WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

Size: px
Start display at page:

Download "WEIGHTED MARKOV-BERNSTEIN INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE"

Transcription

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 ,

SERIES REPRESENTATIONS FOR BEST APPROXIMATING ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

SERIES REPRESENTATIONS FOR BEST APPROXIMATING ENTIRE FUNCTIONS OF EXPONENTIAL TYPE SERIES REPRESENTATIONS OR BEST APPROXIMATING ENTIRE UNCTIONS O EXPONENTIAL TYPE D. S. LUBINSKY School of Mathematics, Georgia Institute of Technology, Atlanta, GA 333-6. e-mail: lubinsky@math.gatech.edu

More information

Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces ABSTRACT 1. INTRODUCTION

Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces ABSTRACT 1. INTRODUCTION Malaysian Journal of Mathematical Sciences 6(2): 25-36 (202) Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces Noli N. Reyes and Rosalio G. Artes Institute of Mathematics, University of

More information

AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS

AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS W. M. SHAH A. LIMAN P.G. Department of Mathematics Department of Mathematics Baramulla College, Kashmir National Institute of Technology India-193101

More information

Mean Convergence of Interpolation at Zeros of Airy Functions

Mean Convergence of Interpolation at Zeros of Airy Functions Mean Convergence of Interpolation at Zeros of Airy Functions D. S. Lubinsky Abstract The classical Erdős-Turán theorem established mean convergence of Lagrange interpolants at zeros of orthogonal polynomials.

More information

An operator preserving inequalities between polynomials

An operator preserving inequalities between polynomials An operator preserving inequalities between polynomials NISAR A. RATHER Kashmir University P.G. Department of Mathematics Hazratbal-190006, Srinagar INDIA dr.narather@gmail.com MUSHTAQ A. SHAH Kashmir

More information

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let HOW FAR IS AN ULTRAFLAT SEQUENCE OF UNIMODULAR POLYNOMIALS FROM BEING CONJUGATE-RECIPROCAL? Tamás Erdélyi Abstract. In this paper we study ultraflat sequences (P n) of unimodular polynomials P n K n in

More information

PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 58 (72), 1995, Slobodan Aljan»cić memorial volume AN ESTIMATE FOR COEFFICIENTS OF

PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 58 (72), 1995, Slobodan Aljan»cić memorial volume AN ESTIMATE FOR COEFFICIENTS OF PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 58 (72), 1995, 137 142 Slobodan Aljan»cić memorial volume AN ESTIMATE FOR COEFFICIENTS OF POLYNOMIALS IN L 2 NORM. II G. V. Milovanović and

More information

GROWTH OF MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS. Abdul Aziz and B. A. Zargar University of Kashmir, Srinagar, India

GROWTH OF MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS. Abdul Aziz and B. A. Zargar University of Kashmir, Srinagar, India GLASNIK MATEMATIČKI Vol. 37(57)(2002), 73 81 GOWTH OF MAXIMUM MODULUS OF POLYNOMIALS WITH PESCIBED ZEOS Abdul Aziz and B. A. Zargar University of Kashmir, Srinagar, India Abstract. Let P (z) be a polynomial

More information

References 167. dx n x2 =2

References 167. dx n x2 =2 References 1. G. Akemann, J. Baik, P. Di Francesco (editors), The Oxford Handbook of Random Matrix Theory, Oxford University Press, Oxford, 2011. 2. G. Anderson, A. Guionnet, O. Zeitouni, An Introduction

More information

WEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN

WEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN WEAK TYPE ESTIMATES FOR SINGULAR INTEGRALS RELATED TO A DUAL PROBLEM OF MUCKENHOUPT-WHEEDEN ANDREI K. LERNER, SHELDY OMBROSI, AND CARLOS PÉREZ Abstract. A ell knon open problem of Muckenhoupt-Wheeden says

More information

NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS

NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS Tamás Erdélyi Dedicated to the memory of George G Lorentz Abstract Let Λ n := {λ 0 < λ < < λ n } be a set of real numbers The collection of all linear

More information

be the set of complex valued 2π-periodic functions f on R such that

be the set of complex valued 2π-periodic functions f on R such that . Fourier series. Definition.. Given a real number P, we say a complex valued function f on R is P -periodic if f(x + P ) f(x) for all x R. We let be the set of complex valued -periodic functions f on

More information

Topics in Harmonic Analysis Lecture 1: The Fourier transform

Topics in Harmonic Analysis Lecture 1: The Fourier transform Topics in Harmonic Analysis Lecture 1: The Fourier transform Po-Lam Yung The Chinese University of Hong Kong Outline Fourier series on T: L 2 theory Convolutions The Dirichlet and Fejer kernels Pointwise

More information

Polarization constant K(n, X) = 1 for entire functions of exponential type

Polarization constant K(n, X) = 1 for entire functions of exponential type Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 35-45 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.252 Polarization constant K(n, X) = 1 for entire functions of exponential type A.

More information

SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART

SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART Georgian Mathematical Journal Volume 4 (27), Number 4, 673 68 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics

More information

A trigonometric orthogonality with respect to a nonnegative Borel measure

A trigonometric orthogonality with respect to a nonnegative Borel measure Filomat 6:4 01), 689 696 DOI 10.98/FIL104689M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A trigonometric orthogonality with

More information

Xiyou Cheng Zhitao Zhang. 1. Introduction

Xiyou Cheng Zhitao Zhang. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 34, 2009, 267 277 EXISTENCE OF POSITIVE SOLUTIONS TO SYSTEMS OF NONLINEAR INTEGRAL OR DIFFERENTIAL EQUATIONS Xiyou

More information

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET PETER BORWEIN, TAMÁS ERDÉLYI, FRIEDRICH LITTMANN

More information

EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS

EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS Tamás Erdélyi Abstract. Let Λ n := {λ,λ,...,λ n } be a set of n distinct positive numbers. The span of {e λ t,e λ t,...,e λ nt } over R

More information

YAN GUO, JUHI JANG, AND NING JIANG

YAN GUO, JUHI JANG, AND NING JIANG LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION YAN GUO, JUHI JANG, AND NING JIANG Abstract. We revisit the classical ork of Caflisch [C] for compressible Euler limit of the Boltzmann equation. By using

More information

Markov s Inequality for Polynomials on Normed Linear Spaces Lawrence A. Harris

Markov s Inequality for Polynomials on Normed Linear Spaces Lawrence A. Harris New Series Vol. 16, 2002, Fasc. 1-4 Markov s Inequality for Polynomials on Normed Linear Spaces Lawrence A. Harris This article is dedicated to the 70th anniversary of Acad. Bl. Sendov It is a longstanding

More information

INEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES

INEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 45, Number 5, 2015 INEQUALITIES FOR SUMS OF INDEPENDENT RANDOM VARIABLES IN LORENTZ SPACES GHADIR SADEGHI ABSTRACT. By using interpolation with a function parameter,

More information

Review: Stability of Bases and Frames of Reproducing Kernels in Model Spaces

Review: Stability of Bases and Frames of Reproducing Kernels in Model Spaces Claremont Colleges Scholarship @ Claremont Pomona Faculty Publications and Research Pomona Faculty Scholarship 1-1-2006 Review: Stability of Bases and Frames of Reproducing Kernels in Model Spaces Stephan

More information

Random Bernstein-Markov factors

Random Bernstein-Markov factors Random Bernstein-Markov factors Igor Pritsker and Koushik Ramachandran October 20, 208 Abstract For a polynomial P n of degree n, Bernstein s inequality states that P n n P n for all L p norms on the unit

More information

lim f f(kx)dp(x) = cmf

lim f f(kx)dp(x) = cmf proceedings of the american mathematical society Volume 107, Number 3, November 1989 MAGNIFIED CURVES ON A FLAT TORUS, DETERMINATION OF ALMOST PERIODIC FUNCTIONS, AND THE RIEMANN-LEBESGUE LEMMA ROBERT

More information

引用北海学園大学学園論集 (171): 11-24

引用北海学園大学学園論集 (171): 11-24 タイトル 著者 On Some Singular Integral Operato One to One Mappings on the Weight Hilbert Spaces YAMAMOTO, Takanori 引用北海学園大学学園論集 (171): 11-24 発行日 2017-03-25 On Some Singular Integral Operators Which are One

More information

INVARIANT SUBSPACES FOR CERTAIN FINITE-RANK PERTURBATIONS OF DIAGONAL OPERATORS. Quanlei Fang and Jingbo Xia

INVARIANT SUBSPACES FOR CERTAIN FINITE-RANK PERTURBATIONS OF DIAGONAL OPERATORS. Quanlei Fang and Jingbo Xia INVARIANT SUBSPACES FOR CERTAIN FINITE-RANK PERTURBATIONS OF DIAGONAL OPERATORS Quanlei Fang and Jingbo Xia Abstract. Suppose that {e k } is an orthonormal basis for a separable, infinite-dimensional Hilbert

More information

arxiv: v1 [math.cv] 13 Jun 2014

arxiv: v1 [math.cv] 13 Jun 2014 M. RIESZ-SCHUR-TYPE INEQUALITIES FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE TAMÁS ERDÉLYI, MICHAEL I. GANZBURG, AND PAUL NEVAI arxiv:1406.3664v1 [math.cv] 13 Jun 2014 Abstract. We prove a general M. Riesz-Schur-type

More information

MORE NOTES FOR MATH 823, FALL 2007

MORE NOTES FOR MATH 823, FALL 2007 MORE NOTES FOR MATH 83, FALL 007 Prop 1.1 Prop 1. Lemma 1.3 1. The Siegel upper half space 1.1. The Siegel upper half space and its Bergman kernel. The Siegel upper half space is the domain { U n+1 z C

More information

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative This chapter is another review of standard material in complex analysis. See for instance

More information

arxiv: v1 [math.ca] 10 Sep 2017

arxiv: v1 [math.ca] 10 Sep 2017 arxiv:179.3168v1 [math.ca] 1 Sep 217 On moduli of smoothness and averaged differences of fractional order Yurii Kolomoitsev 1 Abstract. We consider two types of fractional integral moduli of smoothness,

More information

GROWTH OF THE MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS M. S. PUKHTA. 1. Introduction and Statement of Result

GROWTH OF THE MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS M. S. PUKHTA. 1. Introduction and Statement of Result Journal of Classical Analysis Volume 5, Number 2 (204, 07 3 doi:0.753/jca-05-09 GROWTH OF THE MAXIMUM MODULUS OF POLYNOMIALS WITH PRESCRIBED ZEROS M. S. PUKHTA Abstract. If p(z n a j j is a polynomial

More information

Towards a Theory of Societal Co-Evolution: Individualism versus Collectivism

Towards a Theory of Societal Co-Evolution: Individualism versus Collectivism Toards a Theory of Societal Co-Evolution: Individualism versus Collectivism Kartik Ahuja, Simpson Zhang and Mihaela van der Schaar Department of Electrical Engineering, Department of Economics, UCLA Theorem

More information

A Tilt at TILFs. Rod Nillsen University of Wollongong. This talk is dedicated to Gary H. Meisters

A Tilt at TILFs. Rod Nillsen University of Wollongong. This talk is dedicated to Gary H. Meisters A Tilt at TILFs Rod Nillsen University of Wollongong This talk is dedicated to Gary H. Meisters Abstract In this talk I will endeavour to give an overview of some aspects of the theory of Translation Invariant

More information

EXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS

EXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS EXTENDED LAGUERRE INEQUALITIES AND A CRITERION FOR REAL ZEROS DAVID A. CARDON Abstract. Let fz) = e bz2 f z) where b 0 and f z) is a real entire function of genus 0 or. We give a necessary and sufficient

More information

A RECONSTRUCTION FORMULA FOR BAND LIMITED FUNCTIONS IN L 2 (R d )

A RECONSTRUCTION FORMULA FOR BAND LIMITED FUNCTIONS IN L 2 (R d ) PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3593 3600 S 0002-9939(99)04938-2 Article electronically published on May 6, 1999 A RECONSTRUCTION FORMULA FOR AND LIMITED FUNCTIONS

More information

Section 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44

Section 7.4 #1, 5, 6, 8, 12, 13, 44, 53; Section 7.5 #7, 10, 11, 20, 22; Section 7.7 #1, 4, 10, 15, 22, 44 Math B Prof. Audrey Terras HW #4 Solutions Due Tuesday, Oct. 9 Section 7.4 #, 5, 6, 8,, 3, 44, 53; Section 7.5 #7,,,, ; Section 7.7 #, 4,, 5,, 44 7.4. Since 5 = 5 )5 + ), start with So, 5 = A 5 + B 5 +.

More information

EXACT INTERPOLATION, SPURIOUS POLES, AND UNIFORM CONVERGENCE OF MULTIPOINT PADÉ APPROXIMANTS

EXACT INTERPOLATION, SPURIOUS POLES, AND UNIFORM CONVERGENCE OF MULTIPOINT PADÉ APPROXIMANTS EXACT INTERPOLATION, SPURIOUS POLES, AND UNIFORM CONVERGENCE OF MULTIPOINT PADÉ APPROXIMANTS D S LUBINSKY A We introduce the concept of an exact interpolation index n associated with a function f and open

More information

Interpolation and Cubature at Geronimus Nodes Generated by Different Geronimus Polynomials

Interpolation and Cubature at Geronimus Nodes Generated by Different Geronimus Polynomials Interpolation and Cubature at Geronimus Nodes Generated by Different Geronimus Polynomials Lawrence A. Harris Abstract. We extend the definition of Geronimus nodes to include pairs of real numbers where

More information

96 CHAPTER 4. HILBERT SPACES. Spaces of square integrable functions. Take a Cauchy sequence f n in L 2 so that. f n f m 1 (b a) f n f m 2.

96 CHAPTER 4. HILBERT SPACES. Spaces of square integrable functions. Take a Cauchy sequence f n in L 2 so that. f n f m 1 (b a) f n f m 2. 96 CHAPTER 4. HILBERT SPACES 4.2 Hilbert Spaces Hilbert Space. An inner product space is called a Hilbert space if it is complete as a normed space. Examples. Spaces of sequences The space l 2 of square

More information

Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,

Riemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E, Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of

More information

Introduction to Fourier Analysis

Introduction to Fourier Analysis Lecture Introduction to Fourier Analysis Jan 7, 2005 Lecturer: Nati Linial Notes: Atri Rudra & Ashish Sabharwal. ext he main text for the first part of this course would be. W. Körner, Fourier Analysis

More information

Scaling Limits of Polynomials and Entire Functions of Exponential Type

Scaling Limits of Polynomials and Entire Functions of Exponential Type Scaling Limits of Polynomials and Entire Functions of Exponential Type D. S. Lubinsky Abstract The connection between polynomials and entire functions of exponential type is an old one, in some ways harking

More information

Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting

More information

3 - Vector Spaces Definition vector space linear space u, v,

3 - Vector Spaces Definition vector space linear space u, v, 3 - Vector Spaces Vectors in R and R 3 are essentially matrices. They can be vieed either as column vectors (matrices of size and 3, respectively) or ro vectors ( and 3 matrices). The addition and scalar

More information

Preliminary Exam 2018 Solutions to Morning Exam

Preliminary Exam 2018 Solutions to Morning Exam Preliminary Exam 28 Solutions to Morning Exam Part I. Solve four of the following five problems. Problem. Consider the series n 2 (n log n) and n 2 (n(log n)2 ). Show that one converges and one diverges

More information

THE L p VERSION OF NEWMAN S INEQUALITY FOR LACUNARY POLYNOMIALS. Peter Borwein and Tamás Erdélyi. 1. Introduction and Notation

THE L p VERSION OF NEWMAN S INEQUALITY FOR LACUNARY POLYNOMIALS. Peter Borwein and Tamás Erdélyi. 1. Introduction and Notation THE L p VERSION OF NEWMAN S INEQUALITY FOR LACUNARY POLYNOMIALS Peter Borwein and Tamás Erdélyi Abstract. The principal result of this paper is the establishment of the essentially sharp Markov-type inequality

More information

THE LINEAR BOUND FOR THE NATURAL WEIGHTED RESOLUTION OF THE HAAR SHIFT

THE LINEAR BOUND FOR THE NATURAL WEIGHTED RESOLUTION OF THE HAAR SHIFT THE INEAR BOUND FOR THE NATURA WEIGHTED RESOUTION OF THE HAAR SHIFT SANDRA POTT, MARIA CARMEN REGUERA, ERIC T SAWYER, AND BRETT D WIC 3 Abstract The Hilbert transform has a linear bound in the A characteristic

More information

Piecewise Smooth Solutions to the Burgers-Hilbert Equation

Piecewise Smooth Solutions to the Burgers-Hilbert Equation Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang

More information

1.5 Approximate Identities

1.5 Approximate Identities 38 1 The Fourier Transform on L 1 (R) which are dense subspaces of L p (R). On these domains, P : D P L p (R) and M : D M L p (R). Show, however, that P and M are unbounded even when restricted to these

More information

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative

Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative This chapter is another review of standard material in complex analysis. See for instance

More information

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

More information

Some coefficient estimates for polynomials on the unit interval

Some coefficient estimates for polynomials on the unit interval Some coefficient estimates for polynomials on the unit interval M. A. Qazi and Q.I. Rahman Abstract In this paper we present some inequalities about the moduli of the coefficients of polynomials of the

More information

Math 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:

Math 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find: Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..

More information

Bucket handles and Solenoids Notes by Carl Eberhart, March 2004

Bucket handles and Solenoids Notes by Carl Eberhart, March 2004 Bucket handles and Solenoids Notes by Carl Eberhart, March 004 1. Introduction A continuum is a nonempty, compact, connected metric space. A nonempty compact connected subspace of a continuum X is called

More information

MARKOV-NIKOLSKII TYPE INEQUALITIES FOR EXPONENTIAL SUMS ON FINITE INTERVALS

MARKOV-NIKOLSKII TYPE INEQUALITIES FOR EXPONENTIAL SUMS ON FINITE INTERVALS MARKOV-NIKOLSKII TYPE INEQUALITIES FOR EXPONENTIAL SUMS ON FINITE INTERVALS TAMÁS ERDÉLYI Abstract Let Λ n := {λ 0 < λ 1 < < λ n} be a set of real numbers The collection of all linear combinations of of

More information

Simultaneous Gaussian quadrature for Angelesco systems

Simultaneous Gaussian quadrature for Angelesco systems for Angelesco systems 1 KU Leuven, Belgium SANUM March 22, 2016 1 Joint work with Doron Lubinsky Introduced by C.F. Borges in 1994 Introduced by C.F. Borges in 1994 (goes back to Angelesco 1918). Introduced

More information

Density results for frames of exponentials

Density results for frames of exponentials Density results for frames of exponentials P. G. Casazza 1, O. Christensen 2, S. Li 3, and A. Lindner 4 1 Department of Mathematics, University of Missouri Columbia, Mo 65211 USA pete@math.missouri.edu

More information

Wiener Measure and Brownian Motion

Wiener Measure and Brownian Motion Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u

More information

UNIFORM BOUNDS FOR BESSEL FUNCTIONS

UNIFORM BOUNDS FOR BESSEL FUNCTIONS Journal of Applied Analysis Vol. 1, No. 1 (006), pp. 83 91 UNIFORM BOUNDS FOR BESSEL FUNCTIONS I. KRASIKOV Received October 8, 001 and, in revised form, July 6, 004 Abstract. For ν > 1/ and x real we shall

More information

A Riemann Lebesgue Lemma for Jacobi expansions

A Riemann Lebesgue Lemma for Jacobi expansions A Riemann Lebesgue Lemma for Jacobi expansions George Gasper 1 and Walter Trebels Dedicated to P. L. Butzer on the occasion of his 65-th birthday (To appear in Contemporary Mathematics) Abstract. A Lemma

More information

FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355. Analysis 4. Examiner: Professor S. W. Drury Date: Wednesday, April 18, 2007 INSTRUCTIONS

FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355. Analysis 4. Examiner: Professor S. W. Drury Date: Wednesday, April 18, 2007 INSTRUCTIONS FACULTY OF SCIENCE FINAL EXAMINATION MATHEMATICS MATH 355 Analysis 4 Examiner: Professor S. W. Drury Date: Wednesday, April 18, 27 Associate Examiner: Professor K. N. GowriSankaran Time: 2: pm. 5: pm.

More information

ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT

ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT ON A WEIGHTED INTERPOLATION OF FUNCTIONS WITH CIRCULAR MAJORANT Received: 31 July, 2008 Accepted: 06 February, 2009 Communicated by: SIMON J SMITH Department of Mathematics and Statistics La Trobe University,

More information

Initial Coefficient Bounds for a General Class of Bi-Univalent Functions

Initial Coefficient Bounds for a General Class of Bi-Univalent Functions Filomat 29:6 (2015), 1259 1267 DOI 10.2298/FIL1506259O Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat Initial Coefficient Bounds for

More information

The DMP Inverse for Rectangular Matrices

The DMP Inverse for Rectangular Matrices Filomat 31:19 (2017, 6015 6019 https://doi.org/10.2298/fil1719015m Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat The DMP Inverse for

More information

Bloom Filters and Locality-Sensitive Hashing

Bloom Filters and Locality-Sensitive Hashing Randomized Algorithms, Summer 2016 Bloom Filters and Locality-Sensitive Hashing Instructor: Thomas Kesselheim and Kurt Mehlhorn 1 Notation Lecture 4 (6 pages) When e talk about the probability of an event,

More information

On Some Estimates of the Remainder in Taylor s Formula

On Some Estimates of the Remainder in Taylor s Formula Journal of Mathematical Analysis and Applications 263, 246 263 (2) doi:.6/jmaa.2.7622, available online at http://www.idealibrary.com on On Some Estimates of the Remainder in Taylor s Formula G. A. Anastassiou

More information

ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS. Lecture 3

ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS. Lecture 3 ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS A. POLTORATSKI Lecture 3 A version of the Heisenberg Uncertainty Principle formulated in terms of Harmonic Analysis claims that a non-zero measure (distribution)

More information

arxiv:math/ v1 [math.fa] 26 Oct 1993

arxiv:math/ v1 [math.fa] 26 Oct 1993 arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological

More information

Remarks on the Rademacher-Menshov Theorem

Remarks on the Rademacher-Menshov Theorem Remarks on the Rademacher-Menshov Theorem Christopher Meaney Abstract We describe Salem s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all

More information

8 Singular Integral Operators and L p -Regularity Theory

8 Singular Integral Operators and L p -Regularity Theory 8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation

More information

ON PARABOLIC HARNACK INEQUALITY

ON PARABOLIC HARNACK INEQUALITY ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics THE EXTENSION OF MAJORIZATION INEQUALITIES WITHIN THE FRAMEWORK OF RELATIVE CONVEXITY CONSTANTIN P. NICULESCU AND FLORIN POPOVICI University of Craiova

More information

Outline of Fourier Series: Math 201B

Outline of Fourier Series: Math 201B Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C

More information

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )

A COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 ) Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE

More information

FOURIER SERIES WITH THE CONTINUOUS PRIMITIVE INTEGRAL

FOURIER SERIES WITH THE CONTINUOUS PRIMITIVE INTEGRAL Real Analysis Exchange Summer Symposium XXXVI, 2012, pp. 30 35 Erik Talvila, Department of Mathematics and Statistics, University of the Fraser Valley, Chilliwack, BC V2R 0N3, Canada. email: Erik.Talvila@ufv.ca

More information

On lower bounds of exponential frames

On lower bounds of exponential frames On lower bounds of exponential frames Alexander M. Lindner Abstract Lower frame bounds for sequences of exponentials are obtained in a special version of Avdonin s theorem on 1/4 in the mean (1974) and

More information

THE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE. In memory of A. Pe lczyński

THE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE. In memory of A. Pe lczyński THE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE T. FIGIEL AND W. B. JOHNSON Abstract. Given a Banach space X and a subspace Y, the pair (X, Y ) is said to have the approximation

More information

On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices

On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices On the approximation of real poers of sparse, infinite, bounded and Hermitian matrices Roman Werpachoski Center for Theoretical Physics, Al. Lotnikó 32/46 02-668 Warszaa, Poland Abstract We describe a

More information

Growth of Solutions of Second Order Complex Linear Differential Equations with Entire Coefficients

Growth of Solutions of Second Order Complex Linear Differential Equations with Entire Coefficients Filomat 32: (208), 275 284 https://doi.org/0.2298/fil80275l Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Growth of Solutions

More information

Minimization problems on the Hardy-Sobolev inequality

Minimization problems on the Hardy-Sobolev inequality manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev

More information

Basic relations valid for the Bernstein space B 2 σ and their extensions to functions from larger spaces in terms of their distances from B 2 σ

Basic relations valid for the Bernstein space B 2 σ and their extensions to functions from larger spaces in terms of their distances from B 2 σ Basic relations valid for the Bernstein space B 2 σ and their extensions to functions from larger spaces in terms of their distances from B 2 σ Part 3: Distance functional approach of Part 2 applied to

More information

MATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE

MATH 311: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE MATH 3: COMPLEX ANALYSIS CONTOUR INTEGRALS LECTURE Recall the Residue Theorem: Let be a simple closed loop, traversed counterclockwise. Let f be a function that is analytic on and meromorphic inside. Then

More information

On the local existence for an active scalar equation in critical regularity setting

On the local existence for an active scalar equation in critical regularity setting On the local existence for an active scalar equation in critical regularity setting Walter Rusin Department of Mathematics, Oklahoma State University, Stillwater, OK 7478 Fei Wang Department of Mathematics,

More information

DERIVATIVE-FREE CHARACTERIZATIONS OF Q K SPACES

DERIVATIVE-FREE CHARACTERIZATIONS OF Q K SPACES ERIVATIVE-FREE CHARACTERIZATIONS OF Q K SPACES HASI WULAN AN KEHE ZHU ABSTRACT. We give two characterizations of the Möbius invariant Q K spaces, one in terms of a double integral and the other in terms

More information

Homepage: WWW: george/

Homepage: WWW:  george/ LIST OF PUBLICATIONS of George Gasper (George Gasper, Jr.) (2/16/07 version) Department of Mathematics, Northwestern University, Evanston, Illinois 60208, (847) 491-5592 E-mail: george at math.northwestern.edu

More information

IMPROVED RESULTS ON THE OSCILLATION OF THE MODULUS OF THE RUDIN-SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE. September 12, 2018

IMPROVED RESULTS ON THE OSCILLATION OF THE MODULUS OF THE RUDIN-SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE. September 12, 2018 IMPROVED RESULTS ON THE OSCILLATION OF THE MODULUS OF THE RUDIN-SHAPIRO POLYNOMIALS ON THE UNIT CIRCLE Tamás Erdélyi September 12, 2018 Dedicated to Paul Nevai on the occasion of his 70th birthday Abstract.

More information

The Hilbert transform

The Hilbert transform The Hilbert transform Definition and properties ecall the distribution pv(, defined by pv(/(ϕ := lim ɛ ɛ ϕ( d. The Hilbert transform is defined via the convolution with pv(/, namely (Hf( := π lim f( t

More information

Analytic families of multilinear operators

Analytic families of multilinear operators Analytic families of multilinear operators Mieczysław Mastyło Adam Mickiewicz University in Poznań Nonlinar Functional Analysis Valencia 17-20 October 2017 Based on a joint work with Loukas Grafakos M.

More information

A Proof of Markov s Theorem for Polynomials on Banach spaces

A Proof of Markov s Theorem for Polynomials on Banach spaces A Proof of Markov s Theorem for Polynomials on Banach spaces Lawrence A. Harris Department of Mathematics, University of Kentucky Lexington, Kentucky 40506-007 larry@ms.uky.edu Dedicated to my teachers

More information

THE MUCKENHOUPT A CLASS AS A METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES. Nikolaos Pattakos and Alexander Volberg

THE MUCKENHOUPT A CLASS AS A METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES. Nikolaos Pattakos and Alexander Volberg Math. Res. Lett. 9 (202), no. 02, 499 50 c International Press 202 THE MUCKENHOUPT A CLASS AS A METRIC SPACE AND CONTINUITY OF WEIGHTED ESTIMATES Nikolaos Pattakos and Alexander Volberg Abstract. We sho

More information

3 Orthogonality and Fourier series

3 Orthogonality and Fourier series 3 Orthogonality and Fourier series We now turn to the concept of orthogonality which is a key concept in inner product spaces and Hilbert spaces. We start with some basic definitions. Definition 3.1. Let

More information

THE ZERO-DISTRIBUTION AND THE ASYMPTOTIC BEHAVIOR OF A FOURIER INTEGRAL. Haseo Ki and Young One Kim

THE ZERO-DISTRIBUTION AND THE ASYMPTOTIC BEHAVIOR OF A FOURIER INTEGRAL. Haseo Ki and Young One Kim THE ZERO-DISTRIBUTION AND THE ASYMPTOTIC BEHAVIOR OF A FOURIER INTEGRAL Haseo Ki Young One Kim Abstract. The zero-distribution of the Fourier integral Q(u)eP (u)+izu du, where P is a polynomial with leading

More information

Math 20C Homework 2 Partial Solutions

Math 20C Homework 2 Partial Solutions Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we

More information

arxiv: v2 [math.fa] 17 May 2016

arxiv: v2 [math.fa] 17 May 2016 ESTIMATES ON SINGULAR VALUES OF FUNCTIONS OF PERTURBED OPERATORS arxiv:1605.03931v2 [math.fa] 17 May 2016 QINBO LIU DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY EAST LANSING, MI 48824, USA Abstract.

More information

FURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS

FURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS FURTHER STUDIES OF STRONGLY AMENABLE -REPRESENTATIONS OF LAU -ALGEBRAS FATEMEH AKHTARI and RASOUL NASR-ISFAHANI Communicated by Dan Timotin The new notion of strong amenability for a -representation of

More information

MATH 126 FINAL EXAM. Name:

MATH 126 FINAL EXAM. Name: MATH 126 FINAL EXAM Name: Exam policies: Closed book, closed notes, no external resources, individual work. Please write your name on the exam and on each page you detach. Unless stated otherwise, you

More information

Polynomial approximation via de la Vallée Poussin means

Polynomial approximation via de la Vallée Poussin means Summer School on Applied Analysis 2 TU Chemnitz, September 26-3, 2 Polynomial approximation via de la Vallée Poussin means Lecture 2: Discrete operators Woula Themistoclakis CNR - National Research Council

More information

Harmonic Analysis Homework 5

Harmonic Analysis Homework 5 Harmonic Analysis Homework 5 Bruno Poggi Department of Mathematics, University of Minnesota November 4, 6 Notation Throughout, B, r is the ball of radius r with center in the understood metric space usually

More information

ETNA Kent State University

ETNA Kent State University Electronic Transactions on Numerical Analysis. Volume 35, pp. 148-163, 2009. Copyright 2009,. ISSN 1068-9613. ETNA SPHERICAL QUADRATURE FORMULAS WITH EQUALLY SPACED NODES ON LATITUDINAL CIRCLES DANIELA

More information