Some inequalities for the ”derivatives” of iterated linear operators will be presented, which will be applied for the investigation of degrees of approximation.

Thus, with the application of the Laplacian we improve some classical results concerning the Jackson type estimate, the inverse theorem as well as the saturation phenomeno

Hans-Bernd Knoop

Xinlong Zhou

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 30 (2001) no. 2, pp. 197–206ictp.acad.ro/jnaatINEQUALITIES FOR SOME ITERATED LINEAR OPERATORSAND THEIR APPLICATIONSIN APPROXIMATION THEORYHANS-BERND KNOOP∗ and XINLONG ZHOU∗Dedicated to Professor Dr. W. Haussmannon the occasion of his 60th birthdayAbstract. Some inequalities for the ”derivatives” of iterated linear operatorswill be presented, which will be applied for the investigation of degrees of ap-proximation. Thus, with the application of the Laplacian we improve someclassical results concerning the Jackson type estimate, the inverse theorem aswell as the saturation phenomenon.MSC 2000. 42A05, 41A17, 41A25, 41A35.1. INTRODUCTIONLet Xp,d be either Lp(T d), the Lp-space of periodic functions with T =[−π, π] or Lp(Rd), 1 ≤ p ≤ +∞ (for p = +∞, let L∞(T d) = C(T d) andL∞(Rd) be the set of bounded functions in C(Rd)). Denote by || · ||Xp,d theusual norm on Xp,d. Denote further Di := ∂/∂xi and Dα := Dα1i · · ·Dαdd ,where the multiindex α = (α1, . . . , αd) has non-negative integers αi. Thelength of α is |α| = α1 + . . . + αd. Let πn,d be the set of trigonometricpolynomials of degree ≤ n with respect to each variable xi, i = 1, ..., d.The Peetre K-functional defined on Xp,d is given byKk(f, t)Xp,d := infg∈Wkp,d{||f − g||Xp,d + tk sup|α|=k||Dαg||Xp,d},where W kp,d is the Sobolev space given by∗Gerhard-Mercator-University of Duisburg, Department of Mathematics, D-47048 Duisburg, Germany, e-mail: knoop@math.uni-duisburg.de, zhou@informatik.uni-duisburg.de.198 Hans-Bernd Knoop, Xinlong Zhou 2W rp,d := {f ∈ Xp,d : Dαf ∈ Xp,d, |α| ≤ r}.It is known (see [1] and [4]) that Kk(f, t)Xp,d is equivalent to the modulus ofsmoothness ωk(f, t)Xp,d , defined byωk(f, t)Xp,d := sup|h|≤t,h∈Rd{∥∥∥ k∑j=0(−1)j(kj)f(·+ jh)∥∥∥Xp,d}.Two basic results in Approximation Theory are the so-called Bernstein in-equality and the Jackson inequality.Roughly speaking, in case of approximation by πn,d the Bernstein inequalitytells us that the Lp-norm of the derivative of a polynomial can be estimatedby the product of the Lp-norm of the polynomial and the degree of the corre-sponding polynomial. For example (see [12])(1.1) ||DαTn||Lp(T d) ≤ Cn|α|||Tn||Lp(T d), for 1 ≤ p ≤ ∞, Tn ∈ πn,d,while the Jackson inequality says that the best approximation constant Ep,d(f)for f ∈ Lp(T d) in case of approximation by πn,d can be estimated if thesmoothness of the given function is known. The second result was pointed outby Jackson in 1912 (see [5]) in case d = 1 and p = ∞. To mention his resultletkn,r(t) = αn,r ·sin2r(nt/2)sin2r(t/2),where the constant αn,r is such that ||kn,r||L1(T ) = 1. Let Kn,r,d be the tensor-product of the kernels kn,r, namelyKn,r,d(t) =d∏i=1kn,r(ti), t = (t1, . . . , td),and let Jn,r,d be the corresponding convolution operator with the kernel Kn,r,d,i.e.,Jn,r,df = Kn,r ∗ f,the so-called Jackson operator. Jackson’s result may be formulated as||f − Jn,2,1f ||Lp(T ) ≤ C2ω2(f, 1n)Lp(T ).3 Inequalities for some iterated linear operators and applications 199If we consider the best approximation polynomials as operators of the ap-proximated function, then these operators are in general not linear. Thereare however only a few cases for which one can get the exact form of thebest approximation polynomials. Thus, the linear method or the linear oper-ator approach is a good alternative for the approximation by polynomials, inparticular the linear positive method which has been attracting special inter-est in the past. In connection with the approximation degree for the linearmethod one investigates, among others, the Jackson type estimate (i.e., theupper bound of the approximation error), the inverse theorem and the satura-tion phenomenon. We refer to [2], [11], [12] and [14] for detailed informationconcerning these subjects. In [6] (see also [3] and [7]) we introduced a newtechnique. Using this technique we unify the above mentioned subjects for alarge class of positive linear operators in one form. In this way we obtain forexample the best lower and upper estimate for the Bernstein operator (see [7]and [15]) which completely characterizes the approximation behaviour of thisoperator. In the present paper we shall give a brief view of our investigationconcerning this subject. In Section 2 we shall point out the relation betweenthe derivative of iterated operators and the approximation behaviour of theconsidered operators. In Section 3 we give some applications of our technique(or the results in Section 2) for approximation by trigonometric polynomialsin several variables.2. ITERATION OF CONVOLUTION OPERATORSFor positive convolution operators the following results generalize the theo-rems in [6] (see also [17]). Let Φ ∈ X1,d be a positive kernel. The convolutionoperator L defined by this Φ isLf = f ∗ Φ,which satisfies Lf = 1 for f ≡ 1 and Lf ∈ Xp,d whenever f ∈ Xp,d, 1 ≤ p ≤ ∞.We shall denote by Φj the kernel of Lj , which is given by L1 := L andLjf := L(Lj−1f). The key step of our technique is the so-called iterationinequality. For better understanding we shall present this inequality for thecase Lp(R) and give a much more natural proof. The proof is different fromthe one in [6].Lemma 2.1. [9]. Suppose Φ ∈W 1∞,1. Then for all f ∈ Xp,1, 1 ≤ p ≤ +∞,||(LN (f))′||Xp,1 ≤ 1ε√N ||f ||Xp,1 , ∀f ∈ Xp,1, N = 1, 2, ...,whereε−2 =∫R(Φ′(t))2Φ(t) dt < +∞.200 Hans-Bernd Knoop, Xinlong Zhou 4Proof. We notice that if we can verify this inequality for p = 1 and p = +∞,then, via the Riesz-Thorin theorem (see Chapter XII [18]), this inequality holdsfor all 1 ≤ p ≤ +∞. Furthermore, it is enough to prove it only for p = +∞.To see this, we note that, for all f ∈ X1,1, g ∈ X∞,1 with f∗(x) := f(−x) andg∗(x) := g(−x), it holds that∫R∫Rg(x)f(t)Φ′(x− t)dtdx =∫R∫Rf∗(t)g∗(x)Φ′(t− x)dxdtor ∫R(L(f))′ · g =∫Rf∗ · (L(g∗))′.As LN is again a convolution operator, the last identity implies∣∣∣∣∫R(LN (f))′ · g∣∣∣∣ = ∣∣∣∣∫Rf∗ · (LN (g∗))′∣∣∣∣ ≤ 1ε√N ||f ||X1,1 ||g||X∞,1 ,which yields the assertion for p = 1 by the duality argument. Thus, it remainsto prove our inequality only for p = +∞. To this end, let us observe that onecan write Lf asLf(x) =∫Rf(t+ x)Φ(t)dt.Hence,LNf(x) =∫RNf(t1 + t2 + ...+ tN + x)Φ(t1)Φ(t2) · · ·Φ(tN )dt1dt2 · · · dtN .Writing t1 + t2 + ...+ tN +x = (t1 +x/N) + (t2 +x/N) + ...+ (tN +x/N) andreplacing tj + x/N, j = 1, 2, ..., N in the last formula by tj , j = 1, 2, ..., N, weobtainLNf(x) =∫RNf(t1+t2+...+tN )Φ(t1− xN )Φ(t2−xN ) · · ·Φ(tN−xN )dt1dt2 · · · dtN .It is now clear that the derivative of LNf(x) is(LN (f))′(x) == −∫RNf(t1 + t2 + ...+ tN ) 1NK(t1, t2, ..., tN , x)N∑j=1(Φ(tj − xN ))′Φ(tj − xN )dt1 · · · dtN ,where K(t1, t2, ..., tN , x) = Φ(t1 − xN )Φ(t2 −xN ) · · ·Φ(tN −xN ). Denote5 Inequalities for some iterated linear operators and applications 201lj :=(Φ(tj − xN ))′Φ(tj − xN ).Using Cauchy’s inequality we get, because of the positivity of K,∣∣∣(LN (f))′(x)∣∣∣ ≤ 1N (∫RN |f |2K) 12(∫RNK ·( N∑j=1lj)2) 12.It is clear that the first integral can be estimated by∫RN|f |2K ≤ ||f ||2X∞,1as∫RN K = 1. On the other hand, since (∑Nj=1 lj)2 =∑Nj=1 l2j +2∑i<j lilj and∫R Φ(ti −xN )dti = 1 we conclude that∫RNK ·( N∑j=1lj)2 = N∑j=1∫RNKl2j + 2∑i<j∫RNKlilj= Nε+ 2∑i<j∫R2Φ′(ti − xN )Φ′(tj − xN )dtidtj= Nε.The desired assertion for the case p = +∞ follows from the last two inequali-ties. Lemma 2.1 allows us to improve some well-known inverse results for theapproximation by positive linear operators. Thus, we haveTheorem 2.2. [9], [17]. Let there exist some j ≥ 1 such that Φj ∈ W 1∞,dandε−2 :=∫Md|grad Φj(t)|2Φj(t)dt < +∞ ,where M = R or M = T . Then for fixed k ∈ N there exists a constant Ck > 0,which does not depend on ε, such that for all f ∈ Xp,d and 1 ≤ p ≤ ∞ we haveK2k+1(f, ε)Xp,d ≤ Ck ||(I − L)kf)||Xp,d .To present further results we need the concept of conjugate functions incase d = 1. It is known that the conjugate function f̃ of f is defined in the202 Hans-Bernd Knoop, Xinlong Zhou 6periodic case byf̃(x) := − 12π∫ π0f(x+ t)− f(x− t)tan t2dt(cf. [18]) and for the real line R byf̃(x) :=∫ ∞0f(x+ t)− f(x− t)tdt,respectively. In case 1 < p <∞ one has (see [18])||g̃||Xp,1 ≤ Cp||g||Xp,1 .The above inequality is not true for p = 1 and p = ∞. In fact, the followingestimate for trigonometric polynomials is in general sharp:For p = 1 or p =∞ one has (see [18])||T̃n||Lp(T ) ≤ C lnn||Tn||Lp(T ), ∀Tn ∈ πn,1.On the other hand, it is known that for polynomials and their conjugatefunctions one has the so-called Szegö-inequality (see [18]):For all 1 ≤ p ≤ ∞ there holds||T̃ ′n||Lp(T ) ≤ Cn||Tn||Lp(T ), ∀Tn ∈ πn,1.Now let d be greater than 1, Lig be the conjugate function of g with respectto xi and D̃i := DiLi. In this way we define D̃α = D̃α11 . . . D̃αdd . With thesenotations the Szegö-inequality in multivariable version is(2.1) ||D̃αTn||Lp(T d) ≤ Cαn|α|||Tn||Lp(T d), ∀Tn ∈ πn,d.Using our technique we get a generalization of the Szegö-inequality:Theorem 2.3. [8]. Let 1 ≤ p ≤ ∞, k ∈ N, and the dimension d ≥ 1 befixed. Then there exists a positive constant Ck > 0 such that for all ε > 0, alln and all Tn ∈ πn,d one hassup|α+β|=2k+1||DαD̃βTn||Lp(T d) ≤(2.2)≤ Ck{1ε2k+2sup|α|=2k+1||DαTn||Lp(T d) + εn||∆kTn||Lp(T d)}.7 Inequalities for some iterated linear operators and applications 203The following result characterizes completely the quantitative approxima-tion behavior for some positive linear operators.Theorem 2.4. [6], [9], [10], [17]. Let the condition of Theorem 2.2 be fulfilledand 1 ≤ p ≤ +∞ be fixed. If there exists some constant A > 0 such that forall g ∈W 3p,d(2.3) ||g − Lg − η(ε)∆g||Xp,d ≤ Aε3 sup|α+β|=3||DαD̃βg||Xp,dwith η(ε) ∼ ε2, then for some Ck > 0, which does not depend on ε, the in-equalities(2.4) C−1k Kk∆(f, ε)Xp,d ≤ ||(I − L)kf ||Xp,d ≤ Ck Kk∆(f, ε)Xp,d , ∀f ∈ Xp,dare valid, where Kk∆(f, ·)Xp,d is the K-functional for f , defined byKk∆(f, t)Xp,d = infg∈W 2kp,d(T d){||f − g||Xp,d + t2k||∆kg||Xp,d}with the Laplacian ∆ and the identity operator I.It is known (see e.g [13]) that if 1 < p <∞ thensup|α|=2k||Dαg||Xp,d ≤ Cp,d,k||∆kg||Xp,d .Thus K2k(f, t)Xp,d is equivalent to Kk∆(f, t)Xp,d . For p = 1 or p =∞ howeverone can use the approach of [13] to show that these K-functionals are notequivalent. In general (see [17] for the detail) we have for some C > 0, whichdoes not depend on f ∈ Xp,d and t, the estimate(2.5) C−1K2k+1(f, t)Xp,d ≤ Kk∆(f, t)Xp,d ≤ CK2k(f, t)Xp,d .Obviously, (2.4) unifies the so-called direct estimate, inverse theorem andsaturation theorem in one form for those positive linear operators, which sat-isfy the conditions of Theorems 2.2 and 2.4.3. APPLICATONSIn this section we shall give two applications concerning the approximationby trigonometric polynomials in several variables. We study first the Jacksonoperators Jn,r,d with r ≥ 2. It is not hard to verify that these operators satisfyall the conditions of Theorem 2.3. Thus, the following inequalities give thefinal version for the approximation degree of Jackson operators204 Hans-Bernd Knoop, Xinlong Zhou 8Theorem 3.1. [9]. For fixed d ≥ 1, r ≥ 2 and k ≥ 1 there exists a positiveconstant Cr,d,k > 0 such that for all 1 ≤ p ≤ ∞, for all f ∈ Lp(T d) and alln ≥ 1(3.1) C−1r,d,kKk∆(f, 1n)Lp(T d) ≤ ||(I − Jn,r)kf ||Lp(T d) ≤ Cr,d,kKk∆(f, 1n)Lp(T d).Our method allows us also to improve some classical estimates about theapproximation by trigonometric polynomials. For d = 1 these results maybe found in Chapter 7 of [2]. Thus, the direct and inverse results may beformulated asTheorem 3.2. For fixed d ≥ 1, k ≥ 1 and 1 ≤ p ≤ ∞ there exists C > 0such that the following estimates hold:(3.2) En(f)p,d ≤ Cn2En(∆f)p,d, ∀f ∈W2p (T d);(3.3) En(f)p,d ≤ CKk∆(f, 1n)Lp(T d), ∀f ∈ Lp(Td);(3.4) Kk∆(f, h)Lp(T d) ≤ Ch2k ∑0≤n≤1/h(n+ 1)2k−1En(f)p,d, ∀f ∈ Lp(T d).Proof. The second inequality of (2.5) and the estimate in [12] (see pages197-204 there) imply (3.4), while (3.3) is a simply consequence of (3.2). Thus,we need only to show (3.2). It is known (see page 197 of [12]) that En(f)p,d ≤CK3(f, 1n)p,d. Thus, (2.5) implies En(f)p,d ≤ Cn−2||∆f ||p,d. On the otherhand, as for any Tn ∈ πn,d there holds En(f)p,d = En(f − Tn)p,d, we concludethatEn(f)p,d ≤ Cn−2||∆f −∆Tn||Lp(T d), ∀Tn ∈ πn,d.Assume T ∗n ∈ πn,d to be a best approximation of ∆f , so there exists a Tn ∈ πn,dsuch that ∆Tn = c+ T ∗n with c = −∫T ∗n . We notice∫∆f = 0. Thus, we canestimate c by |c| ≤∫|∆f − T ∗n | ≤ CEn(∆f)p,d. Therefore,En(f)p,d ≤ Cn−2||∆f −∆Tn||Lp(T d) ≤ Cn−2En(∆f)p,d,which gives (3.2). For applications (e.g., in Computer Aided Geometric Design) sometimes oneneeds to know not only the distance between the approximation polynomialand the function which is approximated, but the uniform smoothness of thispolynomial sequence in connection with this function as well. For a bestapproximation sequence {T ∗n} of f one has (see [16], [17])9 Inequalities for some iterated linear operators and applications 2051n2k||∆kT ∗n ||Lp(T d) ≤ CkKk∆(f, 1n)Lp(T d).The following result shows that, in fact, if the distance between the approxima-tion polynomial and the approximated function is bounded by a K-functional,then the uniform smoothness of this sequence can always be estimated by thisK-functional.Theorem 3.3. [9]. Let f ∈ Lp(T d) and the polynomial sequence {Tn},Tn ∈ πn,d satisfy||Tn − f ||Lp(T d) ≤ CkKk∆(f, 1n)Lp(T d).Then there exists C ′k > 0 such that for j = 1, ..., k the following estimates hold:1n2j ||∆jTn||Lp(T d) ≤ C′kKj∆(f,1n)Lp(T d),andKj∆(Tn, t)Lp(T d) ≤ C′kKj∆(f, t)Lp(T d).For d = 1 the above assertions follow from a lemma by Zamansky [2].REFERENCES[1] Bennett, C. and Sharply, R., Interpolation of Operators, Academic Press, New York,1988.[2] DeVore, R. A. and Lorentz, G. G., Constructive Approximation, Springer-Verlag,Berlin, 1993.[3] Gonska, H. H. and Zhou, X. L., The strong converse inequality for Bernstein-Kantorovich operators, Comput. Math. Appl., 30 (3–6), pp. 103–128, 1995.[4] Johnen, H. and Scherer, K., On the equivalence of the K-functional and moduliof continuity and some applications, In: Constructive Theory of Functions of SeveralVariables (Proc. Conf., Oberwolfach 1976), 119–140. Lecture Notes in Mathematics 571,Springer-Verlag, 1977.[5] Jackson, D., On the approximation by trigonometric sums and polynomials, TAMS,13, pp. 491–515, 1912.[6] Knoop, H.-B. and Zhou, X. L., The lower estimate for linear positive operators, I.,Constr. Approx., 11, pp. 53–66, 1995.[7] Knoop, H.-B. and Zhou, X. L., The lower estimate for linear positive operators, II.,Results Math., 25, pp. 315–330, 1994.[8] Knoop, H.-B. and Zhou, X. L., Some inequalities for trigonometric polynomials andtheir derivatives, International Series of Numerical Mathematics, Vol. 132, Birkhäuser-Verlag 1999, pp. 87–94.[9] Knoop, H.-B. and Zhou, X. L., Inequalities for trigonometric polynomials and someintegral means, to appear in J. Mathematical Analysis and Applications.[10] Knoop, H.-B. and Zhou, X. L., On Jackson integral operator, (preprint).[11] Lorentz, G. G., Approximation of Functions, Holt, Rinehart and Winston, New York,1966.206 Hans-Bernd Knoop, Xinlong Zhou 10[12] Nikol’skii, S. M., Approximation of Functions of Several Variables and ImbeddingTheorems, Springer-Verlag, Berlin, 1975.[13] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, PrincetonUniversity Press, Princeton, 1975.[14] Timan, A. F., Theory of Approximation of Functions of a Real Variable, Physics andMathematics Publishing House, Moscow, 1960.[15] Zhou, X. L., Approximation by multivariate Bernstein operators, Results Math., 25,pp. 166–191, 1994.[16] Zhou, X. L., Degree of approximation associated with some elliptic operators and itsapplications, Approx. Theory Appl., 2, pp. 9–29, 1995.[17] Zhou, X. L., Approximationsordnung und Regularität von Differentialoperatoren, Ha-bilitationsschrift, Gerhard-Mercator-Universität Duisburg, 1996.[18] Zygmund, A., Trigonometric Series. Vol. I and II, 2nd ed., Cambridge UniversityPress, Cambridge, 1990.Received December 15, 2000.

Inequalities for some iterated linear operators and their applications in approximation theory

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Inequalities for some iterated linear operators and their applications in approximation theory

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