/2, b] and y ∈ [a, (a + b)/2], respectively, we obtain for any y ∈ [a, b] Remark Estimate (6) of Lemma 1 should be compared to the classical Markov inequality stating that for any algebraic polynomial p of degree ≤ n. Here the "square of the degree" provides an upper bound for the derivatives. On the other hand (6) gives a "degree times dimension" type estimate which in general can be of essentially better order. A standard substitution x = e t leads to a statement for exponential sums on [α, β] ⊂ R similar to the above lemma. Moreover, imposing the separation condition μ j+1 − μ j ≥ δ β−α for exponents e μ j t makes the statement domain independent. We can also iterate the result for higher derivatives as well yielding the next extension of Lemma 1 for exponential sums.
Similarly using substitution x = e −δt we can derive from (5) Now, we proceed by verifying certain Bernstein type inequalities for univariate exponential sums. We will derive the needed inequalities from a basic result given in [11], p. 293, E.4.d according to which for any g(t) = 1≤ j≤n c j e −μ j t , c j ∈ R and Obviously, we can iterate the above estimate for higher derivatives using it consequently for The following Bernstein type inequalities can be derived from estimate (13).

Markov Type Inequalities for Multivariate Exponential Sums
In this section, we will extend Markov type estimates given in the previous section to the derivatives of multivariate exponential sums on convex bodies and polyhedral cones. Let us denote by ∇g the gradient of a differentiable function g.

be a convex body with r K being the radius of its largest inscribed ball. Then, for every exponential sum g(w)
In order to verify the above theorem we need first some technical lemmas. Let us denote by B(0, r ) the ball of radius r in R d centered at the origin. In addition, S d−1 stands for the unit sphere in R d .

Lemma 4
Assume that d = 2 and K ⊂ R 2 is a convex body satisfying B(0, r ) ⊂ K ⊂ B(0, R). Then, for any A ∈ K there exists a triangle ⊂ K with vertices A, B, C such that the angle at A is ≥ r π R and dist(A, Proof We may assume that A = (0, y), y ≥ 0. Consider first the case when A ∈ Consider the exponents e λ j ,w , λ j ∈ R d , 1 ≤ j ≤ n which satisfy the separation condition |λ j − λ k | ≥ δ > 0, j = k. We will need to know in what way this separation condition is preserved when these exponents are considered on certain lines and hyperplanes. Clearly, the orthogonal projections of λ j -s into lines {tu : t ∈ R}, u ∈ S d−1 is given by λ j , u u, where these projections are separated by quantities We will repeatedly use below the following auxiliary proposition which shows that the separation of the exponents can be preserved in some sense after a proper small perturbation of lines and hyperplanes.

This verifies that for any
Now let E (w * ) := {y ∈ S d−1 : |w * − y| ≤ } be the sphere cap centered at w * ∈ S d−1 . Then, clearly α d (E (w * )) ≥ c * Since u ∈ B (w * ) it follows that there exists u 2 ∈ E (w * ), u 2 ⊥u. Thus u ∈ H u 2 and hence using the above lower bound In order to verify the statement of the theorem we need to give a proper upper bound for the derivative of g(w) = 0≤ j≤n c j e μ j ,w when |μ j+1 − μ j | ≥ δ in any direction u ∈ S d−1 and for any w ∈ K . Thus clearly we can consider the problem in the 2-dimensional plane spanned by u, w. This and rotation invariance of the setting yields that it suffices to consider the case d = 2 and estimate ∂ g ∂ x (A) for any given A ∈ K . By Lemma 4 there exists a triangle ⊂ K with vertices A, B, C such that the angle at A is ≥ r π R and dist(A, [B, C]) ≥ r 2 . Let w * 1 , w * 2 ∈ S 1 be unit vectors dividing the angle at vertex A into 3 equal parts. Then, using Lemma 5 (i) with w * = w * 1 and w * = w * 2 when d = 2, = r 12π R yields that there exist D, E ∈ [B, C] such that the angle D AE ≥ r 6π R and the exponential sum g(w) = 0≤ j≤n c j e μ j ,w along each segment [A, D] and [A, E] reduces to univariate exponential sums g(t) = 0≤ j≤n c * j e tλ j , t ∈ R with exponents satisfying max 0≤ j≤n |λ j | ≤ μ * n and separated by In addition, since dist(A, [B, C]) ≥ r 2 we have that |A − D|, |A − E| ≥ r 2 . This and the above separation property yield that we can use estimate (10) of Lemma 2 for the exponential sum g(t) = 0≤ j≤n c * j e tλ j , t ∈ R along segment [A, D] with separation parameter δ 1 := cδr 2 Rn 2 , and c > 0 a proper absolute constant. Thus setting M n := Rn 3 μ * n r 2 δ we obtain by (10) where u is the unit vector in the direction of segment [A, D] and c 0 > 0 is a proper absolute constant. Analogously, where w is the unit vector in the direction of segment [A, E] and c > 0 is an absolute constant. Clearly, we may assume that u = (0, 1). In addition, recalling that the angle D AE ≥ r 3π R , we have ξ ≥ r 3π R where ξ is the angle between u and w. Hence, we easily obtain from (19) and (20) that i.e., we have the next estimate for the gradient ∇g(A) with some absolute constant Now it remains to apply the well-known John Ellipsoid Theorem [12]. Using this theorem one can show that a certain regular linear transformation of norm ≤ d r K maps convex body K ⊂ R d into a convex body K 0 satisfying B(0, 1) ⊂ K ⊂ B(0, d).
Thus we can assume that that r = 1 and R = d in (21), this may increase the upper bound at most by a factor of d r K , see [13], p. 91 for details. This observation together with estimate (21) leads to the final upper bound |∇g(A)| ≤ cd 3 n 3 μ * n r K δ g K .
Next we will obtain a Markov type inequality for multivariate exponential sums on an unbounded polyhedral cone in R d In particular, j being the Kronecker delta. Since polyhedral cones are unbounded sets in R d + naturally we will consider exponential sums spanned by e − μ j ,w , w ∈ R d + with μ j ∈ R d + . As above μ * n := max 0≤ j≤n |μ j |.
Proof Evidently it suffices to verify the theorem for d = 2 since in order to estimate the derivative in any direction u ∈ S d−1 + we can consider the problem in the 2-dimensional plane spanned by x and u. In addition, without the loss of generality we may assume that K = R 2 + . (This can be accomplished by a regular linear transformation of the space.) Now consider any p(w) = 0≤ j≤n c j e − μ j ,w , w ∈ R 2 with μ j ∈ R 2 Applying now Lemma 5 (i) with w * = e 2πi/3 and w * = e 5πi/6 for d = 2 and = π 24 we can easily choose two finite segments I 1 , I 2 ⊂ R 2 + in directions w 1 , w 2 ∈ S d−1 with endpoints on the coordinate axes so that x = I 1 ∩ I 2 , |I 1 |, |I 2 | ≥ |x|, 0 < w 1 , w 2 < cos π 12 , and Using now univariate Markov type inequality (10) on the intervals I j , j = 1, 2 with δ replaced by cδ|I j | n 2 yields the next estimate for the directional derivative |D w s p(x)| ≤ cn 3 μ * n δ|x| p K , s = 1, 2.
Combining the two estimates of the gradient verified above easily yields the statement of the theorem.

Discretization of Uniform Norm for Univariate Exponential Sums
Markov type inequalities given in Sects. 1 and 2 provide a straightforward path to estimating cardinality of meshes which discretize uniform norms of exponential sums. This can be accomplished recalling the notion of fill distance of sets. The fill distance of a subset D in K ⊂ R d is defined by The following proposition provides a simple connection between fill distance and Markov type estimates.

Lemma 6
Let K ⊂ R d be a convex body and assume that for some function g differentiable on K we have ∇g K ≤ M g g K . Then, for every compact set D ⊂ K with ρ(D, K )M g < 1 we have Proof Assume that g K = |g(x)|, x ∈ K . Then, there exists y ∈ D such that |x − y| ≤ ρ(D, K ). By the convexity of K we have that [x, y] ∈ K and thus Above lemma combined with Markov type estimates verified in Sects. 1 and 2 immediately yields certain bounds for the cardinality of discretization meshes for exponential sums. Indeed for any K ⊂ R d one can choose uniformly distributed discrete subsets Y N ⊂ K with ρ(D, K ) = ρ and cardinality N ∼ ρ −d . Then, in view of Lemma 6 if a Markov type inequality holds with a factor M g then relation g K ≤ (1 + τ ) g D can be ensured with discrete sets of cardinality N ∼ ( M g τ ) d . Hence, recalling the univariate Markov type estimate of Lemma 2 for k = 1 leads to discrete meshes of cardinality N ∼ nλ * n δ for univariate exponential sums g(t) = 0≤ j≤n c j e μ j t , c j ∈ R of degree μ * n = max 0≤ j≤n |μ j | with μ j ∈ R, 0 ≤ j ≤ n satisfying μ j+1 − μ j ≥ δ, 0 < δ ≤ 1. Likewise using Theorem 1 leads to discrete meshes on convex bodies K ⊂ R d of cardinality N ∼ ( n 3 μ * n δ ) d for exponential sum p(w) = 0≤ j≤n c j e μ j ,w , w ∈ R d with μ j ∈ R d satisfying |μ k − μ j | ≥ δ, j = k, 0 < δ ≤ 1. However, this rough approach leads to discrete meshes having large cardinality which is far from optimal. The main drawback of above estimates for cardinality consists in the fact that they depend heavily on the degree μ * n of the exponential sums. In Sects. 3 and 4 we will provide a more delicate approach to constructing discretization meshes for exponential sums. This approach will rely not only on Markov but also on Bernstein type inequalities and thus will lead to a finer distribution of meshes around the boundary of the domain considered. This will result in near optimal discretization meshes with only logarithmic dependence of the cardinality on the degree of exponential sums.
The construction of the discrete point sets for univariate exponential sums will be based on the measures and As we will see below the discretization meshes for the exponential sums on finite intervals and on (0, ∞) can be chosen to be equidistributed with respect to the measures μ 1 and μ 2 given by (24) and (25), respectively.

Lemma 7 For any
and thus we obtain Finally, since Now we will apply Bernstein and Markov type inequalities for univariate exponential sums given in Sect. 1 together with Lemma 6 in order to verify the following basic discretization result for univariate n term exponential sums g ∈ 1 (n, δ, M).
with some absolute constant c > 0 so that for every exponential sum g ∈ 1 (n, δ, M) we have Proof The statement of the theorem is clearly dilation and shift invariant, therefore we can assume without the loss of generality that [α, β] = [0, 1]. Applying (10) with k = 2 yields that The construction of the needed discrete point set will be based on the points x j,m , 1 ≤ Thus which is the needed estimate. Now let x * ∈ (x j,m , x j+1,m ) for some 1 ≤ j ≤ N − 1. Then, applying the Taylor formula with integral remainder and Bernstein type inequality (14) for k = 2 and .
It remains now to estimate the cardinality of the discrete point set Y N . Recall that by Lemma 7 Clearly, assumption μ j+1 − μ j ≥ δ, ∀ j means that n ≤ 2M δ , i.e., Finally combining this with the above estimate for N we arrive at The proof of Theorem 3 presented yields an algorithm of explicit construction of the discretization meshes for the univariate exponential sums exhibited by the next proposition.

Corollary 2 Let
τ such that for every univariate n term exponential sum g ∈ 1 (n, δ, M) we have g [0,1] Theorem 3 gives exact discretization meshes of cardinality ≤ cn √ τ ln M δ √ τ for univariate n term exponential sums g ∈ 1 (n, δ, M). A remarkable feature of this upper bound consists in the fact that the degree M and separation parameter δ of exponential sums appear only in the logarithmic term, while n √ τ provides the main part of the bound. Now we are going to show that this bound for cardinality is in general sharp up to the logarithmic term.
with some absolute constant c > 0.
Proof For any 0 < h < 1 consider the univariate algebraic polynomial of degree n given by is the classical Chebyshev polynomial. Then, |q n (x)| ≤ 1 for every h ≤ |x| ≤ 1 and We have constructed a univariate algebraic polynomial q n of degree n such that |q n (x)| ≤ 1 for every h ≤ |x| ≤ 1 and |q n (0)| ≥ 1 + 2n 2 h 2 . Performing a shift and dilation of the variable it is easy to see that for any [a, b] ⊂ R and a < r < r + h < b there exists an algebraic polynomial p n of degree n so that | p n (x)| ≤ 1 for every x ∈ [a, b] \ [r , r + h] and | p n (r + h 2 )| ≥ 1 + cn 2 h 2 with some c > 0 depending only on a, b. Using this observation with [a, b] = [1, e] and substituting x = e t easily yields that for every subinterval [ξ, γ ] ⊂ [0, 1] of length h, γ − ξ = h there exists an exponential sum g * (t) = 0≤ j≤n c j e jt such that |g * (t)| ≤ 1 for every t ∈ [0, 1] \ [ξ, γ ] and |g * ( ξ +γ Thus using the exponential sum g * (t) constructed above for this Near optimal discretization of uniform norm similar to Theorem 3 can be obtained even for exponential sums on unbounded domain [0, ∞). This can be achieved applying Markov and Bernstein type estimates (11) and (15) on half axis for exponential sums with negative exponents. In order to present exponent independent results let us introduce the set of n term exponential sums of variable w ∈ R d + with "negative" exponents separated by δ and bounded by M Now we give an analogue of Theorem 3s for the domain [0, ∞).
Theorem 5 Let n ∈ N, M ≥ 1, 0 < δ ≤ 1. Then, for any 0 < τ < 1 there exist discrete points sets so that for every exponential sum g ∈ 1 + (n, δ, M) we have Proof Set (Note that the norm could be attained at x * = ∞, as well.) If x * = 0 ∈ Z N then the statement of the theorem holds trivially. So we may assume that x * ∈ (0, ∞]. Now we distinguish three cases depending on the location of x * . Case 1. x * ∈ (0, x 1 ). Then, by (11) |g It is easy to see that Therefore, it follows from (11) that Then, x * ∈ [x j , x j+1 ] for some 1 ≤ j ≤ N − 1 and g (x * ) = 0. Now we apply Bernstein type inequality (15) for k = 2 together with the Taylor formula with integral remainder term yielding Summarizing above three cases we can see that independently of the location of x * there always exists an It remains to note now that nδ ≤ M and hence we have with some absolute constant c > 0

Discretization of Uniform Norm for Multivariate Exponential Sums
Now we proceed to the extension of the discretization results for multivariate exponential sums on convex polytopes in R d and polyhedral cones in R d + . This extension will be based on multivariate Bernstein-Markov type inequalities of Sect. 2 and discretization methods for univariate sums developed in Sect. 3. The main building block in the multivariate case is the d-dimensional simplex . . . , 1).
be a convex polytope and consider any 0 < δ, τ < 1, M ≥ 1. Then, given any such that for every exponential sum p = 1≤ j≤n c j e μ j ,w , n ∈ N we have Proof Clearly, it suffices to verify the theorem for the d-dimensional simplex d . We will accomplish this by the induction on the dimension d.
The univariate case d = 1 is covered by Theorem 3. So let us assume that the statement of the theorem holds for the d − 1 dimensional simplex d−1 , d ≥ 2. It should be noted that a dilation of this simplex alters both parameters δ and M of the exponential sums by the same constant factor. This constant factor clearly cancels out in the upper bound for the cardinality of the discrete mesh, while uniform norms are dilation invariant. This shows that Theorem 6 is dilation invariant.
It follows by Lemma 5 (ii) applied to w * := u 0 /|u 0 |, u 0 = (1, . . . , 1) that there exists an perturbation u * = (b 1 , . . . , b d ), |u 0 − u * | < such that exponents e μ j ,w restricted to the hyperplanes {w ∈ R d : u * , w = a}, a ∈ R + satisfy the separation condition (17) with parameter δ 1 := c d δ n 2 and u 2 = u * . The choice of 0 < < 1/3 will be specified somewhat later. Now for any a > 0 consider the d − 1 dimensional simplex 1] be the discretization nodes for univariate exponential sums specified in Corollary 2. According to Corollary 2 given any δ 2 ( to be specified below) these nodes can be chosen so that √ τ and for every univariate exponential sum g ∈ 1 (n, δ 2 , 2 √ d M), n ∈ N we have Now consider the d − 1 dimensional simplices D j := D(x j ), 1 ≤ j ≤ N which are dilated copies of D(1). As noted above whenever 0 < < h/e, i.e., Recall that exponents e μ j ,w restricted to D j , 1 ≤ j ≤ N satisfy the separation condition (17) with parameter δ 1 (and u 2 = u * ). In addition, it is easy to see that these restricted exponents are bounded by the same M. Moreover, since simplex D (1) Obviously, the discretization meshes Z j ⊂ D j ⊂ d are dilated copies of each other for distinct values of j. Now set Since N ≤ c d n √ τ ln M δ 2 √ τ this results in a discrete point set of cardinality Next we extend the discretization result of Theorem 6 to polyhedral cones given by Theorem 7 Let K := K (u 1 , . . . , u m ) ⊂ R d + , d ≥ 2 be a polyhedral cone and consider arbitrary 0 < τ, δ < 1, M ≥ 1. Then, given any such that for every exponential sum p = 1≤ j≤n c j e − μ j ,w we have Proof The proof of this theorem will be based on the discretization for univariate exponential sums on [0, ∞) (Theorem 5), and discretization for multivariate exponential sums on convex polytopes (Theorem 6). In addition, the Markov type estimate provided by Theorem 2 will be applied, as well. First note that for any u ∈ S d−1 + the set so that the new separation parameter for the exponents on perturbed polytopes D a (u * ), u * ∈ S d−1 + , |u * − u 0 | < is δ 1 = c d δ n 2 . We will specify the choice of 0 < < 1 2 √ d somewhat later. In addition, the upper bound M for the size of exponents is also preserved when restricting them to the hyperplanes {w ∈ R d , u * , w = a}. Now we can apply Theorem 6 for d − 1 dimensional convex polytopes D a (u * ), a > 0 with this δ 1 which implies that each of these polytopes possesses discrete point sets Y a ⊂ D a (u * ) of cardinality providing discretization of the uniform norm for the exponential sums p = 1≤ j≤n c j e − μ j ,w on D a (u * ), i.e., p D a (u * ) ≤ (1 + τ ) p Y a , a > 0.
Furthermore by Theorem 5 for any given 0 < δ 2 ≤ 1 (to be specified below) there exist discrete points sets Z N = {a 1 , ..., a N } ⊂ [0, ∞) of cardinality N ≤ cn √ τ ln M δ 2 τ , so that for any exponential sum g ∈ 1 + (n, δ 2 , 2d M) we have other hand imposing some additional regularity assumptions on the distribution of exponents can make it possible to choose proper perturbed directions in Lemma 5 suitable for the corresponding set of exponents. Then, application of above methods would lead to construction of universal exponent independent meshes in multivariate case.
Finally, let us address the question of sharpness of the upper bounds for the cardinality of discrete multidimensional meshes presented in Theorems 6 and 7. It is easy to see that Theorem 4 can be extended to the multivariate case. Namely for any compact set K ⊂ R d of positive Lebesgue measure and any discrete mesh Y N ⊂ K such that p * K ≤ (1 + τ ) p * Y N for every p * (w) = j 1 +···+ j d ≤n c j e j,w , c j ∈ R, j = ( j 1 , . . . j d ) ∈ N d it follows that CardY N ≥ c(K , d) n √ τ d . This shows that the upper bounds of Theorems 6 and 7 are sharp with respect to the parameter τ up to the logarithmic term. However, this leaves open the question of sharpness with respect to the number of terms in exponential sums which is of order ∼ n d for p * as above. On the other hand it should be also mentioned that the number of distinct exponents in d (n, δ, M) and d + (n, δ, M) with, say, uniform spacing is of order ∼ ( M δ ) d . Clearly, the cardinality of discrete meshes in Theorems 6 and 7 might be relatively small compared to this.
Funding Open access funding provided by ELKH Alfréd Rényi Institute of Mathematics.
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