L p Norms and the Sinc Function

Transcription

1 L Norms and the Sinc Function D. Borwein, J. M. Borwein and I. E. Leonard March 6, 9 Introduction The sinc function is a real valued function defined on the real line R by the following eression: sin if sinc) = otherwise This function is imortant in many areas of comuting science, aroimation theory, and numerical analysis. For eamle, it is used in interolation and aroimation of functions, aroimate evaluation of transforms e.g. Hilbert, Fourier, Lalace, Hanel, and Mellon transforms as well as the fast Fourier transform). It is used in finding aroimate solutions of differential and integral equations, in image rocessing it is the Fourier transform of the bo filter and central to the understanding of the Gibbs henomenon []), in signal rocessing and information theory. Much of this is nicely described in [7]. The first elicit aearance of the sinc function in aroimation theory was robably in the use of the Whittaer cardinal functions Cf, h) to aroimate functions analytic on an interval or on a contour. Given a function f which is defined on the real line R, the function Cf, h) is defined by Cf, h) = = fh)s, h) whenever the series converges, where the stesize h > and where S, h)) = sin[ π h)] h π h h), that is, S, h)) = sinc π h h)). See, for eamle, []. The object of this note is to study the behavior and roerties of the following function I) = d for < <. Note that this function is only defined for >, since Indeed d = π while d = +, d is conditionally convergent. see []. This integral arises, for eamle, in the L aroimation of real valued functions by Whittaer cardinal functions, and is imortant in estimating the error made in the aroimation. It also arises in many other comutational roblems, and it is surrising that so little is nown about it. Various roerties of the function I) are nown. For eamle, the behavior of I) for large is nown: I) = sin d =

2 Figure : The function I on [, ] This result, obtained indeendently by A. Meir and I. E. Leonard, is in rincile not new see equation ). We rovide a self-contained roof below as art of our more general result in Theorem. Also, for integer, the integral can be calculated elicitly. In fact, for n we have ) n sin d = n )! ) d π n n n ) = ) n ) n This result is most definitely not new, it can be found in Bromwich [4, Eercise,. 58], where it is attributed to Wolstenholme, and in many other laces including two relatively recent articles on integrals of more general roducts of sinc functions [, ]. Thus, if is an even integer, then we have a closed form eression for I), and in this case the values of I) can be calculated eactly: I) = ) d = )! π ) ) ). ) In articular I) = π/, I4) = π/ and I6) = 6π/4. That said, this sum is very difficult to use numerically for large. Not only are the rational factors growing raidly but it contains etremely large terms of alternating sign and consequently dramatic cancelations. For eamle I6) = π, and I) = Q π where Q is a rational number whose numerator and denominator both have roughly 5 digits. Similarly I) = Q 44 π where Q 44 is comrised of 4 digit integers. We also note that numerical integration of I) even to single recision is not easy and so ) rovides a very good confirmation of numerical integration results. We challenge the reader to numerically confirm the it at infinity to 8 laces. The behavior of I) for intermediate values of is not fully established. It had been conjectured that I) had a global minimum at = 4, however, very) recent comutations using both Male and Mathematica suggest that the global minimum, and unique critical oint, is at aroimately =.6... as illustrated in Figure. Although it is nown that I) = +, and that I) = +, it is not nown recisely how the asymtote y = is aroached, although both numerical and grahical evidence strongly suggest the following conjecture: =

3 Figure : The function I and its iting value on [, ] Conjecture I is increasing for above the conjectured global minimum near.6 and concave for above an inflection oint near This is shown in Figure in which the dashed line has height. Moreover, in Theorem we shall rove I) > + > ), ) for all >. We conclude this introduction by observing that one can derive the eistence of an asymtotic eansion for I) from a general result of Olver [] on asymtotics of integrals using critical oint theory and contour integration. Secialized to our case, [, Theorem 7.,. 7] with q = and = logsin)/) on [ π, π]) establishes the eistence of real constants c s such that I) π sin) π d + c s s + ) as. From this one may deduce that I) is concave and increasing for sufficiently large values of consistent with our stronger conjecture as ) may be differentiated termwise. Our Main Results In order to study the roerties of the function I), we consider first the functions ϕ n ) = log sin n ) d for > and n a nonnegative integer. We write ϕ) = ϕ ) = s= In Lemma below we confirm that ϕ) is analytic in a region containing, ) and that its n-th derivative for > is given by ϕ n) ) = ϕ n ). Then in Theorem we shall use induction to rove the following result for n a nonnegative integer: n+ ϕ n) ) = ) n Γ n + ). d.

4 The base case, n =, for our induction is established in Lemma below. It uses Lalace s method for determining asymtotic behavior of an integral for large values of a arameter, see, e.g., [6,. 6]. Lemma For > z >, ϕ z) = ) n ϕ n ) zn n!. n= In articular, ϕ) is analytic in a region containing, ) and its n-th derivative for > is given by ϕ n) ) = ϕ n ). Proof. We have ϕ z) = = z d = d log n ) z n 4) n! n= log n ) sin z n n! d = ) n ϕ n ) zn n!, 5) n= n= the inversion of sum and integral in 4) being justified as follows: Case i. > z. All the terms involved are nonnegative. Case ii. > z >. By Case i ϕ z ) = d log n ) z n n! n= <. Thus 5) yields the Taylor series for ϕ z) at z =, and the final conclusion follows. Lemma Proof. Let a >, then for > we have I) = d = We show first that I) = ϕ) =. 6) a a d + sin It suffices to consider the case < a < ; since for a, we have b a d b a d = as. Now, for a < <, we have and it follows that < a < sin < sina a a d. d =. 7) <, d a d + a) sin a a + a d 4

5 as. This establishes 7). We net use the following easily roved results [9, 8]: for all real, 8) and π u ) Γ + ) du = Γ ). + 9) where the equality is a secial case of a beta-function evaluation see also [, Theorem 7.69]). It follows from 8) and 9) that 6 sin d 6 u ) du = Γ + ) Γ ), + ) and hence that inf I) Γ + ) Γ ) +. ) Now, in order to get an aroriate inequality for the su, we note that for any w > such that W = 5 ) 6, w we have whence W sin sin 6w d 6w W 6w u ) du 6w for < < W, ) w u ) Γ + ) du = Γ ). + ) It follows from 7) and ) that and therefore from ) and 4), for w > we have w Γ + ) su I) Γ ) +, 4) Γ + ) Γ ) + inf I) su Letting in 5), since for any a >, we have a from [8, Problem,. 45] or ), we obtain inf a Γ a + Γa + ) ) I) =, I) su I) for all w >. Finally, letting w +, we get the desired equation 6). We are now ready for our more general result. w Γ + ) Γ ) +. 5) w, 6) 5

6 Theorem For all natural numbers n we have n+ ϕ n) ) = n+ log = ) n Γ n + sin n ) d ). 7) Proof. The first equality was noted above. We roceed to establish equation 7) by induction. The roof of the base case was given in Lemma. For the inductive ste of the roof, we assume that for a given nonnegative integer n, we have n+ ϕ n) ) = ) n Γ n + ). It is easily verified that < log ) < t for all but countably many values of t. for < <, and setting = < log sint t t < t For q > +, multilying these inequalities by the nonnegative term ) log n sint n t ) q t, we have ) ) log n n t t q t +q) < ) n log t n+ ) ) < ) log n n t t sint t t q, that q sint t for the same values of t, and integrating over, ) yields ) ) ) ϕ n n) q) ϕ n) + q) ) n ϕ n+) q) ) n ϕ n) q ) ϕ n) q), q ) and hence ) n qn+ ϕ n) q) q n+ + q)n+ ϕ n) + q) ) n qn++ ϕ n+) q) + q) n+ ) n q )n+ ϕ n) q ) q ) n+ qn+ q n++ ϕ n) q). 8) q n+ Now let q =, where > is fied, then 8) becomes ) n q n+ ϕ n) q) + q)n+ ϕ n) + q) + )n+ ) n q n++ ϕ n+) q) ) n q )n+ ϕ n) q ) q n+ ϕ n) q). )n+ 6

7 Net let q, eeing > fied, so that and q = ). It follows from the inductive hyothesis that q qn+ and therefore + Since ϕ n) q) = q ) n+ it follows from 9) that + q)n+ Γ n + ) + ) n+ ϕ n) + q) = q = inf q q )n+ )n+ q n++ q )n+ q n++ ϕ n+) q) = ) n+ and this comletes the roof of the inductive ste. Final Remars Our roof of Theorem shows both that ) n+ ϕ n) q ) = ) n Γ n + ϕ n+) q) su q ), ) n+ q n++ ϕ n+) q) Γ n + ). 9) + t) n = = n + t t, n + ) Γ n + ) = Γ n + + ), n+ ϕ n) ) = a n ) eists and determines the value of a n. If we now in advance that the it eists for every nonnegative integer n, then we can use Lemmas and to write ϕ + )) = n+ ϕ n) ) n / n! = + n= for > >, and then justify the echange of it and sum, and eand the final term to obtain n= n a n n! = )n Γ ) n + n n!. n= Comaring coefficients of the above two eonential generating functions yields the desired valuation a n = )n Γ ) n +. ) In fact, to justify the echange by means of the series version of Lebesgue s theorem on dominated convergence one needs to establish something lie n+ ϕ n) ) n! M with M a ositive constant indeendent of n and, and this requires an inequality such as the right-hand side of 9) with q relaced by and n by n ) used in the given roof of Theorem. 7

8 Another way of determining the value of a n in ) if we now it eists for every n, is to roceed via L Hosital s rule as follows: ϕ n ) ) a n = = n+ ϕ n) ) n ) n = a n n, whence, by Lemma, which is ) again. a n = ) n a n = ) = )n Γ ) n + One advantage of our elicit roof of Lemma over Olver s asymtotic result in ) is that it is easily eloited to establish ). Theorem For all > we have I) > + > ). ) Proof. For > and < s <, Abromowitz and Stegun [] records as 5.6.4) in the new web version) that s Γ + ) < Γ + s) < + ) s. ) Hence, from ) and ) we obtain for > that I) > > Γ + ) Γ Γ + ) ) + = + Γ ) + ). + > Here, for the enultimate inequality, we have used the left-hand inequality in ) with =, s = /. Note that ) imlies that sinc > ) / 6 π + when sinc is viewed as a function in L [, ]). We finish by observing that the lower bound is asymtotically of the correct order, and leave as an oen question whether similar elicit techniques to those in Theorem can be used to establish the second-order term in the asymtotic eansion ) or the concavity roerties conjectured in the introduction. References [] M. Abramowitz and I.A. Stegun, Handboo of Mathematical Functions, NBS now NIST), 965. See also htt://dlmf.nist.gov/. [] D. Borwein and J. M. Borwein, Some remarable roerties of sinc and related integrals, Ramanujan J., 5 ), 7 9. [] D. Borwein, J. M. Borwein, and B. Mares, Multi-variable sinc integrals and volumes of olyhedra, Ramanujan J., 6 ), [4] T. J. Bromwich, Theory of Infinite Series, First Edition 98, Second Edition 96, Blacie & Sons, Glasgow. [5] H. S. Carslaw, An Introduction to the Theory of Fourier s Series and Integrals, Third Revised Edition, Dover Publications Inc., New Jersey, 95. [6] N. G. de Bruijn, Asymtotic Methods in Analysis, Second Edition, North-Holland Publishing Co., Amsterdam, 96. 8

9 [7] W. B. Gearhart and H. S. Schultz, The function sin), The College Mathematics Journal, 99), [8] P. Henrici, Alied and Comutational Comle Analysis Volume, John Wiley & Sons, Inc., New Yor, 977. [9] I. E. Leonard and James Duemmel, More and Moore Power Series without Taylor s Theorem, The American Mathematical Monthly, 9 985), [] F. W. J. Olver, Asymtotics and Secial Functions AKP Classics), Second Edition, AK Peters, Nattic, Mass, 997. [] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Sringer Series in Comutational Mathematics, Sringer Verlag, New Yor, 99. [] K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, Belmont, CA, 98. Acnowledgements We wish to than Amram Meir and David Bailey for very useful discussions during the rearation of this note. 9

10 David Borwein obtained two B.Sc. degrees from Witwatersrand University, one in engineering in 945 and the other in mathematics in 948. From University College London UK) he received a Ph.D. in 95 and a D.Sc. in 96. He has been at the University of Western Ontario since 96 with an emeritus title since 989. His main area of research has been classical analysis, articularly summability theory. Deartment of Mathematics, The University of Western Ontario, London, ONT, N6A 5B7, Canada. dborwein@uwo.ca Jonathan M. Borwein received his B.A. from the University of Western Ontario in 97 and a D.Phil. from Oford in 974, both in mathematics. He currently holds a Laureate Professorshi at University of Newcastle and a Canada Research Chair in the Faculty of Comuter Science at Dalhousie University. His rimary current research interests are in nonlinear functional analysis, otimization, and eerimental comutationally-assisted) mathematics. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 8, Australia and Faculty of Comuter Science and Deartment of Mathematics, Dalhousie University, Halifa NS, BH W5, Canada. jonathan.borwein@newcastle.edu.au Isaac E. Leonard received his B.A. and M.A. from the University of Pennsylvania in 96 and 96, both in hysics. He received his M.Sc. and Ph.D. from Carnegie-Mellon University in 969 and 97, both in mathematics; and a B.Sc. in Comuter Science from the University of Alberta in 994. He currently holds a osition as a Sessional Lecturer with the Deartment of Mathematical and Statistical Sciences at the University of Alberta, and has taught courses in the Comuting Science and Electrical and Comuter Engineering Deartments at the University of Alberta. His current research interests are in analysis of algorithms and numerical analysis. Deartment of Mathematical and Statistical Sciences, The University of Alberta, Edmonton, AB, T6G G, Canada isaac@math.ualberta.ca