WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE
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1 WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNTIONS OF EXPONENTIAL TYPE DORON S. LUBINSKY A. We prove eighted Markov-Bernstein inequalities of the form f x p x dx σ + p f x p x dx Here satisfies certain doubling type properties, f is an entire function of exponential type σ, p >, and is independent of f and σ. For example, x = + x 2 α satisfies the condtions for any α R. lassical doubling inequalities of Mastroianni and Totik inspired this result. Entire functions of exponential type, Bernstein inequalities 425 In honor of the retirement of Giuseppe Mastroianni. I The classical Markov-Bernstein inequality for the unit circle asserts that for polynomials P of degree n, and < p,. P LpΓ n P L pγ. Here Γ is the unit circle, and if p <, P LpΓ = π π P e iθ p dθ /p. Of course, it as earlier proved for p, and later for < p <, by Arestov []. There is a close cousin for entire functions f of exponential type σ, and < p :.2 f LpR σ f L pr. It too as earlier proved for p, and later for < p <. See [5]. In fact, these inequalities are equivalent, and can be derived from each other - as follos, for example, from the methods of [] here there is a similar equivalence beteen Marcinkieicz-Zygmund and Plancherel-Polya inequalities. These are yet more illustrations of the classical link beteen approximation theory for polynomials and that for entire functions of exponential type, amply explored in the memoir of Ganzburg [8], and in the books of Timan [7], and Trigub and Belinsky [8], for example. There is a vast literature on Markov-Bernstein inequalities, both for polynomials [5], [2], [4], and entire functions of exponential type. For the latter, there are Szegő type inequalities, and sharp inequalities for various subclasses of entire functions ith special properties - see [4], [6], [6]. In another direction, eighted Bernstein Date: December 8, 23. Research supported by NSF grant DMS82 and US-Israel BSF grant 28399
2 2 DORON S. LUBINSKY inequalities involving inner functions, and model spaces have been investigated by Baranov [2], [3]. For polynomials, one of the most beautiful results involves doubling eights, and is due to Mastroianni and Totik [3]. Recall the setting: let W : [ π, π] [, be measurable. Extend W as a 2π periodic function to the real line. We say that W is doubling if there is a constant L called a doubling constant for W such that for all intervals I, e have W L W. 2I I Here 2I is the interval ith the same center as I, but ith tice the length. A typical doubling eight is k W t = h t t β γ, = here h is bounded above and belo by positive constants, and all { } β are distinct and lie in [ π, π], hile all γ >. An immediate consequence of Theorem 4. in [3, p. 45] is that for p <, π π P e iθ p W θ dθ n p π π P e iθ p W θ dθ, valid for n and all polynomials P of degree n. This as extended to < p < by Erdelyi [7]. The constant depends only on p and the doubling constant L, not on the particular W. In this paper, inspired by the results of Mastroanni, Totik, and Erdelyi, e prove eighted Markov-Bernstein inequalities. Our most general result follos: Theorem Let σ, p >, r, ], and let : R [, be a measurable function satisfying the folloing: I The one-sided doubling condition about : there exists L >, such that for a r, 2a a.3 L. a II The groth condition about integers: there exist B, β such that for k and max { } 2k +, r,.4 r+r r a/2 kr+r B + r k β. Assume also the analogous condition for k <. For t R, let.5 r t = 2r t+r t r kr s ds. Then for entire functions f of exponential type σ, e have.6 f t p r t dt σ + p f t p r t dt, provided the right-hand side is finite. Here depends on B, β, p and L, but is independent of σ, r, f, and the particular.
3 3 orollary 2 Let p >. Assume that all the conditions of Theorem hold for some r,, and all r, r, ith L, B and β independent of r. Then for σ >, and entire functions f of exponential type σ, e have.7 f t p t dt σ + p f t p t dt, provided the right-hand side is finite. Here depends on B, β and L, but is independent of σ, f, and the particular. orollary 3 Let σ, p >, and let : R, be a measurable function satisfying the folloing: for some M, e have for both 2 s t 2 and s t 2,.8 s / t M, for t R\ {}. M Then for entire functions f of exponential type σ, e have.9 f t p t dt σ + p f t p t dt, provided the right-hand side is finite. Here depends on M, but is independent of σ, and f. orollary 4 Let σ, p >, and α R. Then for entire functions f of exponential type σ, e have. f t p + t 2 α p dt σ + f t p + t 2 α dt, provided the right-hand side is finite. Here is independent of σ and f. To the best of our knoledge, even the inequalities in orollary 4 are ne. Almost all existing inequalities in the literature are uneighted, though they involve sharp constants as in.2. We note that if = λ < λ 2 <..., and f x = m = c λ ix, e used orthogonal Dirichlet polynomials in [] to prove f x 2 + x 2 dx /2 {log λ m + log λ m /2} f x 2 + x 2 dx /2. Here one cannot replace log λ m +log λ m /2 by any factor smaller than log λ m + for some >. This inequality reflects the fact that f is entire of type log λ m. We prove the results in Section 2. Throughout,,, 2,... denote positive constants independent of f, σ, r. The same symbol does not necessarily denote the same constant in different occurrences. Throughout, e let 2. Proof of The Results S t = sin πt πt
4 4 DORON S. LUBINSKY { } denote the sinc kernel. We ll use the bounds S t min, π t. We begin by applying.2 to [ 2. g t = f t S here l is a fixed positive integer. This yields: t t + is l l + ] l, 2 Lemma 2. Let l, p >, and f be entire of exponential type σ. Then 2.2 f t p + t lp dt σ + p f t p + t lp dt, here is independent of f and σ. Proof Let t t h t = S + is l l +, 2 so that g t = f t h t l. First note that for real t, { } l 2.3 h t min 2, π t + l + t, π t + l/2 here depends only on l. By.2, and some simple calculations, also, 2.4 h t + t, here again depends only on l. In the other direction, e see that 2.5 h t 2 = 2 sin π t 2 l cos π t l π t + l π t l + 2 sin π t 2 l + cos π t 2 l π t l t 2. 2 Then, recalling 2., f t h t l g = t f t lh t l h t 2.6 g t + f t + t l, by 2.3 and 2.4. No g is entire of exponential type σ +, and 2.3 shos that g t p dt f t p + t lp dt <, so applying.2 to g gives g t p dt σ + p g t p dt σ + p f t p + t lp dt.
5 5 Together ith 2.6, and 2.5, this yields So e have the result. From this e deduce: f t p + t lp dt f t h t l p dt {σ + p + } g t p + f t p + t lp dt f t p + t lp dt. Lemma 2.2 Let σ, p >, l, and let : R [, be a measurable function. Let 2.7 H t = x dx, t R, lp + x t and assume that this is finite for t R. Then for entire functions f of exponential type σ for hich the right-hand side is finite, 2.8 f t p H t dt σ + p f t p H t dt, here depends only on l and p. In particular, it is independent of f, σ,, H. Proof For a given x, and f, apply Lemma 2. to the function f + x, so that e are translating the variable. Making a substitution s = t + x yields f s p ds σ + p lp + s x f s p ds + s x lp. No multiply by x and integrate over all real x, and then interchange the integrals. The convergence of the right-hand side in 2.8, and the non-negativity of the integrand ustifies the interchange of integrals. Our final lemma before proving Theorem involves upper and loer bounds on r : Lemma 2.3 Assume the hypotheses of Theorem. Then for some, 2 > that depend on L, B, β,,, but not on r, t, nor on the particular, t β r t + t log 2 L. Proof We first establish the loer bound. Let us assume first that t and choose such that r t + r. Note that then 2. r t r and + r t + r; r t r and + 2 r t + r.
6 6 DORON S. LUBINSKY Then, using.4, t+r t r +r r r B + r β r B + t β B + t β + B + 2 β r r, again by.4. Thus for t, and some depending only on B, β, 2. r t + t β r. Next, using.4, [ r ] = +r r r [ r ] B + r β = r [ r ]+ B + sr β ds = B r B r r r r[ r ]+r + y β dy 2 + y β dy. A similar estimate holds for, so for some depending only on B, β, 2.2 r. Together ith 2., this establishes the loer bound for t, and of course t < is similar. We turn to the upper bound. Again, e assume t, and that is as above. We see using 2., and then.4, that t+r t r +2r r + 2B2 β +r. r
7 7 We continue this using.4, as + 2 β+ B +[ ] r [ ] B + k r β r + k= + 2 β+ B +[ B2 β r ]+r r. r k+r kr Thus e have shon that r t r+2 r, here is independent of r, t, but depends on B and β. We continue this using 2. as [ + = + t+2 ] klog 2 t+2 + L log 2 t+2 2 k+ klog 2 t+2 t + 2 log 2 L. 2 k L k+ This gives the upper bound for t, and the case t < is similar. Proof of Theorem hoose l so large that 2.3 log 2 L + β lp 2. Note that this choice does not depend on. Let H be as in Lemma 2.2. We estimate H above and belo. Let us assume first that t and choose such that r t < + r, so that 2. holds. Split 2.4 H t = + max{2+r,} = I + I 2 + I 3. + max{2 +r,} s + s t lp ds We start ith the central integral I 2 as it ill contribute to both our upper and loer bounds. We use our groth condition.4 as ell as that fact that for s [r, + r], e have s t r r 2 r if 2. If
8 8 DORON S. LUBINSKY 2, observe that r 2. Thus 2.5 I 2 max{2 +,[ r ]} = max{2 +,[ r ]} = +r r s + s t 4 + r lp lp ds +r r s ds +r max{2+,[ r ]} 4 lp B s ds r = + r lp β t+r 4 lp B s ds t r + k r lp β 4 lp+3 Br r t r t. k= + s r lp β ds Here depends on B, β, l, p but is independent of r and. We have also used 2.3 and L to ensure the convergence of the integral in the second last line. Note that e could not simply use the upper bound in Lemma 2.3 for r, as e need the last right-hand side of 2.5 to involve r t. In the other direction, e see from.4 that 2.6 I 2 max{2 +,[ r ]} +r = + r + r lp s ds r +r max{2+,[ r ]} B s ds r = 2 + r lp+β +r max{+,[ r ] } B s ds r k= Here, using our groth condition.4, and then 2., and + 2B2 β +r s ds r max{ +,[ r ] } k= +2r r 2 + kr lp+β 2 = 2r 2r s ds t+r t r max{+,[ r ] } 2 + kr lp+β. s ds = 2r r t, 2 + tr max{+r,r[ r ] r} /2 2 + s lp+β ds, lp+β dt 2 + s lp+β ds
9 since if + r 2, then r [ ] r r r r 2. Substituting the last to inequalities in 2.6, and using 2.5, e have shon that for t, r t I 2 2 r t, here and 2 depend on l, p, β, B, but not on r or the particular. Next, our doubling condition.3 gives I 2.8 = s + s + t lp ds + + t lp 2 = t lp s ds t lp L + = t lp + + t lp + t lp L + + L log 2 +t >log 2 +t log2 +t + t lp Llog 2 L +t + 2 lp + t log 2 L lp, L 2 lp + + t lp by 2.3. Here depends only on p, l, L. Next, let N = log 2 max {[2 + r], }, and let N, and s [ 2, 2 +]. We claim that s t 3 2. If first =, then N = and t < r, so + s t + 2 = 2. If, then + r r 2 3 2N, so s t 2 + r N 3 2.
10 DORON S. LUBINSKY Thus e have 2.9. Then our doubling hypothesis.3 gives 2 + s I 3 ds lp 2 + s t =N 2 + =N 3 2 lp s ds 2 =N 3 2 L+ lp L 3 lp L 2 lp =N N L 2 3 lp L 2 lp max {[2 + r], } log 2 L lp + t log 2 L lp. In the third last line, e used L/2 lp /4, as follos from 2.3. In the last line, e used 2.. Together ith 2.4, 2.7, and 2.8, e have shon that for t, 2 r t H t r t + + t log 2 L lp. Next, from 2. and 2.2, e can continue this as 2 r t H t r t + + t log 2 L lp+β 3 r t, by 2.3. The case t < is similar. No the result follos from Lemma 2.2. We note that at least for p, one can use the Markov-Bernstein inequalities in Theorem to prove that there exists δ, such that for σ >, and nonidentically vanishing entire functions f of exponential type σ, e have 2 f t p δ/σ+ t dt/ f t p t dt 3 2. This gives one ay to prove orollary 2. Hoever, e use a different method belo: Proof of orollary 2 First note that Lemma 2.3 and our hypotheses imply that for some >, t β r t + t log 2 L, r, r and t R. Here is independent of r and t. Let σ > and f be entire of type σ, ith the integral in the right-hand side of.7 finite. Let kp β + log 2 L + 2, ε > and g t = f t S εt k. By Lebesgue s differentiation theorem, e have for a.e. t R, lim r t g t p = t g t p. r +
11 Next, 2.2 shos that for r, r and all real t r t g t p + t log 2 L f t p min + t log 2 L kp f t p { } kp, πε t + t log 2 L kp+β t f t p t f t p, by Lemma 2.3 and our choice of k. Here and are independent of r, f but depend on ε and. Since t f t p is independent of r and integrable by.7, Lebesgue s Dominated onvergence Theorem gives lim r + r t g t p dt = t g t p dt. Next, for each given R >, as g is bounded in each finite interval, and r is bounded independently of r, lim r + R R r t g t p dt = R R t g t p dt. Then as g has exponential type σ + kεπ, Theorem and the last to limits yield R p t d f t S εt R dt dt σ + kεπ + p t f t S εt p dt σ + kεπ + p t f t p dt, recall that S. We can no let ε +, and use the fact that S εt converges uniformly for t in compact subsets of to S =. A similar statement then holds for the derivatives. We deduce that R R Finally, let R. t f t p dt σ + p t f t p dt. Proof of orollary 3 We choose r = in Theorem. Our condition.8 shos that for some > and all t R, 2.2 M t / t M. That condition also gives for a, and similarly, 2a a M2a a 4M 2 a a 2a 4M 2 a/2. a/2
12 2 DORON S. LUBINSKY So e can choose L = 4M 2 in.3. Next, let k and max { } 2k +, r = 2k +. We have to sho that.4 holds for the given, k and ith r =. Firstly if = or,.8 gives M 2. So no let us consider 2k +. Let us first suppose that k, and choose n log 2 k such that k 2 n+ k 2 n. Then by repeated use of.8, + k M M 2 2 n 2.23 Here M n = 2 n log 2 k M = + k k+2 M n+2 k M n+3. log2 M log2 M k + k log 2 M. ombined ith 2.22 and 2.23, e have shon that for k, + Next, if k < 2k +, + M 2 k k+ M 5 + k log 2 M. k+ k+ M 3 M 3 + k log 2 M. k In summary, e have established.4 ith B = M 5 and β = log 2 M. Then, recalling 2.2, Theorem. gives the result. Proof of orollary 4 It is easy to see that x = + x 2 α satisfies.8, ith, for example, M = 7 α. R [] V.V. Arestov, On Integral Inequalities for Trigonometric Polynomials and Their Derivatives, Math. USSR-Izv., 8982, -7. [2] A.D. Baranov, Weighted Bernstein-Type Inequalities, and Embedding Theorems for the Model Spaces, St. Petersburg Math. J., 524, [3] A.D. Baranov, Bernstein-type Inequalities for Shift-oinvariant Subspaces and their Applications to arleson Embeddings, J. Funct. Anal., 22325, [4] R.P. Boas, Entire Functions, Academic Press, Ne York, 954. [5] P. Borein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer, Ne York, 995. k k
13 3 [6] D.P. Dryanov, M.A. Qazi, Q.I. Rahman, Entire Functions of Exponential Type in Approximation Theory, in onstructive Theory of Functions, ed. B. Boanov, Darba, Sofia, 23, pp [7] T. Erdelyi, Notes on Inequalities ith Doubling Weights, J. Approx. Theory, 999, [8] M. Ganzburg, Limit Theorems of Polynomial Approximation ith Exponential Weights, Memoirs of the Amer. Math. Soc., 9228, Number 897. [9] B. Ja Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, American Mathematical Society, Providence, 996. [] D. S. Lubinsky, On Sharp onstants in Marcinkieicz-Zygmund and Plancherel-Polya Inequalities, to appear in Proceedings of the American Math. Soc. [] D. S. Lubinsky, Orthogonal Dirichlet Polynomials ith Arctan Density, to appear in Journal of Approximation Theory. [2] G. Mastroianni, G.V. Milovanovic, Interpolation Processes: Basic Theory and Applications, Springer, Berlin, 28. [3] G. Mastroianni and V. Totik, Weighted Polynomial Inequalities ith Doubling and A Weights, onstr. Approx., 62, [4] G. V. Milovanović, Dragoslav S. Mitrinović, T. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 994. [5] Q.I. Rahman, G. Schmeisser, L p Inequalities for Entire Functions of Exponential Type, Trans. Amer. Math. Soc., 3299, 9-3. [6] Q.I. Rahman, Q.M. Tariq, On Bernstein s inequality for entire functions of exponential type, J. Math. Anal. Appl , [7] A.F. Timan, Theory of Approximation of Functions of a Real Variable, Translated by J. Berry, Dover, Ne York, 994. [8] R.M. Trigub and E.S. Belinsky, Fourier Analysis and Approximation of Functions, Kluer, Dordrecht, 24. S M, G I T, A, GA ,
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