Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation

Transcription

1 652 J. Opt. Soc. Am. A/Vol. 15, No. 3/March 1998 Barakat et al. Zero-order Hankel transformation algorithms based on Filon quadrature philosophy for diffraction optics and beam propagation Richard Barakat Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138, and Electro-Optics Technology Center, Tufts University, Medford, Massachusetts Elaine Parshall* Electro-Optics Technology Center, Tufts University, Medford, Massachusetts Barbara H. Sandler Gordon McKay Laboratory, Harvard University, Cambridge, Massachusetts Received February 20, 1997; revised manuscript received September 15, 1997; accepted September 22, 1997 Our purpose is to bring to the attention of the optical community our recent work on the numerical evaluation of zero-order Hankel transforms; such techniques have direct application in optical diffraction theory and in optical beam propagation. The two algorithms we discuss (Filon Simpson and Filon-trapezoidal) are reasonably fast and very accurate; furthermore, the errors incurred are essentially independent of the magnitude of the independent variable. Both algorithms are then compared with the recent (fast-fourier-transform-based Hankel transform algorithm developed by Magni, Cerullo, and Silvestri (MCS algorithm) [J. Opt. Soc. Am. A 9, 2031 (1992)] and are shown to be superior. The basic assumption of these algorithms is that the term in the integrand multiplying the Bessel function is relatively smooth compared with the oscillations of the Bessel function. This condition is violated when the inverse Hankel transform has to be computed, and the Filon scheme requires a very large number of quadrature points to achieve even moderate accuracy. To overcome this deficiency, we employ the sampling expansion (Whittaker s cardinal function) to evaluate numerically the inverse Hankel transform Optical Society of America [S (98) ] OCIS codes: , , INTRODUCTION Zero-order Hankel transforms are of common occurrence in diffraction optics and in beam propagation. Only rarely can the integrations be carried out analytically; numerical approaches are generally required. A survey of some of the older numerical algorithms for evaluating the zero-order Hankel transform are outlined in Ref. 1. The popularization of the fast Fourier transform (FFT) in the late 1960 s spawned the obvious idea of using it to numerically evaluate finite-range Fourier integrals. 2,3 As pointed out in Ref. 1, using the FFT in such a fashion leads only to a rectangular-rule quadrature scheme (not even a trapezoidal quadrature scheme). Nevertheless, the FFT has served a very valuable purpose provided that one does not require much accuracy. Siegman 4 was evidently the first to apply FFT concepts to the evaluation of zero-order Hankel transforms. By using an exponential transform of the dependent and independent variables, he cast the Hankel transform into a one-dimensional crosscorrelation integral, which is then evaluated by use of the FFT. Following this work, some investigators 5 8 developed other algorithms for evaluating the zero-order Hankel transform. Murphy and Gallagher 5 provide a valuable survey of such studies, in addition to their own algorithm. Very recently Magni et al. 9 have produced yet another FFT-type of algorithm, which we will comment on in Section 4. A different problem arises in diffraction by fractal objects, such as the random Sierpinski carpet. 10 Here, because of the self-replicating issues, reasonably high numerical accuracy [error O(10 6 )] is required. The purpose of this paper is to bring to the attention of the optics community two recent numerical algorithms, based on Filon quadrature philosophy, for evaluating zero-order Hankel transforms. 11,12 Both algorithms are reasonably fast and very accurate; they are outlined in Section 2. See also Ref. 13 for the first-order Hankel transform. In actual practice, we do not evaluate the infinite-range Hankel transform Hr h pj 0 rppdp. (1.1) 0 We replace the infinite limit by a finite limit, b Hr h pj 0 rppdp, (1.2) a as we can always find a value b that approximates infinity for a particular ; here a /98/ $ Optical Society of America

2 Barakat et al. Vol. 15, No. 3/March 1998/J. Opt. Soc. Am. A 653 In a seminal paper, Filon 14 (see also Ref. 15) studied the numerical evaluation of the finite Fourier transform b Fx f pexpixpdp. (1.3) a [In fact he studied the cosine and sine transforms independently, but there is no particular difficulty in studying Eq. (1.3) directly.] For large values of x, the graph of the integrand consists of positive and negative areas of nearly equal size. The addition of these two equations of areas results in substantial loss of accuracy. Filon conceived the idea of retaining Simpson s rule but required that only f( p) be fitted to a quadratic over the basic subinterval instead of the entire integrand f( p)exp(ixp). The fact that only f( p) has to be approximated as a quadratic means that we can take the number of subintervals to be relatively small in many cases of practical interest. Filon s work on the finite Fourier transform, suggests that it is possible to attack Eq. (1.2) by similar methods. There are profound differences between Eqs. (1.2) and (1.3) in that exp(ixp) is periodic and translationally invariant on the real line, whereas J 0 (rp) is an almost periodic, decaying function. Nevertheless, Filon s approach can be carried through, as we will show. We wish to stress that this paper is a methodology paper, and we hope that it will be of use to others in optics. Some direct optical applications will be discussed in separate publications. 2. FILON SIMPSON AND FILON- TRAPEZOIDAL HANKEL TRANSFORM ALGORITHMS Although we must refer to Refs. 11 and 12 for the full details of both algorithms, we briefly sketch the essentials of them here. The first algorithm is the Filon Simpson. 11 The function is assumed to be approximated by a quadratic over the quadrature points p 2k2, p 2k1, and p 2k (which are a subset of the integration interval a, b, h p c 1 c 2 p p 2k1 c 3 p p 2k1 c 3 p p 2k1 2, (2.1) where ( p 2k p p 2k2 ). The c s can be determined by setting p p 2k, p 2k1, and p 2k2, in Eq. (2.1) and solving the three simultaneous linear equations. We also require the first two derivatives of ; the explicit expressions are in Ref. 11. Next consider the integral p2k2 H K r h pj 0 rppdp, (2.2) p 2k which is effectively a double panel of three quadrature points: p 2k2, p 2k1, and p 2k, where p 2k2 p 2k1 p 2k1 p 2k. (2.3) These panels are a subset of the larger interval a, b, and is approximated by Eq. (2.1). Upon integrating H k (r) by parts twice, noting that x yj 0 ydy xj 1 x, (2.4) we obtain an expression for H k (r) [see Eq. (2.8) of Ref. 11.] To evaluate the integral over a, b, i.e., Eq. (1.2), divide the interval, as in Simpson s method, into an even number of subintervals N, each of length, so that b a N. Consequently, Hr N2/2 k0 H k r. (2.5) We can show that [where h( p 2k2 ) h 2k2, etc.] where Hr 1 r h N J 1 rp N p N h 0 J 1 rp 0 p 0 1 2r 3 h N2 4h N1 3h N $ 0 rp N 1 N2/2 2r 3 Q k $ 0 rp 2k k0 1 N2/2 2 r 4 h 2k2 2h 2k1 k0 h 2k $ 1 rp 2k, rp 2k2, (2.6) Q 0 h 2 4h 1 3h 0, Q k h 2k2 4h 2k1 6h 2k 4h 2k1 h 2k2. (2.7) The functions $ 0 (x) and $ 1 (x 1, x 2 ) are defined by x $ 0 x J 1 yydy, (2.8) 0 $ 1 x 1, x 2 x 1 x 2$0 ydy. (2.9) The evaluation of these integrals is discussed in Appendix A of Ref. 11. Equation (2.6) is the basic expression for the Filon Simpson Hankel algorithm. As with Filon s original approach to Fourier integrals, the error incurred in Eq. (2.6) is proportional to the derivatives of itself rather than to the whole integrand, and hence the errors are relatively independent of r. In some situations, we do not require such great accuracy for the Hankel transforms but need to maintain a given accuracy, more or less uniformly, independent of the magnitude of r. In Ref. 12, a second algorithm, termed the Filon trapezoidal, was developed to answer this need. We now sketch the algorithm and refer to Ref. 12 for details. Consider two points p k and p k1, which are a subset of a, b. Now is approximated as a straight line between p k and p k1 : where p k p p k1. h p A Bp, (2.10) Here

3 654 J. Opt. Soc. Am. A/Vol. 15, No. 3/March 1998 Barakat et al. A 1 p k1h k p k h k1, B 1 h k1 h k, (2.11) and ( p k1 p k ). Now consider the integral p k1h H k r pj0 rppdp, (2.12) p k and integrate by parts, using Eq. (2.4). Consequently, H k r 1 r h k1j 1 rp k1 h k J 1 rp k 1 r 3 h k1 h k $ 0 rp k1 $ 0 rp 2k, (2.13) where $ 0 is given by Eq. (2.8). To evaluate over a, b, we again let b a N so that N Hr k1 H k r 1 r h N J 1 rp N h 0 J 1 rp 0 1 N r 3 h k1 h k $ 0 rp k1 $ 0 rp k. k0 (2.14) Note that this expression does not contain $ 1 (x 1, x 2 ); this is both a blessing and a curse. It is a blessing because its absence in Eq. (2.14) speeds up the computation;it is a curse because accuracy is lost. As with the Filon Simpson algorithm, the error incurred in Eq. (2.14) is proportional to the derivatives of itself rather than to the whole integrand; hence the errors are relatively independent of r. Both Eqs. (2.6) and (2.14) are valid for r 0. For r 0, b H0 h ppdp, (2.15) a which can be evaluated separately by standard quadrature schemes. We should re-emphasize that the Filon-trapezoidal algorithm is not meant to be a direct competitor of the Filon Simpson algorithm with respect to accuracy. Rather its main use is to maintain a given, but moderate, accuracy more or less uniformly independent of the magnitude of r, where speed of execution is of importance. A reviewer has raised the question as to the usefulness of Gauss quadrature (see Ref. 1 and references therein) for the evaluation of these oscillatory integrals. When r is not too large (say, r 8), the Bessel function has not oscillated appreciably and Gauss quadrature is effective because the entire integrand of Eq. (1.2) can be approximated as a polynomial of fairly high degree. Gauss quadrature using N points is exact for polynomial integrands of degree (2N 1). However, when r becomes large, the Bessel function oscillates appreciably and the integrand is poorly approximated as a polynomial. Consequently, Gauss quadrature is not very effective when r is large (but then none of the standard quadrature schemes are). 3. NUMERICAL EXAMPLES As illustrative examples that possess exact solutions, let us consider one from optical diffraction theory and one from beam-propagation theory. The pair h p 2 arccos p p1 p2 1/2, 0 p 1, (3.1) Hr 2J 2 1r r, 0 r, (3.2) arises in optical diffraction theory, 16 where is the optical transfer function of an aberration-free optical system with a circular aperture and H(r) is the corresponding point-spread function. Calculations with the Filon Simpson algorithm were carried out for N 100 and N 200 with r 1(1)100. The maximum and minimum errors over this range are max error for N 100, and min error max error min error at r 2, at r 64 at r 2 at r 76 for N 200. In addition, we calculated the average value of the absolute error, error 1 M9 error at r m, (3.3) 10 mm over a block of 10 values (i.e., M 1, 10, 20,...). The results are summarized in Table 1. Two points to make about the table are that (1) there is essentially an order of Table 1. Averaged Absolute Error (10 8 ) over Blocks of Ten Values between Exact Results [Eq. (3.2)] and the Numerical Computations (Filon Simpson Algorithm) r m N 100 N

4 Barakat et al. Vol. 15, No. 3/March 1998/J. Opt. Soc. Am. A 655 magnitude difference in the averaged error between N 100 and N 200 and (2) the averaged error is reasonably constant irrespective of r. In both cases, the largest errors occur when r is reasonably small. Now consider the corresponding Filon-trapezoidal scheme, but now it is for N 100, 200, and 300 again for r 1(1)100. The maximum and the minimum errors are for N 100, max error for N 200, and min error max error min error max error min error at r 2, at r 96 at r 2 at r 96 at r 2 at r 85 for N 300. As with the Filon Simpson algorithm, the maximum error is relatively small. We also carried out computations of the averaged absolute error, and they seem to behave somewhat like those in Table 1. Unlike the Filon Simpson algorithm, increasing the number of quadrature points does not decrease the error significantly. After all, the local curvature of has been neglected, and we cannot expect to gain accuracy without this vital information, irrespective of the number of quadrature points. As our second numerical example, consider 17 h p 1 p 2, 0 p 1. (3.4) Here is real and positive. Note that is bell shaped about p 0 and tends to mimic a Gaussian; in fact, 1 p 2 expp 2. (3.5) The corresponding expression for H(r) is 17 Hr 2 1 J 1r r 1. (3.6) As a representative value, we choose to illustrate the computations for 3/2. As with the other numerical example, the largest error is for small r. Tables 2 and 3 show the averaged absolute errors for the respective algorithms; they need no detailed comment. Other values of exhibited roughly the same behavior. In these two numerical examples, we indulged in overkill by using N 100 and 200 when in fact we could have employed N 25 to secure reasonable accuracy [i.e., errors of O(10 4 )too(10 5 )] with the resulting speedup of computations, especially for beam propagation. Some aspects of beam propagation using both Gaussian beams and supergaussian beams will be discussed in a separate publication that is in preparation. Table 2. Averaged Absolute Error (10 9 ) over Blocks of Ten Values between Exact Results [Eq. (3.6)] and the Numerical Computations (Filon Simpson Algorithm) r N 100 N Table 3. Averaged Absolute Error (10 7 ) over Blocks of Ten Values between Exact Results [Eq. (3.6)] and the Numerical Computations (Filon-Trapezoidal Algorithm) r N 100 N COMPARISON OF FILON SIMPSON AND FILON-TRAPEZOIDAL ALGORITHMS WITH THE MAGNI CERULLO SILVESTRI ALGORITHM As noted in Section 1, Magni, Cerullo, and Silvestri (MCS) 9 have recently developed a FFT-based zero-order Hankel transform, which we will term the MCS algorithm. They compared their algorithm with that of Siegman 4 for the function (in our notation) with b 1 and a 0. to this is h p p 2, (4.1) The exact solution corresponding Hr 1 r 3 r2 4J 1 r 2rJ 0 r. (4.2) On the basis of numerical evidence they conclude that their algorithm is superior to that of Siegman (also an FFT-based algorithm) in accuracy; this is a conclusion with which we concur. Now for the comparison of our algorithms with theirs. Our independent variable r is related to their independent variable y by r 2N f y, (4.3)

5 656 J. Opt. Soc. Am. A/Vol. 15, No. 3/March 1998 Barakat et al. where N f is the Fresnel number. Although 0 y, they confine their attention to 0 y 1 and N f 10, 200. Thus the maximum values that they encounter are r 20 (62.83), and r 400 (1256.6). From our viewpoint, the MCS algorithm does not perform particularly well, either in accuracy or in speed of execution (measured in function evaluations), compared with our algorithms. The function p 2 is quadratic in p and thus should yield an exact solution (irrespective of the magnitude of r) by virtue of satisfying Eq. (2.1) if we employ the Filon Simpson quadrature. The only error induced in the Filon Simpson algorithm is in the numerical evaluation of $ 0 and $ 1. It is not our intention to produce a large amount of numerics for the comparison; suffice it to say that where r 25, the error is only for N 60 and for N 100. In fact, the errors are O(10 8 ) for all values of r that we calculated. Even at the maximum of the r values (i.e., r 400), the error is It would be of some interest to see how the MCS algorithm performs on functions such as in Eq. (3.1). Now for the Filon-trapezoidal algorithm. Clearly, we cannot expect such accuracy, but in calculations that we need not reproduce, we easily achieved errors of O(10 6 ) for N 100; the details are omitted. An integrand similar to Eq. (4.1) appears in the diffraction theory of random Sierpinski carpets, 10 namely, Fig. 1. Plot of exact solution given by Eq. (3.6) with 1.5 (dashed curve) versus direct numerical calculation of Eq. (5.1) with Filon Simpson with N 200 points (solid curve). h p p, (4.4) where is related to the fractal nature of the carpet and is generally not an integer. This integrand is now under investigation. 5. INVERSION ISSUES In many problems, especially beam propagation, it is necessary to calculate, given H(r), h p HrJ 0 prrdr. (5.1) 0 Now because is relatively smooth with respect to the Bessel-function oscillations (basic assumption of the two algorithms discussed in Section 2), then H(r) is generally an oscillating function of diminishing magnitude as r is made to increase. We consider two different approaches to evaluating Eq. (5.1). In the first approach, a brute force one, we directly attack Eq. (5.1). The upper limit of infinity can be replaced by a finite limit, call it b, which depends on the particular H(r) in question. Since H(r) is an oscillating function, then a very large number of quadrature points are needed to evaluate to even moderate accuracy. To show how serious the situation can be, consider the functions given in Eqs. (3.4) and (3.6) with 3/2. After some initial numerical experimentation, we found that b 120 was sufficient for our particular example. The results of evaluating, given H(r), are best presented in the form of graphs rather than in tables. Figures 1 3 show the inversion (by use of Filon Simpson) for N 200, 400, and 800 quadrature points (solid curves) and the exact Fig. 2. Fig. 3. Same data as for Fig. 1, but with N 400 points. Same data as for Fig. 1, but with N 800 points.

6 Barakat et al. Vol. 15, No. 3/March 1998/J. Opt. Soc. Am. A 657 Table 4. Exact Solution of the Inverse Hankel Transform Compared with the Sampling Expansion Evaluation for Eqs. (3.4) and (3.6) with Exact Value of b 1 for N 50 and 30 p Exact N 50 Error N 30 Error Table 5. Exact Solution of the Inverse Hankel Transform Compared with the Sampling Expansion Evaluation for Eqs. (3.4) and (3.6) with Estimated Value of b > 1(b1.1) for N 50 and 30 p Exact N 50 Error N 30 Error Table 6. Exact Solution of the Inverse Hankel Transform Compared with the Sampling Expansion Evaluation for Eqs. (3.4) and (3.6) with Estimated Value of b < 1(b0.9) for N 50 and 30 p Exact N 50 Error N 30 Error

7 658 J. Opt. Soc. Am. A/Vol. 15, No. 3/March 1998 Barakat et al. function (dashed curves). Note that N 200 is virtually useless; even N 800 still shows a noticeable discrepancy! Given these depressing computational efforts, we turn to the second (and preferable) approach. The fact that 0 for p b allows us to employ the sampled Fourier Bessel expansion 18 : h p 2 b 2 n1 H n b J 1 n 2 J 0 n p b. (5.2) for 0 p b. Here the n are the positive zeros of J 0, i.e., J 0 n 0. (5.3) The n are easily calculated from the asymptotic series, 19,20 provided that n 3. The first three n are not accurately calculated from the asymptotic series, but accurate values of 1, 2, and 3 are available in Table VII of Ref. 19. Only a relatively small number of values of H(r) are needed in Eq. (5.2), say, terms in the series. This approach to the inversion problem goes back to Barakat, 21,22 who employed Eq. (5.2) in the optical diffraction theory as a computational tool. For an excellent pedagogic review of sampling expansions, see Jerri. 22 We again employ Eqs. (3.4) and (3.6) for numerical calculations, using the sampling expansion. Numerical results are shown in Table 4 for N 50 and 30 along with the absolute error and are self-explanatory. Needless to say that the sampling expansion approach is superior to the direct quadrature approach when the integrand is oscillating as rapidly as the Bessel function. We must realize that there is a hidden assumption in using the sampling expansion, namely, that b is known exactly. If b is not known exactly, then numerical errors are going to occur. This situation is bound to happen in beam propagation because H(r) is numerically evaluated, and it is very hard to determine b exactly. To this end, we now see what happens when b is somewhat smaller or larger than the exact numerical value. Table 5 contains the numerical results when b is greater than the exact value (i.e., b 1.1); however, the absolute errors are still small. When b is smaller than the exact value (i.e., b 0.9), the numerical results in Table 6 are decidedly in error as p approaches unity. When p is small, the error is small; but for p 0.85, the absolute errors increase very rapidly and the resultant answers are virtually useless. Our conclusion based on this (and other examples) is that the approximate b must be larger than the exact b for useful numerical results. If the approximate b is smaller than the exact b, then the sampling expansion inversion of the Hankel transform leads to very large errors. 6. SUMMARY In view of the density of equations, it seems desirable to summarize our findings. First, we gave an overview of the algorithms for the numerical evaluation of zero-order Hankel transforms (work previously derived elsewhere), using Filon quadrature philosophy. In this approach, the integrand is separated into the product of the function being integrated and the Bessel-function kernel. The basic assumption is that the function being integrated is slowly varying compared with the Bessel-function oscillations. Two versions of the algorithm are discussed: Filon Simpson and Filon-trapezoidal. The former is much more accurate than the latter. The error incurred in both versions of the algorithm depends mainly on the behavior of the smooth portion of the integrand with only a weak dependence on the oscillating Bessel function. Thus, for all practical purposes, the error is relatively independent of the magnitude of the independent variable, and hence the algorithms can be employed to obtain high accuracy for large values of the independent variable without having to use asymptotic methods. As expected, the trapezoidal version is less accurate than the Filon Simpson version, but it is quicker. Furthermore, the trapezoidal version tends to saturate in that increasing materially the number of quadrature points does not yield significantly greater accuracy. Both versions of the algorithm are compared with the FFT-based MCS algorithm for evaluating zero-order Hankel transforms. The Filon approach is shown to be more accurate and requires far fewer quadrature points. Finally, the Filon approach is used to invert the Hankel transform where now the function to be integrated is itself oscillatory (violating the basic assumption of the algorithm). Numerical results show the scheme to yield poor accuracy while requiring a very large number of quadrature points. To circumvent this difficulty, we use the sampling expansion. Only a moderate number of terms in the series yields accuracy sufficient for physical problems. As noted in the introduction, this paper is concerned with methodology. Optical applications will be discussed separately. * Present address, Electro-Optic Group, Polaroid Corporation, Cambridge, Massachusetts REFERENCES AND NOTES 1. R. Barakat, The numerical evaluation of diffraction integrals, in The Computer in Optical Research, R. Frieden, ed. (Springer, New York, 1980), Chap L. Bingham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Englewood Cliffs, N. J., 1988). 3. D. Elliott and K. Rao, Fast Transforms (Academic, Orlando, 1982), Chaps. 4 and A. Siegman, Quasi-fast Hankel transform, Opt. Lett. 1, (1977). 5. P. Murphy and N. Gallagher, Fast algorithm for the computation of the zero-order Hankel transform, J. Opt. Soc. Am. 73, (1983). Contains references to other FFT-based Hankel transform algorithms. 6. G. Agrawal and M. Lax, End correction in the quasi-fast Hankel transform for optical propagation problems, Opt. Lett. 6, (1981). 7. A. Oppenheim, G. Frisk, and D. Martinez, An algorithm for the numerical evaluation of the Hankel transform, Proc. IEEE 66, (1978).

8 Barakat et al. Vol. 15, No. 3/March 1998/J. Opt. Soc. Am. A S. Candel, An algorithm for the Fourier Bessel transform, Comput. Phys. Commun. 23, (1981). 9. V. Magni, V. Cerullo, and S. Silvestri, High-accuracy fast Hankel transform for optical beam propagation, J. Opt. Soc. Am. A 9, (1992). 10. D. Berger, S. Chamaly, M. Perreau, D. Mercier, P. Monceau, and J. Levy, Optical diffraction of fractal figures: random Sierpinski carpets, J. Phys. I (Paris) 1, (1991). 11. R. Barakat and E. Parshall, Numerical evaluation of zeroorder Hankel transforms using Filon quadrature philosophy, Appl. Math. Lett. 9, (1996). 12. R. Barakat and B. Sandler, Filon trapezoidal schemes for Hankel transforms of orders zero and one, Appl. Math. Lett. (to be published). 13. R. Barakat and B. Sandler, Numerical evaluation for firstorder Hankel transforms using Filon quadrature philosophy, Appl. Math. Lett. (to be published). 14. L. Filon, On a quadrature formula for trigonometric integrals, Proc. R. Soc. Edin. 49, (1928). 15. C. Trantner, Integral Transforms in Mathematical Physics (Methuen, London, 1966), Chap. 6. This is the only book that contains full details of Filon s work. 16. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1965). 17. I. Gradshteyn and I. Ryzhk, Tables of Integrals, Series, and Products (Academic, San Diego, Calif., 1980). 18. I. Sneddon, Fourier Transforms (Dover, New York, 1995), Chap G. Watson, Theory of Bessel Functions (Cambridge, London, 1944). 20. F. Oliver, ed., Royal Society Mathematical Tables: Vol. 7, Bessel Functions, Part III, Zeros and Associated Values (Cambridge University Press, Cambridge, 1960). 21. R. Barakat, Application of the sampling theorem to optical diffraction theory, J. Opt. Soc. Am. 54, (1964). 22. R. Barakat, Solution to an Abel integral equation for bandlimited functions by means of sampling theorems, J. Math. Phys. (Cambridge, Mass.) 43, (1964). 23. A. Jerri, The Shannon sampling theorem its various extensions and applications: a tutorial review, Proc. IEEE 65, (1977).