On discretizing integral norms of exponential sums

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

On discretizing integral norms of exponential sums

András Kroó¹

AlfrédRényiInstituteofMathematicsandBudapestUniversityofTechnologyandEconomics,Budapest, Hungary

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received5January2021 Availableonline21October2021 Submittedby S.Tikhonov

Keywords:

Marcinkiewicz-Zygmund BernsteinandMarkovtype inequalitiesforgeneralexponential sums

DiscretizationofLpnorm Multivariateexponentialsums Exponentialsumswithnonnegative coefficients

InthispaperwestudyLp Marcinkiewicz-Zygmundtypeinequalities

c₁

1≤j≤N

w_j|g(x_j)|^p≤ g^pLp(K)≤c₂

1≤j≤N

w_j|g(x_j)|^p

forgeneralexponentialsumsoftheformg(x)=

1≤j≤naje^λj,x, x,λj∈R^d,aj∈ R.Oneof themainresultsofthepaperassertsthatwhen1≤p<∞,K= [a,b]

andtheexponentssatisfyrelationsλj+1−λj≥,1≤j≤n−1, max_1≤j≤n|λj|≤ Λ then Marcinkiewicz-Zygmund type inequalities hold for certain point sets of cardinality

N≤cnln^1p+1Λ .

Since the dependence of cardinality on the parameters,Λ appears only in the logarithmic term, this bound is “almost” degree and separation independent.

Moreover,thediscretemeshesxjandweightswj=xj+1−xjaregivenexplicitlyand theyareuniversalinthesensethattheyworkforanyexponentialsumasabove.This resultwillrelyonsomenewdegreeindependentBernstein-Markovtypeinequality for exponentialsums.Moreoverwewillextendour considerationstomultivariate exponentialsums.Inaddition,itwillbeshownthatmuchstrongerresultsholdfor exponentialsumswithnonnegativecoefficients.

©2021TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticle undertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

ConsiderthespaceL_p(K),1≤p≤ ∞endowedwith someprobabilitymeasure onthecompactset K⊂ R^d. Then givenasubspaceU ⊂Lp(K) theMarcinkiewicz-Zygmundtype problemfor 1≤p<∞consists

E-mailaddress:kroo.andras@renyi.mta.hu.

1 SupportedbytheNKFIH- OTKAGrantK128922.

https://doi.org/10.1016/j.jmaa.2021.125770

0022-247X/©2021TheAuthor(s). PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

infindingdiscretepointsetsY_N={x₁,...,x_N}⊂K andcorrespondingpositiveweights w_j >0,1≤j≤N suchthatforanyg∈U wehave

c₁

1≤j≤N

w_j|g(x_j)|^p≤ g^p_Lp(K)≤c₂

1≤j≤N

w_j|g(x_j)|^p (1)

with someconstants c₁,c₂ >0 depending onlyon p,dand K. Incasewhen K = [a,b] ⊂R isan interval on thereal line anatural choiceof theweights associated with the pointsa < x1 < x2 < ... < xN < b is givenbywj=xj+1−xj,1≤j≤N−1.InthisrespecttheclassicalMarcinkiewicz-Zygmundinequalityfor trigonometricpolynomialsstatesthat(1) holdsforthespaceU=Tnoftrigonometricpolynomialsofdegree

≤nand uniformlydistributed pointsontheperiodwith wj= _n¹ andN = 2n+ 1.Clearly thecardinality ofthediscretepointsetN = 2n+ 1 isoptimalhere.Thisequivalencerelationturnedouttobeaneffective tool usedforthediscretizationoftheL_pnormsoftrigonometricpolynomialswhichiswidelyappliedinthe study oftheconvergenceofFourierseries,LagrangeandHermiteinterpolation,positivequadratureformu- las,scattereddatainterpolation,seeforinstance[9] forasurveyontheunivariateMarcinkiewicz-Zygmund typeinequalities.AnimportantgeneralizationoftheclassicalMarcinkiewicz-Zygmundinequalityfortheso called doubling weights wasgivenbyMastroianniand Totik[10].Various extensionsof theMarcinkiewicz- Zygmund inequality for multivariate algebraic and trigonometric polynomials can be found in [1], [4]

and [2].

Inthispaper wewillconsiderproblem (1) forgeneralrealexponentialpolynomials oftheform

g(x) =

1≤j≤n

aje^λj,x, aj ∈R, x∈R^d (2)

with arbitrarygiven λj ∈R^d. Ingeneral,resultsrelated tothese exponential sumsdependonthe number n of terms in the sums and the size of λj-s. The “degree” of these exponential polynomials is given by max1≤j≤n|λj|. Here and throughout the paper |· | denotes the usual Euclidian norm of vectors in R^d. Naturally oneshould tryto aim for discrete meshes with possibly smallestcardinality. A particularly in- teresting problem consists inobtaining discrete meshesof cardinality depending onlyon the dimensionn of theexponential sums(2), thatis independent of their degree. For the trigonometric exponential sums with λ_j-s being of the form λ_j =in_j,n_j ∈Z this question is discussed in detailin the survey paper [2].

In particular in[12] Theorem1.1 itis shownthatfor generaltrigonometric exponential sumsonecanget meshes ofcardinality∼n.Intrigonometric casethebasicproperty whichisthemaintool ofinvestigation isthepairwiseorthogonalityofexponentials.Inthegeneralalgebraiccasetheexponentsin(2) donothave this crucial feature. So instead of orthogonalityour considerations will rely heavily on Bernstein-Markov type inequalities for the size of derivatives of exponential sums. We will verify in this paper some new degree independentBernstein-Markov typeinequalitiesforexponential sums.This will allowus toextend Marcinkiewicz-Zygmund typeinequalitiesto generalrealexponential sums.Inparticular, itwill beshown (see Theorem1below) thatthere existmeshesY_N={x₁,...,x_N}⊂[a,b]⊂Rofcardinality

N ∼nln^1p+1 Λ

(3)

so that (1) holds with weights wj := xj+1−xj for every 1≤ p <∞ and every exponential polynomial (2) withany λj ∈ R satisfying λj+1−λj ≥ >0,1≤j ≤n−1 andmax1≤j≤n|λj| ≤Λ. Animportant featureof thisresultis thefactthatthe cardinalityofthe discretemesh isofoptimal ordern, whiletheir degreeΛ andtheseparationparameteroftheexponentsessentiallyappearonlyinthelogarithmicterm.

So inthis sense ourbound is“almost”degree andexponent independent.The explicitlygivendiscrete sets usedinTheorem1areuniversalinthesensethattheyworkforallexponentialsumsasabove.Wewillalso includesimilar resultsfor multivariateexponentialsums.Finally,we willpresentsomenewdimensionand

degreeindependentBernstein-Markovtypeinequalitiesformultivariateexponential sumswithnonnegative coefficients.ThiswillleadtoaconsiderableimprovementofthecorrespondingMarcinkiewicz-Zygmundtype inequalitiesformultivariateexponentialsumswithnonnegativecoefficients.Itshouldbealsonotedthatall discretemeshesareconstructedinthispaperexplicitly.

2. Marcinkiewicz-Zygmundtype inequalitiesforgeneralexponentialsums

ThenexttheoremwhichisoneofthemainresultsofthispaperpresentsaMarcinkiewicz-Zygmundtype inequalityforgeneralreal univariateexponentialsumsbasedonpointsetsofcardinalityN ≤cnln^{p1+1 Λ}. Theorem 1.Let 1 ≤ p < ∞,[a,b] ⊂ R,0 < ≤ 1,n ∈ N,Λ > 1. Then there exist discrete points sets YN={x1< ...< xN}⊂(a,b)of cardinality

N ≤cpnln^1p+1Λ

wherec>0isanabsolute constant,sothat forevery exponentialsum (2) witharbitrary λj ∈R satisfying λj+1−λj ≥

b−a,1≤j≤n−1, max

1≤j≤n|λj| ≤Λ wehave

1 2

1≤j≤N−1

(xj+1−xj)|g(xj)|^p≤ g^p_Lp([a,b])≤2

1≤j≤N−1

(xj+1−xj)|g(xj)|^p. (4)

Remark1.Oneshouldnote the factthatthe discrete nodesprovided byTheorem 1are rather universal, they work for all exponential sums satisfying the separation condition λj+1−λj ≥ _b−a,1 ≤ j ≤ n−1 and the upper bound max_0≤j≤n|λ_j| ≤ Λ. In addition the dependence of cardinality on the parameters ,Λ appearsonly inthe logarithmic term, so the bound on cardinalityis “almost” degreeand separation independent. It shouldbe alsomentioned thattheconstants ¹₂ and2in(4) canbe replaced by1−ξ and 1+ξ, respectively,with anarbitrarily small 0< ξ <1.Of course, onewouldhave topay aprice forthis inthe sense that this will make theconstant inthe upper bound forthe cardinality of Y_N dependent on ξ. The proof ofTheorem 1 given belowindicatesthatconstants of order −^ln_ξ^ξ canbe used for theupper boundof cardinality.Inarecentpaper[8] theprecise dependenceofthis constantonparameterξ >0 was givenintheclassicalunivariate trigonometriccase.

TheproofoftheaboveMarcinkiewicz-ZygmundinequalitywillbebasedonanewdegreeindependentL_p Bernsteintypeinequalityforderivativesofexponentialsums(2).WewillalsousethefollowingL_p,1≤p<

∞Bernsteintypeinequalityforderivativesofunivariateexponential sumsfn(x)=

1≤j≤ncje^λjxgivenin [5],Theorem3.4

f_nLp[−1+δ,1−δ]≤ 2n−1

δ fnLp[−1,1], 0< δ <1. (5) ThiselegantresultprovidesexponentindependentupperboundsforL_pnormsofderivativesofexponential sums inside the interval. The drawback of the above estimate is the appearance of the term ¹_δ in the upperestimate whichleadsto largerthan requireddiscretizationsets inLp Marcinkiewicz-Zygmund type inequalities. Our next lemma shows that introducing a weight 1−x² into the Lp norms of derivatives of exponential sums allows to replace ¹_δ by a substantially smaller term ln²_δ. This improvement will be subsequentlyusedinordertoverifynearoptimaldiscretizationmeshes.

Lemma 1.Let1 ≤p< ∞,0 < δ < 1,n ∈ N. Then forany distinct real numbers λ₁,...,λ_n ∈ R and any exponential sum f_n(x)=

1≤j≤nc_je^λjx,∀c_j ∈R wehave

(1−x²)f_n(x)Lp[−1+δ,1−δ]≤9nln^1p 2

δfnLp[−1,1]. (6)

Proof. Using(5) withany 0< t<1 andmultiplyingthep-thpowerofthisestimatebyt^p−1 yields

t^p−1 1−t

−1+t

|f_n(x)|^pdx≤ (2n−1)^p t

1

−1

|fn(x)|^pdx, 0< t <1.

Integratingaboveinequalitywithrespect tot∈[^δ₂,1] wehave 1

δ 2

1−t

−1+t

t^p−1|f_n(x)|^pdxdt≤(2n−1)^p 1

δ 2

dt t

1

−1

|fn(x)|^pdx= (2n−1)^pln2 δ 1

−1

|fn(x)|^pdx.

Furthermore applyingFubinitheoremfortheintegralontheleft handsideofaboveestimateimplies 1

δ 2

1−t

−1+t

t^p−1|f_n(x)|^pdxdt=

1− ^δ2

−1+^δ₂ 1−|x|

δ 2

t^p−1|f_n(x)|^pdtdx= 1 p

1− ^δ2

−1+^δ₂

|f_n(x)|^p((1− |x|)^p−(δ 2)^p)dx

≥1 p

1−δ

−1+δ

|f_n(x)|^p((1− |x|)^p−(δ

2)^p)dx≥ 1 p

1−δ

−1+δ

|f_n(x)|^p((1− |x|)^p−((1− |x|)/2)^p)dx

≥ 1 p2^p+1

1−δ

−1+δ

|f_n(x)|^p(1−x²)^pdx.

Thus combiningthelast twoestimatesabovewearriveat

1−δ

−1+δ

|f_n(x)|^p(1−x²)^pdx≤p2^2p+1n^pln2 δ 1

−1

|fn(x)|^pdx.

Finally takingthep-throotaboveyieldstherequiredestimatewithaconstantc≤4e^2e ≤9.

ByastandardlineartransformationLemma1canbeextendedtoanyinterval [a,b]⊂R.

Corollary1.Let1≤p<∞,[a,b]⊂R,0< δ <^b−₂^a,n∈N.Thenforanydistinctrealnumbersλ1,...,λn∈R and anyexponential sum f_n(x)=

1≤j≤nc_je^λjx,∀c_j∈Rwe have (b−x)(x−a)f_n(x)Lp[a+δ,b−δ] ≤9(b−a)

2

lnb−a δ

¹_p

nfnLp[a,b]. (7) AnimportantfeatureofLemma1consistsinthefactthatitprovidesanestimateforthederivativesof exponential sums

1≤j≤na_je^λjx, x,a_j ∈R whichis independentof theexponentsλ_j. Infactthesize of thederivativesessentiallydependsjustonn,thenumberoftermsintheexponentialsum.Whatmakesthis

happenis thepresence oftheweight(b−x)(x−a) inthenorm ofthe derivatives.This weightenforces a lineardeclineattheendpointsoftheinterval.Anaturalquestionarisesinthisrespect:cananupperbound similar toLemma1holdwithaweight(b−x)(x−a)^1−ε forsomeε>0?Thenextsimpleexampleshows thatingeneral,theanswertothisquestionisnegative,i.e.,lineardeclineoftheweightiscrucialhere.

Example.Let n = 1 and consider the exponent g(x) = e^λx,λ > 4. Clearly with any 0 < ε < 1 and 0< δ <1/3 wehave

x(1−x)^1−εg(x)²L2[δ,1−δ]≥

1−δ

1 2

x²(1−x)^2−2ελ²e^2λxdx≥λδ^2−2ε

8 (e^2λ(1−δ)−e^λ).

Thussettingδ=_λ¹ implieswhenλ>4

x(1−x)^1−εg(x)²L2[δ,1−δ]≥ λ^2ε−1e^2λ

8 (e⁻²−e^−λ)≥ce^2λλ^2ε−1. Sinceinaddition,

g(x)²L2[0,1] ≤e^2λ λ wearriveat

x(1−x)^1−εg(x)²_L2[¹

λ,1−¹λ]≥ce^2λλ^2ε−1≥cλ^2εg(x)²L2[0,1].

Note thatwhen n = 1 and δ = ¹_λ we get an upper bound of magnitude O(logλ) in the Bernstein type estimate(6) whichmeansthatnomatterhowsmallisε>0 anupperboundsimilarto(6) isimpossiblefor theweight(b−x)(x−a)^1−ε.Obviouslythesameconclusionholdsfortheweight(b−x)^1−ε(x−a),aswell.

ThenextlemmaprovidesanexplicitconstructionofnodesusedforthediscretizationoftheLpnormsof theexponentialsums.Essentially thesenodesarechosento beequidistributedwithrespect tothemeasure

μ₁(E) :=

E

dx

x(1−x), E⊂(0,1) appearingintheBernsteintypeinequality(6).

Lemma2.Forany 0< h<1/2andm≥1set

x_j,m:= 1

1 + ^1−h_h e^−(j−1)/m, 1≤j≤N =N_m:= [2mln1−h h ] + 2.

Then1−h≤x_Nm,m≤1−h/e and

xj+1,m−xj,m≤4x(1−x)

m , x∈(xj,m, xj+1,m), 1≤j≤Nm−1. (8) Proof. Clearly

2 ln1−h

h ≤N_m−1

m ≤2 ln1−h h + 1

and thusweobtain

1−h= 1

1 +^1−h_h e^{−2 ln1−hh} ≤x_Nm,m ≤ 1

1 +_1−h^h e⁻¹ ≤1−h e.

Sethj:= ^1−h_h e^−(j−1)/m.Thenevidently xj+1,m−xj,m

1−xj+1,m

= hj−hj+1

hj+1(1 +hj)≤ hj

hj+1 −1 =e^m1 −1≤ 2 m.

Hence

xj+1,m−xj,m≤ 2

m(1−x), x∈(xj,m, xj+1,m).

Similarly,

x_j+1,m−x_j,m xj,m

= 1 +h_j

1 +hj+1 −1≤ h_j

hj+1 −1 =e^m1 −1≤ 2 m,

yielding that

x_j+1,m−x_j,m≤ 2

mx_j,m≤ 2

mx, x∈(x_j,m, x_j+1,m).

Thus forany x∈(xj,m,xj+1,m) wehavethat x_j+1,m−x_j,m≤ 2

mmin{1−x, x} ≤ 4x(1−x)

m .

WewillalsoneedbelowanL_∞Markovtypeinequalityverifiedin[7] formultivariateexponentialsumson convexbodies.Letusdenoteby∇gthegradientofadifferentiablefunctiong,|∇g|standsforitsEuclidian norm which isused indefining∇gL_∞(K):= sup_x∈K|∇g(x)|. Inaddition, c will denote possiblydistinct positive absoluteconstants.

Lemma 3.Let K⊂R^d,d≥1be aconvex body with rK being theradius of its largestinscribed ball. Then for every exponential sum g(w) =

1≤j≤ncje^λj,w,w ∈ R^d with λj ∈ R^d satisfying max1≤j≤n|λj| ≤ Λ, |λ_j+1−λ_j|≥ _rK,j=k,0< ≤1

∇gL_∞(K)≤ cd³n³Λ

gL_∞(K).

In addition,when d= 1thestronger inequality

gL_∞[a,b] ≤cnΛ

gL_∞[a,b]

holds.

AstandardconsequenceofanL_∞MarkovtypeinequalityisacorrespondingNikolskiitypeupperbound forsupnormsviatheL_p norms.NamelywecaneasilydeducefromLemma3thenext

Corollary2. Forany convexbody K⊂R^d,d≥1and every exponentialsum g(w)=

1≤j≤nc_je^λj,w,w∈ R^d with λ_j ∈ R^d satisfying |λ_j+1−λ_j| ≥ _rK,j = k,0 < ≤ 1, max_1≤j≤n|λ_j| ≤ Λ we have for any 1≤p<∞and someconstant c(K,d)>0depending onK,d

gL_∞(K)≤c(K, d) n³Λ

^d_p

gLp(K). (9) Whend= 1 wehave thestrongerestimate

gL_∞[a,b] ≤c(a, b) nΛ

¹_p

gLp[a,b]. (10)

Proof. FirstweneedtoobservethatwhenK⊂R^disaconvexbodythenforanyddimensionalballBr⊂R^d ofradius rcenteredinKits intersectionwith K willhaveLebesguemeasure atleast c(K,d)r^d with some fixed constantdepending only onK and d. Furthermore, assuming thatgL_∞(K) = 1=g(y),y∈ K we have by Lemma3 thatg(x) ≥ ¹₂ whenever x ∈ K∩B_r with B_r being the ball centered at y and radius r:= _2cd3n³Λ.Hence

g^p_Lp(K)≥

K∩Br

|g|^p≥2^−pc(K, d)r^d≥2^−pc(K, d) n³Λ

d

withaproperc(K,d)>0.Thisclearlyimplies(9).Thesecond claimfollowsanalogously.

Nowwehaveallthetoolsneededinorder toproveTheorem 1.

Proof of Theorem1. Clearly,itsufficestoverifythetheoremwhen[a,b]= [0,1],thegeneralcasewillthen followbyastandardlineartransformation.Wewill usetheelementaryestimate

b a

g(x)dx−(b−a)g(a)

≤(b−a) b a

|g(x)|dx

applied to the function g(x) = |f(x)|^p on the interval [x_j,m,x_j+1,m] where x_j,m,1 ≤ j ≤ N_m−1 is the discretepointset specifiedinLemma2.Thenitfollowsthat

xj+1,m

xj,m

|f(x)|^pdx−(x_j+1,m−x_j,m)|f(x_j,m)|^p

≤(x_j+1,m−x_j,m)

xj+1,m

xj,m

p|f(x)|^p−1|f(x)|dx.

Applyingestimate(8) ofLemma2wehave

xj+1,m

xj,m

|f(x)|^pdx−(x_j+1,m−x_j,m)|f(x_j,m)|^p ≤4p

m

xj+1,m

xj,m

x(1−x)|f(x)|^p−1|f(x)|dx, 1≤j≤N_m−1.

Summingupaboveupperboundsfor1≤j ≤N_m−1 wearriveat

x_Nm

x1,m

|f(x)|^pdx−

1≤j≤Nm−1

(xj+1,m−xj,m)|f(xj,m)|^p ≤4p

m

x_Nm

x1,m

x(1−x)|f(x)|^p−1|f(x)|dx.

Moreover, applyingtheHölderinequalitytotheintegralontherighthandside itfollowsthat

xNm

x1,m

|f(x)|^pdx−

1≤j≤Nm−1

(xj+1,m−xj,m)|f(xj,m)|^p ≤ 4p

mf^p−1_Lp[0,1]x(1−x)f(x)Lp[x1,m,xNm]. Thus

f^p_Lp[0,1]−

1≤j≤Nm−1

(x_j+1,m−x_j,m)|f(x_j,m)|^p ≤ 4p

mf^p_L⁻_p[0,1]¹ x(1−x)f(x)Lp[x1,m,x_Nm]

+

x1,m

0

|f(x)|^pdx+ 1 xNm

|f(x)|^pdx. (11)

Now weneedtoestimatethethreetermsontherighthandsideof(11).

InordertoestimatethefirsttermrecallthatbyLemma2wehave1−h≤x_Nm,m≤1−h/eandx_1,m=h.

Therefore

x(1−x)f(x)Lp[x1,m,x_Nm]≤ x(1−x)f(x)Lp[h/e,1−h/e]. Furthermore, applyingestimate(7) with[a,b]= [0,1] andδ:= ^h_e yields

x(1−x)f(x)Lp[h/e,1−h/e] ≤9

2(1−lnh)^1pnfLp[0,1]. Using thelastupperboundforthefirsttermontherighthandsideof (11) weobtain

4p

mf^p−1_Lp[0,1]x(1−x)f(x)Lp[x1,m,x_Nm]≤ 18p

m (1−lnh)^1pnf^p_Lp[0,1].

Forthesecond andthirdtermsontherighthandsideof(11) wecanproceed using(10) asfollows

x1,m

0

|f(x)|^pdx+ 1 xNm,m

|f(x)|^pdx≤2hf^p_L∞[0,1]≤c₁h nΛ

¹_p

f^p_Lp[0,1].

Substitutingthelast twoboundsinto (11) wearriveat

f^p_Lp[0,1]−

1≤j≤Nm−1

(x_j+1,m−x_j,m)|f(x_j,m)|^p ≤18p

m (1−lnh)^1pnf^p_Lp[0,1]+c₁h nΛ

¹_p

f^p_Lp[0,1]. (12) Now settingh:=ξ

nΛ

¹_p

andm:= ^pn(1−lnh)

1 p

ξ evidentlyyields

f^p_Lp[0,1]−

1≤j≤Nm−1

(xj+1,m−xj,m)|f(xj,m)|^p

≤(18 +c1)ξf^p_Lp[0,1].

Since 0< ξ <1 canbe chosen arbitrarily thelast estimate obviously impliestherequired Marcinkiewicz- Zygmundtypeinequalityafterproperchoiceoftheconstantξ.

Itremainsnowto checkthecardinalityN_m= [2mln^1−h_h ]+ 2 ofthediscretepointset.Clearly N_m≤cmln 1

h ≤cpnln^1p+1 1

h ≤cpnln^1p+1nΛ . Recallingthat|λj+1−λj|≥,j =k andmax_1≤j≤n|λj|≤Λ it followsthat

(n−1)≤2Λ (13)

i.e.,ln^nΛ ≤cln^Λ.ThisclearlyyieldsthatNm≤cpnln^{1p+1 Λ}.

Remark 2.Theorem 1 provides explicit discrete meshes xj of cardinality ∼ nln^{1p+1 Λ} for the weighted Marcinkiewicz-Zygmund type inequalitieswith weights x_j+1−x_j. The nodes x_j are equidistributedwith respect to themeasure μ₁(E)=

E

x(1−x)dx , E ⊂(0,1).It shouldbe noted thatTheorem 1remainsvalid for any sequence of nodes satisfying (8) together with theproper restrictions onthe first and last nodes.

In addition, therestrictions of theexponents imposedin Theorem 1wereused only for thefirst and last intervals ofthe partition.This meansthat“near”discretization ofthe normfLp[δ,1−δ] ispossible using orderofnlog¹_δ nodeswithouttheseparationandthedegreerequirementsontheexponents.

The argumentsgiven in theproof of Theorem 1canbe used to derive aMarcinkiewicz-Zygmund type resultwith equalweights 1foruniformly distributeddiscrete meshesx₀:=h,x_j :=h+^1−2h_m j,0≤j ≤m leadingtothenext

Proposition 1.Let 1 ≤ p< ∞,0 < ≤ 1,n ∈ N,Λ > 1. Then given equidistributed discrete points sets x₀:=h,x_j :=h+¹⁻_m^2hj,0≤j≤m, h=c

nΛ

¹_p

of cardinality

m≤c₁n Λ

²_p

wherec,c₁>0areproper absoluteconstants, wehavethat 1

2m

1≤j≤m−1

|g(xj)|^p≤ g^p_Lp[0,1] ≤ 2 m

1≤j≤m−1

|g(xj)|^p (14)

forevery exponentialsum (2) satisfyingλ_j+1−λ_j ≥,1≤j≤n−1andmax_1≤j≤n|λ_j|≤Λ.

Proof. SincetheproofofthepropositionusesthesametechniqueasTheorem1wegiveabriefoutline.The BernsteintypeinequalityofLemma1canbereplacedbyestimate(5).Thensimilarlyto(12) wewillarrive at

f^p_Lp[0,1]− 1 m

0≤j≤m−1

|f(xj,m)|^p ≤ cpn

mhf^p_Lp[0,1]+c1h nΛ

¹_p

f^p_Lp[0,1]. Now setting h :=ξ

nΛ

¹_p

and m := ^pn_ξh with a proper 0< ξ <1 will lead to aMarcinkiewicz-Zygmund typeresultwithequalweights 1anduniformly distributedmeshofcardinality

m≤cn nΛ

_p¹

≤cn Λ

_p²

. Notethatthelastestimateabovefollowsfromtheinequality(13).

Remark 3.As noted at the end of the proof of Theorem 1under given assumptions on exponents λ_j we have that(n−1) ≤ 2Λ. Thus the quantity ^Λ appearing both in Theorem 1 and Proposition 1 ideally is of order n for proper distributions of exponents λj. This results in discrete meshes of cardinality ∼ nln^p1+1n, 1 ≤p<∞ inTheorem 1. Similarly inProposition 1when all weights equal 1the cardinality will be of order ∼ n^1+2p for every 1 ≤ p < ∞. In a recent paper [3], Theorem 2.3 the authors verified a weighted Marcinkiewicz-Zygmund type inequality for L^p,1 ≤ p ≤ 2 norm with meshes of cardinality

∼ nln³n and arbitrary n-dimensionalsubspaces of L^∞. This is a rather general beautiful result, but in contrast to Theorem 1 theweights andmeshes are notgiven explicitly. In additionin [3], Theorem 2.2 a Marcinkiewicz-Zygmund typeresultwith weights1isverifiedfor L^p,1≤p≤2 normswiththecardinality ofthediscretemeshbeingeffectedbythequantityappearinginarequiredNikolskiitypeinequality.Onthe other handit isknownthatforgeneral exponentialsumsthe magnitudeof thefactorsinaNikolskii type inequality depends onthe exponentsλ_j and it canbe verylarge, see [6], Theorem 1. Henceeven though Theorem 1and Corollary 3are lessgeneral incomparison to [3] in termsof subspacesconsideredfor the exponential sums our results provide a compensation in terms of explicit nature of meshes and weights, theiruniversality,smallercardinalityofdiscretemeshesand theirvalidityforevery1≤p<∞.

We canextend Marcinkiewicz-Zygmundtype inequalitiesfor generalmultivariate exponential sums, as well. Similarly to the univariate case this can be done by applying the Bernstein-Markov type estimates verifiedearlier.Inaddition,wewillhavetoovercomeatechnicaldifficultyrelatedtotheseparationcondition

|λj+1−λj|≥δ whichneeds tobe imposedon theexponents.Forthis end thenextauxiliaryproposition, see [7], Lemma5willberequired.

Lemma 4.Letλ_j ∈R^d,1≤j ≤ n,d ≥2satisfy the separationcondition |λ_j −λ_k| ≥δ > 0,j =k. Then for any w∈R^d,w=0any every >0there existu∈R^d,|w−u|≤, |w|=|u| sothat with some cd >0 depending only ondwe have

|λ_j−λ_k,u| ≥ cdδ^d−1

|w|^d−2n², ∀j=k. (15) The above lemma shows that the separation condition |λj −λk| ≥ δ > 0,j = k is preserved in a certain form when the exponentsare restricted to small perturbations of arbitrarily chosen lines. Clearly the orthogonal projections of λ_j-s into lines {tu : t ∈ R},u ∈ S^d−1 are given by λ_j,uu, where these projectionsareseparatedbyquantities|λj−λk,u|.

Consider the unit cube in R^d given by I^d := [0,1]^d. Using Bernstein-Markov type estimates verified aboveandseparationLemma4we canprovethenextMarcinkiewicz-Zygmundtypeinequalityforgeneral multivariateexponentialsumsonthecube.WeshallusebelowthenotationA∼Binordertoindicatethat c1(d,p)A≤B≤c2(d,p)Awithsomepositiveconstants c1(d,p),c2(d,p) dependingonlyonp,d.

Theorem 2.Let1≤p<∞,d,n∈N,Λ>1,0< δ <1,andconsiderany λj ∈R^d,1≤j≤nsatisfying

|λ_j−λ_k| ≥δ >0, j=k, max

1≤j≤n|λ_j| ≤Λ.

Then thereexistpositiveweights b1,...,bN anddiscretepointsetsYN={w1,...,wN}⊂I^d of cardinality N ≤c(d, p)n^dln^dp+dΛ

δ, so thatforevery exponential sum g(w)=

1≤j≤nc_je^λj,w,w∈R^d wehave

g^p_Lp(I^d₎∼

1≤i≤N

b_i|g(w_i)|^p. (16)

Proof. The proof of the theorem would be a straightforward product type argument if the separation condition|λj−λk|≥δ >0,j =k wastruefortheorthogonal projectionsofλj-sto everycoordinateaxis in R^d. Indeed, if this was the case then successive integration plus standard inductionwould accomplish theproof. Thus if|λ_j−λ_k,e_s|≥δ^∗ >0,j =kwith someδ^∗ >0 and e_s := (δ_s,i)_1≤i≤d ∈R^d,1≤s≤d being thestandard basis in R^d then for certain positive weights b₁,...,b_N and a discrete point set Y_N = {x1,...,xN}⊂I^d ofcardinalityN ≤c(d,p)n^dln^dp+d_δ^Λ∗

g^p_Lp(Id)∼

1≤i≤N

bi|g(xi)|^p, g(x) =

1≤j≤n

cje^λj,x. (17)

Nowwewill applyLemma4tocertainproperlychosenvectors. Forany< _4d¹ set e^∗_s:= (1−)e_s+

k=s

e_k, 1

2 <|e^∗_s|<2, 1≤s≤d.

Note that each coordinate of e^∗_s is not smaller than . By Lemma 4 for every 1 ≤ s ≤ d there exist u_s∈R^d,|e^∗_s−u_s|≤,|e^∗_s|=|u_s| sothat

|λ_j−λ_k,u_s| ≥ cdδ^d−1

n² , ∀j =k. (18)

Sinceallcoordinatesofe^∗_s are≥thecondition|e^∗_s−u_s|≤impliesthat u_s∈R^d₊:={x= (x₁, ..., x_d)∈R^d:x_j≥0,|x|= 1}. ConsidernowtheparallelepipedinR^d givenby

J^d:={t₁u₁+...+t_du_d, 0≤t_s≤ 1

1 + 2d, 1≤s≤d}. Sinceu_s∈R^d₊ and|e^∗_s−u_s|≤,1≤s≤ditiseasyto checkthatJ^d ⊂I^d.

Denote by T : R^d → R^d theregular matrixtransformation definedby T(e_s)= _1+2d^us ,1≤s≤d. Then evidently,T(I^d)=J^d.Furthermore,notethat

us

1 + 2d−e_s ≤

e^∗_s

1 + 2d−e_s

+≤ |e^∗_s−es|

1 + 2d + (2d+ 1)≤(3d+ 1), 1≤s≤d.

Evidently,thismeansthatforeveryx=

1≤s≤dtses∈I^d,0≤ts≤1 wehave

|x−T(x)|=

1≤s≤d

t_se_s−

1≤s≤d

t_s us

1 + 2d =

1≤s≤d

t_s

e_s− us

1 + 2d

≤d(3d+ 1).

Henceforanyx∈I^d\J^dwehavethatT(x)∈J^dand|x−T(x)|≤d(3d+ 1).ObviouslythismeansthatI^d iscontainedina1+d(3d+ 1)dilationofJ^d abouttheoriginwhichinturnimpliesthatμ(I^d\J^d)≤cd whereμstandsfortheLebesguemeasure.

Considernowanyexponentialsumg(w)=

1≤j≤nc_je^λj,w,w∈R^d with|λ_j−λ_k|≥δ >0,j=k.Since J^d⊂I^d andμ(I^d\J^d)≤c_dwehaveby(9)