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D. Studies on More General Interpolation ProblemsBy the beginning of the 20th century, the problem ofinterpolation by finite or divided differences had beenstudied by astronomers, mathematicians, statisticians, andactuaries, 21 and most of the now well-known variants ofNewton’s original formulae had been worked out. This isnot to say, however, that there are no more advanced developmentsto report on. Quite to the contrary. Already in 1821,Cauchy [65] studied interpolation by means of a ratio of twopolynomials and showed that the solution to this problem isunique, the Waring–Lagrange formula being the special casefor the second polynomial equal to one. 22 Generalizationsfor solving the problem of multivariate interpolation inthe case of fairly arbitrary point configurations began toappear in the second half of the 19th century, in the worksof Borchardt and Kronecker [68]–[70].A generalization of a different nature was published in1878 by Hermite [71], who studied and solved the problemof finding a polynomial of which also the first few derivativesassume prespecified values at given points, where theorder of the highest derivative may differ from point to point.In a paper [72] published in 1906, Birkhoff studied the evenmore general problem: given any set of points, find a polynomialfunction that satisfies prespecified criteria concerningits value and/or the value of any of its derivatives for eachindividual point. 23 Hermite and Birkhoff type of interpolationproblems—and their multivariate versions, not necessarilyon Cartesian grids—have received much attention inthe past decades. A more detailed treatment is outside thescope of this paper, however, and the reader is referred torelevant books and reviews [70], [79]–[83].21 Many of them introduced their own system of notation and terminology,leading to confusion and researchers reformulating existing results. Thepoint was discussed by Joffe [21], who also made an attempt to standardizeyet another system. It is, however, Sheppard’s notation [64] for central andmean differences that has survived in most later publications.22 It was Cauchy also who, in 1840, found an expression for the errorcaused by truncating finite-difference interpolation series [66]. The absolutevalue of this so-called Cauchy remainder term can be minimized bychoosing the abscissae as the zeroes of the polynomials introduced later byTchebychef [67]. See, e.g., Davis [26], Hildebrand [31], or Schwarz [34] formore details.23 Birkhoff interpolation, also known as lacunary interpolation, initiallyreceived little attention, until Schoenberg [73] revived interest in the subject.The problem has since usually been stated in terms of the pair (E;X), whereX = fx g is the set of points or nodes and E =[e ] is theso-called incidence matrix or interpolation matrix, with e =1for those iand j for which the interpolating polynomial P is to satisfy a given criterionP (x )=c and e =0, otherwise. Several special cases had beenstudied earlier and carry their own name: if E is an (m +1)2 1 columnmatrix with e =1for all i =0; ...;m, we have the Waring–Lagrangeinterpolation problem. If, on the other hand, it is a 1 2 (k +1)row matrixwith e =1, for all j =0; ...;k, we may speak of a Taylor interpolationproblem [26]. If E is an (m+1)2(k +1)matrix with e =1, for all i =0; ...;m, and j =0; ...;k , with k k, we have Hermite’s interpolationproblem, where the case k = k for all i =0; ...;m, with usually k =1,is also called osculatory or osculating interpolation [25], [26], [31]. Theproblem corresponding to E = I, with I the (m+1)2(m+1)unit matrix,was studied by Abel [74] and later by Gontcharoff and others [26], [75],[76] and, finally, we mention the two-point interpolation problem studied byLidstone [26], [76]–[78] for which E is a 2 2 (k +1)matrix with e =1for all j even.E. Approximation Versus InterpolationAnother important development from the late 1800s is therise of approximation theory. For a long time, one of the mainreasons for the use of polynomials had been the fact that theyare simply easy to manipulate, e.g., to differentiate or integrate.In 1885, Weierstrass [84] also justified their use forapproximation by establishing the so-called approximationtheorem, which states that every continuous function on aclosed interval can be approximated uniformly to any prescribedaccuracy by a polynomial. 24 The theorem does notprovide any means of obtaining such a polynomial, however,and it soon became clear that it does not necessarily apply ifthe polynomial is forced to agree with the function at givenpoints within the interval, i.e., in the case of an interpolatingpolynomial.Examples of meromorphic functions for which theWaring–Lagrange interpolator does not converge uniformlywere given by Méray [86], [87] and later Runge [88]—especiallythe latter has become well known and can be foundin most modern books on the topic. A more general resultis due to Faber [89], who, in 1914, showed that for anyprescribed triangular system of interpolation points thereexists a continuous function for which the correspondingWaring–Lagrange interpolation process carried out onthese points does not converge uniformly to this function.Although it has later been proven possible to constructinterpolating polynomials that do converge properly for allcontinuous functions, e.g., by using the Hermite type ofinterpolation scheme proposed by Fejér [90] in 1916, thesefindings clearly revealed the “inflexibility” of algebraicpolynomials and their limited applicability to interpolation.IV. THE INFORMATION AND COMMUNICATION ERAWhen Fraser [19], in 1927, summed up the state of affairsin classical interpolation theory, he also expressed his expectationsconcerning the future and speculated: “The 20thcentury will no doubt see extensions and developments ofthe subject of interpolation beyond the boundaries markedby Newton 250 years ago.” This section gives an overviewof the advances in interpolation theory in the past century,proving that Fraser was right. In fact, he was so right thatour rendition of this part of history is necessarily of a morelimited nature than the expositions in the previous sections.After having given an overview of the developments that ledto the two most important theorems on which modern interpolationtheory rests, we focus primarily on their later impacton signal and image processing.A. From Cardinal Function to Sampling TheoryIn his celebrated 1915 paper [91], Whittaker noted thatgiven the values of a function corresponding to an infinitenumber of equidistant values of its argument , ,, from which we can construct a table of differencesfor interpolation, there exist many other functionswhich give rise to exactly the same difference table. He then24 For more detailed information on the development of approximationtheory, see the recently published historical review by Pinkus [85].324 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 3, MARCH 2002
considered the Newton-Gauss formula (9) and set out to answerthe question of which one of the cotabular functions isrepresented by it. The answer, he proved, is that under certainconditions it represents the cardinal function(12)which he observed to have the remarkable properties that,apart from the fact that it is cotabular with the original functionsince , for all , it hasno singularities and all “constituents” of period less thanare absent.Although Whittaker did not refer to any earlier works, itis now known 25 that the series (12) with andhad essentially been given as early as 1899 by Borel [94],who had obtained it as the limiting case of the Waring–Lagrangeinterpolation formula. 26 Borel, however, did not establishthe “band-limited” nature of the resulting function nordid Steffensen in a paper [100] published in 1914, in whichhe gave the same formula as Borel, though he referred toHadamard [102] as his source. De la Vallée Poussin [103], in1908, studied the closely related case where the summationis over a finite interval, but the number of known functionvalues in that interval goes to infinity by takingand . In contrast with the Waring–Lagrange polynomialinterpolator, which may diverge as we have seen inthe previous section, he found that the resulting interpolatingfunction converges to the original function at any point in theinterval where that function is continuous and of boundedvariation.The issue of convergence was an important one in subsequentstudies of the cardinal function. In establishingthe equivalence of the cardinal function and the functionobtained by the Newton-Gauss interpolation formula,Whittaker had assumed convergence of both series expansions.Later authors showed, by particular examples,that the former may diverge when the latter converges.The precise relationship was studied by Ferrar [104], whoshowed that when the series of type (12) converges, (9)also converges and has the same sum. If, on the other hand,(9) is convergent, then (12) is either convergent or has ageneralized sum in the sense used by de la Vallée Poussinfor Fourier series [105], [106]. Concerning the convergenceof the cardinal series, Ferrar [96], [104],[107] and laterWhittaker [76], [108], [109] studied several criteria. Perhapsthe most important is that of, where, being a sufficient condition for having25 For more information on the development of sampling theory, the readeris referred to the historical accounts given by Higgins [92] and Butzer andStens [93].26 Since Borel [94], the equivalence of the Waring–Lagrange interpolationformula and (12) in the case of infinitely many known function valuesbetween which to interpolate has been pointed out and proven by many authors[76], [92], [93], [95]–[99]. Apart from the Newton-Gauss formula, asshown by Whittaker [91], equivalence also holds for other classical interpolationformulae, such as Newton’s divided difference formula [76], [100]or the formulae by Everett et al. [101], discussed in the previous section.Given Borel’s result, this is an almost trivial observation, since all classicalschemes yield the exact same polynomial for a given set of known functionvalues, irrespective of their number.absolute convergence—a criterion that had also been givenby Borel [94]. It was Whittaker [76], [109] who gave morerefined statements as to the relation between the cardinalseries and the truncated Fourier integral representation of afunction in the case of convergence—results that also relateto the property called by Ferrar the “consistency” of theseries [76], [96], [107],[109], which implies the possibilityof reproducing the cardinal function as given in (12) byusing its values , with .It must have been only shortly after publication ofWhittaker’s works [76], [108], [109] on the cardinal seriesthat Shannon recognized their evident importance to thefield of communication. He formulated the now well-knownsampling theorem, which he first published [110] withoutproof in 1948 and the subsequent year with full proof ina paper [111] apparently written already in 1940: “If afunction contains no frequencies higher than cps[cycles per second], it is completely determined by giving itsordinates at a series of points spaced seconds apart.”Later on in the paper, he referred to the critical samplingintervalas the Nyquist interval correspondingto the band , in recognition of Nyquist’s discovery [112]of the fundamental importance of this interval in connectionwith telegraphy. In describing the reconstruction process, hepointed out that “There is one and only one function whosespectrum is limited to a band and which passes throughgiven values at sampling points separated secondsapart. The function can be simply reconstructed from thesamples by using a pulse of the type…Mathematically, this process can be described as follows.Let be the th sample. Then the function is representedby(13)As pointed out by Higgins [92], the sampling theoremshould really be considered in two parts, as done above: thefirst stating the fact that a bandlimited function is completelydetermined by its samples, the second describing how toreconstruct the function using its samples. Both parts ofthe sampling theorem were given in a somewhat differentform by Whittaker [76], [108], [109] and before him also byOgura [113], [114]. They were probably not aware of thefact that the first part of the theorem had been stated as earlyas 1897 by Borel [115]. 27 As we have seen, Borel also usedaround that time what became known as the cardinal series.However, he appears not to have made the link [92]. In lateryears, it became known that the sampling theorem had beenpresented before Shannon to the Russian communicationcommunity by Kotel’nikov [118]. In more implicit verbalform, it had also been described in the German literature byRaabe [119]. Several authors [120], [121] have mentionedthat Someya [122] introduced the theorem in the Japanese27 Several authors, following Black [116], have claimed that this first partof the sampling theorem was stated even earlier by Cauchy in a paper [117]published in 1841. However, the paper of Cauchy does not contain such astatement, as has been pointed out by Higgins [92].MEIJERING: A CHRONOLOGY OF INTERPOLATION 325
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