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The classical inequalities such as the inequalities of Bernstein, Bohr, Nikol’skii and others play an important role in Analysis, the Approximation Theory and Applications (see 8,14 16). Recall now the BohrFavard inequality: Let σ > 0, 1 ≤ p ≤ ∞, f ∈ C m(R), supp ˆf ∩ (−σ, σ) = ∅, where ˆf is the Fourier transform of f, and Dmf ∈ Lp(R). Then f ∈ Lp(R) and (see 2, 5, 6, 7)
A NEW BOHR-NIKOL’SKII INEQUALITY HA HUY BANG & VU NHAT HUY Abstract. In this paper we give a new inequality of the Bohr-Nikol’skii type. 1. Introduction The classical inequalities such as the inequalities of Bernstein, Bohr, Nikol’skii and others play an important role in Analysis, the Approximation Theory and Applications (see [8],[14] - [16]). Recall now the Bohr-Favard inequality: Let σ > 0, 1 ≤ p ≤ ∞, f ∈ C m (R), suppfˆ ∩ (−σ, σ) = ∅, where fˆ is the Fourier transform of f , and Dm f ∈ Lp (R). Then f ∈ Lp (R) and (see [2, 5, 6, 7]): f p ≤ σ −m Km Dm f p, where the Favard constant Km are best possible and have the following properties 4 π 1 = K0 ≤ K2 < ... < < ... < K3 ≤ K1 = . π 2 The Bohr-Favard inequality has applications to the Approximation Theory: For example, let g ∈ C m (R), Dj g ∈ Lp (R), j = 0, 1, ..., m and we approximate g by a function h in some subclass of Lp (R). Putting g − h p = and applying the Bohr-Favard inequality, we get the following estimates for the errors if inf{|x| : x ∈ suppF (g − h)} = σ > 0: Dj g − Dj h p ≥ σ j Kj−1 , j = 1, 2, ..., m. In this paper we give a new inequality, which combines the inequality of BohrFavard and the Nikol’skii idea of inequality for functions in different metrics. 2. main result Let f ∈ L1 (R) and fˆ = F f be its Fourier transform +∞ 1 fˆ(ξ) = √ e−ixξ f (x)dx. 2π −∞ The Fourier transform of a tempered generalized function f is defined via the formula F f, ϕ = f, F ϕ , Let K be an arbitrary compact set in R and and K := {x ∈ R : ∃ξ ∈ K : |x − ξ| ≤ }. ϕ ∈ S(R). > 0. Denote by Dm f = (−i)m f (m) Now, we state our main theorem Key words and phrases. Lp - spaces, Bohr-Favard inequality, Nikol’skii inequality. 2010 AMS Subject Classification. 26D10. 1 2 HA HUY BANG & VU NHAT HUY Theorem 1. Let 1 ≤ q < p ≤ ∞, m ≥ 3, f ∈ Lp (R), σ > 0 and suppfˆ∩(−σ, σ) = ∅. Then there exists a constant C > 0 not depending on f, m, σ such that Dm f (1) q ≥ Cmλ σ m−λ f where λ= p, 1 1 − > 0. q p Before giving the proof of Theorem 1, we need the following result (see [17]): Young inequality. Let f ∈ Lp (R) and g ∈ Lq (R), where 1 ≤ p, q ≤ ∞ and 1 + 1q ≥ 1. Then f ∗ g ∈ Lr (R) and p f ∗g where r ≤ f p g q, 1 1 1 = + − 1. r p q Proof of Theorem 1. Let us first prove (1) for the case σ = 1. Indeed, put K := (−∞, −1] ∪ [1, +∞) and 1 η(ξ) = where C1 is chosen such that (φm (ξ))m∈N via the formula R if |ξ| < 1, if |ξ| ≥ 1, C1 e ξ2 −1 0 η(ξ)dξ = 1. We define the sequence of functions φm (ξ) = (1K3/(4m) ∗ η1/(4m) )(ξ), where η1/(4m) (ξ) = 4mη(4mξ). Then η1/(4m) (ξ) = 0 for all ξ ∈ [−1/(4m), 1/(4m)], R η1/(4m) (ξ)dξ = 1. Hence, for all m ∈ N we have φm (ξ) ∈ C ∞ (R), and φm (ξ) = 1 ∀ξ ∈ K1/(2m) , φm (ξ) = 0 ∀ξ ∈ / m m m K1/m . So, it follows from suppD f ⊂ K that φm (−ξ)D f = D f . Therefore, since Dm f = ξ m fˆ, we get φm (−ξ)Dm f = ξ m fˆ, and then Dm f φm (−ξ)/ξ m = fˆ. Hence, for m ≥ 3 f = (2π)−1/2 (Dm f ) ∗ F −1 (φm (−ξ)(ξ)m ) = (2π)−1/2 (Dm f ) ∗ F (φm (ξ)/(−ξ)m ). Therefore, since Young inequality, we have the following estimate for m ≥ 3 (2) f where p = (2π)−1/2 (Dm f ) ∗ F (φm (ξ)/ξ m ) p ≤ (2π)−1/2 Dm f 1 1 1 = + − 1. p q r q F (φm (ξ)/ξ m ) r , A NEW BOHR-NIKOL’SKII INEQUALITY 3 Since 1 ≤ q < p ≤ ∞, we have 1 < r ≤ ∞. We define for m ≥ 3 km := 1 + 1 , gm (ξ) = φm (km ξ), Φm (ξ) = φm (ξ) − gm (ξ). m Hence, (F (gm (ξ)/ξ m ))(x) = (km )m (F (φm (km ξ)/(km ξ)m )(x) = (km )m−1 (F (φm (ξ)/ξ m ))(x/km ). So, F (gm (ξ)/ξ m ) 1 r = (km )m−1+ r F (φm (ξ)/ξ m ) . r 1 Then it follows from (km )m−1+ r ≥ (km )m−1 = (1 + F (gm (ξ)/ξ m ) ≥ r 1 m−1 ) m ≥ 3 2 that 3 F (φm (ξ)/ξ m ) . 2 r Therefore, since Φm (ξ) = φm (ξ) − gm (ξ) we get F (Φm (ξ)/ξ m ) (3) r ≥ F (gm (ξ)/ξ m ) r − F (φm (ξ)/ξ m ) r 1 ≥ F (φm (ξ)/ξ m ) . 2 r From (2)-(3) we obtain (4) f p ≤ 2(2π)−1/2 Dm f q F (Φm (ξ)/ξ m ) r . Now, we will prove that there exists a constant C such that for all m ∈ N, m ≥ 3 F (Φm (ξ)/ξ m ) 1 r ≤ m−1+ r C(2π)1/2 /2. Indeed, put C2 = max{ η (j) 1 , j ≤ 3}. Since η1/(4m) (x) = 4mη(4mx), we obtain (j) η1/(4m) (x) = (4m)j+1 η (j) (4mx) and then (j) η1/(4m) 1 = (4m)j η (j) 1 ≤ C2 (4m)j , ∀j ≤ 3. Therefore, (5) φ(j) m (j) ∞ = (1K3/(4m) ∗ η1/(4m) ) (j) ∞ ≤ η1/(4m) 1 ≤ (4m)j C2 , ∀j ≤ 3. Note that φm (ξ) = 1 ∀ξ ∈ (−∞, −1 + (1/2m)] ∪ [1 − (1/2m), +∞) and φm (ξ) = 0 ∀ξ ∈ [−1 + (1/m), 1 − (1/m)]. So, if |ξ| < 1−(3/m) then |ξ| < |km ξ| < 1−(1/m) and then φm (ξ) = φm (km ξ) = 0, which implies Φm (ξ) = 0. Further, if |ξ| > 1 then |km ξ| > |ξ| > 1 and then φm (ξ) = φm (km ξ) = 1, which implies Φm (ξ) = 0. From these we have (6) suppΦm ⊂ [1 − (3/m), 1] ∪ [−1, (3/m) − 1]. Now, for ξ ∈ [1 − (3/m), 1] ∪ [−1, (3/m) − 1] we get (7) ξ − km ξ = ξ 1 ≤ . m m 4 HA HUY BANG & VU NHAT HUY From (5) and (7) we have the following estimate for ξ ∈ [1−(3/m), 1]∪[−1, (3/m)−1] Φm (ξ) = φm (ξ) − gm (ξ) = φm (ξ) − φ(km ξ) (8) ≤ ξ − km ξ . φm ∞ ≤ 1 4mC2 = 4C2 , m and (9) Φm (ξ) = φm (ξ) − gm (ξ) = φm (ξ) − φm (km ξ) = φm (ξ) − km φm (km ξ) ≤ φm (ξ) − φm (km ξ) + (1 − km )φm (km ξ) ≤ ξ − km ξ . φm ∞ + 1 − km . φm ∞ 1 ≤ (4m)2 C2 + 1 − km 4mC2 m ≤ 20mC2 . Put Hm (x) = (F (Φm (ξ)/ξ m ))(x). Then 1 Hm (x) = √ 2π e−ixξ Φm (ξ)/ξ m dξ. R Therefore, since (6), we have 1 sup Hm (x) ≤ √ 2π x∈R 1 Φm (ξ)/ξ m dξ = √ 2π Φm (ξ)/ξ m dξ 3 1− m ≤|ξ|≤1 R and it follows from (5) that (10) sup Hm (x) ≤ x∈R 3 6 24e4 C √ sup Φm (ξ) (1 − )−m ≤ √ 2 . m m 2π ξ∈R m 2π We also obtain that 1 sup xHm (x) = √ sup 2π x∈R x∈R e−ixξ ( mΦm (ξ) Φm (ξ) − )dξ ξ m+1 ξm R 1 ≤√ 2π mΦm (ξ) Φm (ξ) − dξ. ξ m+1 ξm R Therefore, since (5)-(6), we have (11) 1 sup xHm (x) ≤ √ 2π x∈R mΦm (ξ) Φm (ξ) − dξ ξ m+1 ξm 3 ≤|ξ|≤1 1− m ≤ ≤ 6 √ m 2π sup 3 1− m ≤|ξ|≤1 mΦm (ξ) Φm (ξ) − ξ m+1 ξm 6 3 3 √ sup Φm (ξ) m(1 − )−m−1 + sup Φm (ξ) (1 − )−m m m m 2π ξ∈R ξ∈R A NEW BOHR-NIKOL’SKII INEQUALITY 5 6 144e4 C2 √ 4C2 me4 + 20C2 me4 = √ . m 2π 2π We see for 1 < r < ∞ ≤ Hm r r r (12) Hm (x) dx Hm (x) dx + = r |x|≥m |x|≤m r 1 dx xr r ≤ sup Hm (x) 1dx + sup xHm (x) x∈R x∈R |x|≤m |x|≥m −r+1 r = 2m sup Hm (x) + x∈R 2m sup xHm (x) r − 1 x∈R r From (10)-(12), we obtain that 24e4 C2 r 2m−r+1 144e4 C2 r e4 C2 r 144r √ √ + 96r ), + = 2m−r+1 √ ( r − 1 r − 1 r m 2π 2π 2π and then for 1 < r < ∞ 1 1 1 e4 C2 144r 2 Hm ≤ √ ( (13) + 96r 2) r m−1+ r = m−1+ r /C. r 2π r − 1 √ 1 r 2 where C = 2π/e4 C2 ( 144 + 96r 2) r . r−1 Further, if r = ∞ then Hm ∞ = sup Hm (x) and so, r Hm ≤ 2m x∈R √ 4 ≤ 24e C /(m 2π). 2 ∞ Hm (14) Using (13) and (14), we obtain for 1 < r ≤ ∞ a constant C > 0 not depending on f, m, σ such that (15) F (Φm (ξ)/ξ m ) 1 ≤ m−1+ r (2π)1/2 /(2C). r = Hm p ≤ m p − q Dm f q /C r From (4) and (15) we have f 1 1 and then 1 1 1 1 Dm f q ≥ Cm q − p f p . So, (1) have been proved for the case σ = 1. Next, we prove (1) for any σ > 0. Put x g(x) = f ( ). σ ˆ Then it follows from suppf ∩ (−σ, σ) = ∅ that suppˆ g ∩ (−1, 1) = ∅. Therefore, Dm g (16) q ≥ Cm q − p g p . Since g(x) = f ( σx ), we have g 1 p = σp f p, Dm g 1 q = σ −m+ q Dm f q . Hence, it follows from (16) that 1 σ −m+ q Dm f 1 q 1 1 ≥ Cm q − p σ p f p. 6 HA HUY BANG & VU NHAT HUY Hence, Dm f 1 q 1 1 1 ≥ Cm q − p σ m+ p − q f p. The proof is complete. Corollary 2. Let 1 ≤ q < p ≤ ∞, f ∈ Lp (R), σ > 0 and suppfˆ ∩ (−σ, σ) = ∅. Then for all a < 1q − p1 we have the following limit lim m→∞ Dm f q /(ma σ m ) = +∞. In comparison, using Bohr-Favard inequality, we get that the sequence { Dm f q /σ m }∞ m=1 is separated with the origin while by Corollary 2 we have the stronger result: limm→∞ Dm f q /(ma σ m ) = +∞ for all a < 1q − p1 . In the following theorem, we give a result for the sequence of Lp (R)−norm of primitives of a function (see the notion of the primitive of functions in Lp (R) in [3], [17]): Theorem 3. Let 1 ≤ q < p ≤ ∞, σ > 0, f ∈ Lq (R), suppfˆ ∩ (−σ, σ) = ∅ and 0 m m−1 f for m = 1, 2, . . . . {I m f }∞ m=0 ⊂ Lp (R), where I f = f , I f is a primitive of I Then for a < 1q − p1 we have the following limits lim ma σ m I m f m→∞ p = 0, In particular, lim σ m I m f m→∞ p = 0. Proof. It was shown in [3] that suppI m f = suppfˆ ∀m ∈ N. Therefore, suppI m f ∩ (−σ, σ) = ∅ and then it follows from Theorem 1 that f 1 q 1 1 1 ≥ Cm q − p σ m+ p − q I m f p. Hence, lim ma σ m I m f m→∞ p = 0. The proof is complete. From Theorem 1 and the Bohr-Favard inequality we have the following result: Corollary 4. Let 1 ≤ q < p ≤ ∞, σ > 0. Denote Nσ,p := {f ∈ Lp (R) : suppfˆ ⊂ (−∞, −σ] ∪ [σ, +∞)} and γm := inf f ∈Nσ,p Dm f σm f Then γm ≤ π2 γm+1 and lim γm = ∞. m→∞ q p . A NEW BOHR-NIKOL’SKII INEQUALITY Note that if p = q then γm = Km does not hold. 7 ∀m ∈ N and the conclusion limm→∞ γm = ∞ Consecutively applying Theorem 1 to each variable, we get the following result for the n-dimensional case: Theorem 5. Let 1 ≤ q < p ≤ ∞, σj > 0, α = (α1 , . . . , αn ) ∈ Zn+ , αj ≥ 3 (j = 1, . . . , n), f ∈ Lp (Rn ) and suppfˆ ⊂ nj=1 ((−∞, −σj ] ∪ [σj , +∞)). Then there exists a constant C > 0 not depending on f, α, σ such that n α D f q α −λ αjλ σj j ≥C f p, j=1 where λ= 1 1 − > 0, q p σ = (σ1 , σ2 , . . . , σn ). Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.32. A part of this work was done when the authors were working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the VIASM for providing a fruitful research environment and working condition. References [1] Akhiezer N.I., Theory of Approximation, F. Ungar (1956) (Translated from Russian). [2] Bang H.H., An inequality of Bohr and Favard for Orlicz spaces, Bull. Polish Acad. Sci. Math. 49 (2001), 381 - 387. [3] Bang H.H. and Huy V.N., Behavior of the sequence of norms of primitives of a function, J. Approximation Theory 162 (2010), 1178 -1186. [4] Baskakov A. G. and Sintyaeva K. A., The Bohr-Favard inequalities for operators, Russian Mathematics 53 (2009), Issue 12, 11-17. [5] Bohr H., Ein allgemeiner Satz u ¨ber die integration eines trigonometrischen Polynoms, Prace Matem.-Fiz, 43(1935), 273-288. [6] Favard J., Application de la formule sommatoire d’Euler a la d´emonstration de quelques propri´et´es extr´emales des integrales des fonction p´eriodiques et presquep´eriodiques, Mat. Tidsskr. M, (1936), 81-94. [7] H¨ ormander L., A new generalization of an inequality of Bohr, Math. Scand. 2 (1954), 33-45. [8] Mitrinovic D. S., Pecaric J. E. and Fink A. M., Inequalities Involving Functions and Their Integrals and Derivatives, Dordrecht, Netherlands: Kluwer, (1991), 71-72. [9] Northcott D. G., Some Inequalities Between Periodic Functions and Their Derivatives, J. London Math. Soc. 14 (1939), 198-202. [10] Nessel R. J., Wilmes G., Nikol’skii - type inequalities in connection with regular spectral measures. Acta Math. 33(1979), 169 - 182. [11] Nessel R. J., Wilmes G., Nikol’skii - type inequalities for trigonometric polynomials and entire functions of exponential type. J. Austral. Math. Soc. 25(1978), 7-18. [12] Nikol’skii S. M., Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables. Trudy Steklov Inst. Mat. 38(1951), 244278. [13] Nikol’skii S. M., Some inequalities for entire functions of finite degree and their application, Dokl. Akad. Nauk SSSR, 76(1951), 785-788 . 8 HA HUY BANG & VU NHAT HUY [14] Nikol’skii S. M., Approximation of Functions of Several Variables and Imbedding Theorems, Berlin: Springer, 1975. [15] Tikhomirov V. M., Approximation Theory, In Analysis II. Convex Analysis and Approximation Theory (Ed. R. V. Gamkrelidze). New York: Springer-Verlag, (1990), 93-255. [16] Triebel H., Theory of Function Spaces, Basel, Boston, Stuttgart: Birkh¨auser, 1983. [17] Vladimirov V.S., Methods of the theory of Generalized Functions, Taylor & Francis, London, New York, 2002. HA HUY BANG Institute of Mathematics, Vietnamese Academy of Science and Technology, 18 Hoang Quoc Viet Street, Cau Giay, Hanoi, Vietnam E-mail address: hhbang@math.ac.vn VU NHAT HUY Department of Mathematics, College of Science, Vietnam National University, 334 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam E-mail address: nhat huy85@yahoo.com ... Science and Technology, 18 Hoang Quoc Viet Street, Cau Giay, Hanoi, Vietnam E-mail address: hhbang@math.ac.vn VU NHAT HUY Department of Mathematics, College of Science, Vietnam National University,... J Approximation Theory 162 (2010), 1178 -1186 [4] Baskakov A G and Sintyaeva K A. , The Bohr-Favard inequalities for operators, Russian Mathematics 53 (2009), Issue 12, 11-17 [5] Bohr H., Ein allgemeiner... of finite degree and their application, Dokl Akad Nauk SSSR, 76(1951), 785-788 HA HUY BANG & VU NHAT HUY [14] Nikol skii S M., Approximation of Functions of Several Variables and Imbedding Theorems,