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Applied Mathematics, 2010, 1, 439-445 doi:10.4236/am.2010.16058 Published Online December 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM On Approximating Two Distributions from a Single Complex-Valued Function William Dana Flanders, George Japaridze Department of Epidemiology, Rollins School of Public Health, Atlanta, USA Department of Physic s, Clark At l ant a Uni ver si t y, Atlanta, USA E-mail: flanders@sph.emory.edu Received May 1, 2010; revised July 10, 2010; accepted July 15, 2010 Abstract We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function and its Fourier transform. We show that this problem always has a solution within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions. We describe the algorithm and construction of and provide examples of approximating several pairs of distributions using the algorithm. Keywords: Probability Distribution, Sinc Approximation, Cardinal Function, Fourier Transform, Wavelet 1. Introduction In this paper we consider the problem of reconstructing two different probability distributions from a single com- plex-valued function with a specified degree of accuracy. This problem was suggested by the interpretation of the wave function which is a solution of the Schrödinger equation. As is well known x x ★ is the proba- bility density in a position space and ˆˆ pp ★ is the corresponding probability density in a momentum space, where x ★ is the complex conjugate of x and ˆp is the Fourier transform of x [1]. We raise the related question as to whether any two distributions can be approximated within a specified error from a single function by calculating its modulus or that of its Fourier transform, calculations like those used in Quantum Mechanics. Our answer to this question is affirmative. We prove this assertion by constructing a complex- valued function which approximates two given distri- butions within a given error, provided the two distribu- tions satisfy some regularity requirements specified below. In particular, we show that the cumulative distri- butions can be approximated pointwise using the indefi- nite integral of t he m odul us o f or of ˆ . The paper is organized as follows. In the next section we list assumptions, conv entio ns and notation . In Section 3 we first introduce an expression for evaluating the error in the approximation of the two given distributions from the modulus of or its Fourier transform. We then show how to construct this function. In the next section we list 4 pairs of distributions approximated using cons- truction described in Section 3. In Section 5 we discuss the method, limitations and possible generalizations. In the appendix we sketch a proof that the function approximates any two given distributions on subintervals as claimed and a proof that the cumulative distributions can be approximated pointwise (uniformly on closed intervals) using the indefinite integral of the modulus of or of ˆ . 2. Definitions, Conventions and Assumptions 2.1. Assumptions 1) X and P are any two variables; X f x is the probability density function for X with corresponding cumulative distribution X f x; P f p is the probabi- lity density function for P with corresponding cumula- tive distribution P F p 2) X f x and P f p have finite variances and they (and their square roots) are continuously differen- tiable with Fourier transforms in 2 L and in 1 L (for definitions of 1 L and 2 L see [2]; throughout, we denote the norm in 2 L as 2) 3) for every >0 , there exist b X and b P such that: a) for all , bb x XX and for all , bb pPP , >0 X fx and >0; P fp and, b) </16 Xb FX , W. D. FLANDERS ET AL. Copyright © 2010 SciRes. AM 440 </16 Pb FP , >1 /16 Xb FX , and >1 /16 Pb FP . 2.2. Definitions and Notations 1) Let n be any integer, set =2 n J, =/ b x XJ, and =/, b pPJwhere b X and b P are given in assumption 3; if necessary, increasen( and/orb P) so that /2 >1/16; Pb FP p 2 J is the number of intervals of width x and pin , bb X Xand /2,/2 ; bb PpPp 2) s is a number defined in the Construction below; it indicates the number of subintervals into which each interval x is divided. It is used to control the size of the error in the approximations ; 3) =/ts xp ; 4) j and k are integers, between s J, and 1sJ ; l, m and m are integers between J , and 1 J or J , unless noted othe rwis e; 5) s x is a function of the form used in the “sinc approximation” [3,4]: 1 = / =/ , / sJ sX jsJ x jxs xfjxssinc xs where sin x sinc x x ; 6) max , xX Qfx for ,; bb xXX min 0, xX Pfx for ,; bb x XX min() >0, pP Pfp for /2, ; bb pPpP max/, xX Qdfxdx for ,; bb xXX 7) 1/2 / jX x afjxs s for =, 1,,1jsJsJ sJ 8) 2 ;, =, jz x jz m mj pr a for 0;zsJj 9) ; p l pr = 1lp P lp f zdz for 1;JlJ 10) ˆ(.) f denotes the Fourier transform of a function (.), f *(.)fdenotes the complex conjugate of a function (.); f 11) j M is a function, defined recursively in the constr- uction, mapping the integers ,1,,1jsJsJ sJ to the numbers /,1/,,1/,/ ; s JxsJx sJxsJx 12) =/ jj mMxs ; j m is an integer; 13) :,1,,1 l SjjsJsJ sJ and =/ j M ls x for =, 1,,lJJ J. Equivalently, 1/, lM Sflsx where 1(.) M f is the inverse mapping. 3. Construction Given , XP f xf p, and n, and b X and b P with the properties described in Assumptions we cons- truct a complex-valued function which satisfies the following two inequalities 1 12* 1 =2 ˆˆ < lp J P lJ lp tqt qtfqdq (1) and 11* =< Jlx X lx lJ xxfxdx (2) where t is a scale parameter defined as = s t x p (3) and s is defined to be the smallest integer so that Inequalities (4)-(8) are satisfied: <4 x xQ s J (4) 1/2 3 4 4322< 42 1 x xQlogsJ sJ (5) 4 2</82 1 sX xfx J (6) 22 4 ||22 242 1 x xx xQ JQlogsJ sP J (7) 1 = 11 1/ / == /= /8 sJ XbXX XbjsJ sJ sJ jxs XX jxs jsJ jsJ x fxdx fjxs s x fxdxfjxs s (8) Many of the functions already defined, such as () s x , or to be defined, such as () x and j M , depend on s and n. For brevity we suppress explicit notation of that dependency. The next step in the construction of is to define subintervals. First we split the interval , bb X X into 2 J intervals of length : x ,1 ,lxl x =, 1,,1lJJ J . Next we further divide each interval ,1lxlx into s subintervals and W. D. FLANDERS ET AL. Copyright © 2010 SciRes. AM 441 number them from =,1,,jsJsJ to 1sJ . Define =, 1,,1jsJsJ sJ 1/2 = jX xjx af ss (9) We continue the construction by introducing the function j M which maps the integers ,1,,1jsJsJ sJ to the numbers /,1/,,1/,/ , s JxsJ xsJ xsJx using the following recursive algorithm: initialize j to s J and l to J . Then proceed with the following steps: 1) Set ;, =0 xjz pr for =1z and set 2 ;, = =jz x jz m mj pr a for each integer 0,1,,,zsJj and set 1 2 ;1 2 lp pl P lp prfz dz for 1JlJ 2) If an integer z exists between 1 and s Jj such that the inequalities ;;, 0/221, plx jz pr prJ and ;,1; >0 xjzpl pr pr hold, then set =kz ; otherwise, set =ksJj 3) If 0k then set all ,, j jk MM equal to /ls x 4) Set =1jjk and =1ll; if =,jsJ stop, otherwise go to step 5) 5) If 1=lJ, set 1 ,1,, j jsJ MM M all to / s Jx an d stop; otherwise return to step 1). Step 2) always returns a value of k. The iterative algorithm continues until stopping, either at step 4) or 5). Properties of j M that will be used subsequently include: for each ,, 1jsJ sJ , the mapping j M is a non-decreasing function of j; each / j M sx is an integer (i.e., /= j j M xs m for some integer ,, j mJJ ). Using this definition of j M , we can now define x appearing in Equations (1)-(2) as: 12 = 12 = / ()= / / =/ sJ iMx j j jsJ sJ iMx j X jsJ sxjxs xasinc e xxs jxx jxs fsince sxs (10) Apart from exp 2j iMx and the s Jth term in the series (10), the function x represents the Whittaker-Shannon interpolation formula [5,6] for X f x. In addition to approximations given in (1)-(2) we also prove uniform approximations to the cumulative distri- butions: let us define /2 =p PP Pp b Fp fqdq * /2 ˆˆ =p PPp b F ptqtqt dq (11) and =x XX Xb F xfzdz * =x XXb F xzzdz (12) Then, it can be shown that with , XP f xf p, and * given and with b X and b P having the properties described in Assumptions, one can choose n large enough that the complex-valued function given by the construction can be used to uniformly approximate the cumulative distributions X F x and P F p for , bb x XX , and for /2, /2 bb pPpPp : * XX Fx Fx (13) and * PP Fp Fp (14) For derivation of (1) and (13) see Appendix. Approximations (2) and (14) are derived in a similar fashion. 4. Examples To illustrate the approximation, we compare the integral of the probability density function over a series of intervals with the correpsonding approximation generated by from (1), (2) and (10). Specifically, in P space we plot /2 /2 pp P pp f qdq and the correspon- ding approximation /2 * /2 ˆˆ pp pp tqt qtdq . Similarly, in X space we plot /2 /2 xx X xx f qdq and the corres- ponding approximation /2 * /2 xx xx qqdq ; we app- roximate the last integral by * x xx , as x is small. We applied the method to four pairs of distributions; for each pair, we use a single (and its Fourier tran- sform) to approximate them with n = 7, s = 64 and n ==5/20.039xp : 1) Figure 1 illustrates the approximation using Equation (10), for the distributions 22 11 88 22 4 = 0.750.25 2 xx X fxe e W. D. FLANDERS ET AL. Copyright © 2010 SciRes. AM 442 Figure 1. Approximation to the integrals 2 2 pp p pp f qdq (top) and 2 2 xx X xx f qdq (bottom) based on Equation (10) for the pair of distributions 2 exp 2pfpp , and 22 11 88 22 40.75 0.25 2 xx X fx e e . 2/2 1 and=2 p P fp e 2) Figure 2 illustrates the approximation using Equation (10), for the distributions 2/2 =,0 and=12 xp XP fx exfpe 3) Figure 3 illustrates the approximation using Equation (10), for the distributions 2/2 1 =and=,0 2 xp XP fxefp ep 4) Figure 4 illustrates the approximation using Figure 2. Approximation to the integrals 2 2 pp p pp f qdq (top) and 2 2 xx X xx f qdq (bottom) based on Equation (10) for the pair of distributions 22 1 2 p p fp e and x X f xe for 0x and 0Xfx for <0x. Equation (10), for the distributions 22 /2 /2 11 =and = 22 xp XP fx efpe 5. Discussion Based on the algorithm described in Section 3 we have shown how to approximate two given distributions from a single complex-valued function with a specified accuracy. The main result of our paper is given in expressions (1-2), (10) and (13-14). Expression (10) can be viewed as an extension of well-known numerical methods based on the sinc function [3], providing a W. D. FLANDERS ET AL. Copyright © 2010 SciRes. AM 443 Figure 3. Approximation to the integrals 2 2 pp p pp f qdq (bottom) and 2 2 xx X xx f qdq (top) based on Equation (10) for the pair of distributions 22 X1 2 x fx e , and expP f pp for 0p and 0Pfp for <0p. Figure 4. Approximation to the integrals 2 2 xx X xx f qdq (bottom) and 2 2 xx X xx f qdq (top) based on Equation (10) for the pair of distributions 22 1 2 p p fp e , and 22 1 2 x X fx e . method for approximating two distributions simulta- neously. The construction and arguments given here show that there always exists a function of the form given by (10) which approximates two given distributions with aspe- cified degree of accuracy provided our assumptions (1-3) are satisfied. For successively higher degrees of accuracy, the method allows construction of a sequence of appro- ximating functions but this sequence may not necessarily converge, say in 2 L, to a limiting function. A modification of our approach or a similar method based not on the sinc function but on some other basis or wavelet series [7] might be used to approximate the two distributions and converge to a limiting function. For example, one might attempt to use the Hermite poly- nomials multiplied by the appropriate Gaussian function as a basis. It remains speculative, however, whether our approach can be modified so as to approximate arbitrary pairs of distributions with a basis other than the sinc functions. Further speculation about this potential generalization is beyond the scope of this paper. We would like to stress that the main goal of the present work is not to find the best numeric approximation (though we can always meet an increasing demand in accuracy) but to establish the existence of an algorithm allowing uniform approxi- mation of the cumulative distributions. 6. References [1] J. von Neumann, “Mathematical Foundations of Quantum W. D. FLANDERS ET AL. Copyright © 2010 SciRes. AM 444 Mechanics,” Princeton University Press, Princeton, 1955. [2] W. Rudin, “Functional Analysis,” McGraw-Hill, New York, 1991. [3] F. Stenger, “Summary of Sinc Numerical Methods,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, September 2000, pp. 379-420. [4] P. L. Butzer, J. R. Higgins and R. L. Stens, “Classical and Approximate Sampling Theorems; Studies in the 2RL and the Uniform Norm,” Journal of Approximation Theory, Vol. 137, No. 2, December 2005, pp. 250-263. [5] J. M. Whittaker, “Interpolation Function Theory,” Cam- bridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge University Press, Cambridge, 1935. [6] C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of Institute of Radio Engineers, Vol. 37, No. 1, January 1949, pp. 10-21. [7] I. Doubechis, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series for Applied Mathematics, 1992. Appendix We sketch the derivation of (1) and (13). First we write down the Fourier transform of x : 2 122 = 12 = ˆ= / =/ = ixq sJ iMx ixq j j jsJ x sJ ijqM j s jj jsJ qxedx sxjxs aesince dx xxs xx aRqMe ss (1) In (1) R is a rectangular function: 11 1when<q<, 22 11 whenq= , =22 0otherwise Rq (2) We consider 1 * 2 1 2 ˆˆ , lp lp I ltqtqt dq (3) the integral of * ˆˆ tqt qt over the interval 11 , 22 lplp ,for each ,1,,1lJJJ . Using expression (1), rescaling the integration variable, and using definition of integer :=/ jj j mmxM s, for this integral we obtain: 11 2 1,= 2 2 ()= sJ l'' j kj ljk sJ ' ijkqjmkm jk 'k I ldqaaRqm Rqme where 1/2 =// jX axfjxss . Property (2) of the rectangular function R implies that each ,jk term in the integrand is either 0 for the full range of integration or vanishes for all but a subinterval of length 1. The latter holds only if = j k mm and we have: 11 2 j lml if and only if = j lm, for some integer j m. After interchanging the order of integration and summation, each integral in the sum is 0 due to the factor 2' ijkq e , unless =jk. On the other hand, if =jk the integral is just 2 j a. Thus, we can write =, 1,,1lJJ J : 1 *2 2 1 2 ˆˆ ==, lp j lp jS l I ltqtqt dqa (4) where we recall the definition of l S: l S is the set of all ,1,,jsJsJ sJ for which /= j M xs l . There are 2 J terms I l of the form given in (4); the approximation given b y Inequality (1) will be proven if we can show that for =, 1,,1jJJ J the following relation holds: 1 22 2 ;1 2 =2 lp jpl jP lp jS jS ll apr afzdz J (5) Either Inequality (5) holds for=, 1,,1lJJ J , and we are done, or there is a smallest l violating (5), say 1 l, such that 1 22 1 12 ||>/2. lp jp jS llp afzdxJ This implies that 11jsJ , where 1=max :l jjjS. Indeed, if this were not the case (i.e., 1<1jsJ ), than 1 1 j M would have been defined by the algorithm described in section 3 as 1 l since 1 1/21 j aJ and 11j would also have been in 1 l S. This would be a contradiction since 1 j was supposed to be the largest element of 1 l S. Therefore, 11jsJ. W. D. FLANDERS ET AL. Copyright © 2010 SciRes. AM 445 By choice of s in section 3, requirement (8): 1/ 12 =/ jxs J jX lJ jSjxs lafxdx is less or equal /8, and because X f x integrates over the interval , bb X X to at least 1/16 (Assumption 3), we have: 12 =1/16/8>1/4 sJ j jsJ a (6) Since () P f p is a probability density function and by Assumption 3 and the algorithm described in section 3: 1 12 12 11 11 22 2 1 === 2 1= =1/4. lp P Pp b ll sJ lp Pjj lp lJlJjS jsJ l fzdz fzdza a (7) The last inequality above is simply Inequality (6). By the Construction, 1 2 2 1 2 lp P j jS l lp f zdza for every l. This fact and (7) imply: 112 21= 2 1 12 2 1 =2 = 1 4 l lp Pj Pp blJjS l llp Pj lp lJ jS l fzdza fzdza (8) Further, the upper bound of 1 in expression (6) implies that: 1 112 2 11 1=1 22 1 3 =4 Jlp Pp b PP lp lp ll fzdz fzdz (9) Now l S is empty for 1 >ll since 11 = sJ M l and j M is non-decreasing. In other words 2=0 j jS la for 1 >ll. This fact and the inequality in Expression (9) yield: 1 11 122 1 =1=1 211 3 =0 4 JJ lp lp Pj P lp lp lljS ll l fzdz afzdz (10) Finally, Inequalities (8) and (10) lead to the desired result, proving Inequality (1). Proof of Inequality (2) requires no new qualitative features but straightforward tedious calculations; we do not cite it here. To prove the uniform approximation for the cumulative distributions, Inequality (13), choose n large enough that * =/2</2 n xxb QxQX , constr- uct () x so that Inequalities (1) and (2) hold with */2 and let =XX x Fx Fx . We now show that * x for all ,( 1) x jx jx and for =, 1,,1jJJ J . Suppose the maximum of XX F xFxoccurs at 0 =, X xfor 0,1 x jx jx . Let F j M denote max,,1. X F xx jxjx If 00 > XX F xFx , then FX j x MFjx (11) since () X F x is non-decreasing. Further, the mean value theorem implies FX x j M FjxQ x since x Q is th e maximum of X f x. Using this result in (11) gives ** * /2 /2= XxX xFjxQxFjx (12) since */2 XX FjxFjx follows from (1) and * ||/2 x Qx by choice of n. On the other hand , if 00 < XX F xFx , then * ** * 11 /2 /2 /2= XXX XXx X x Fj xFjxFj x FjxFjxQx Fjx (13) The chain of estimates above follows from the fact that X F x is non-decreasing, x Qn is the maximum of X f x, and by choice of n. The proof for PP F pFp is similar. |