It is shown that the Caratheodory—Fejer extension of a finite geometric series can be given explicitly up to a simple polynomial equation in an auxiliary variable. This result allows us to analyse the Caratheodory-Fejer approximation method in the case where the quotients of successive Maclaurin coefficients of the given function tend to a limi

ELLACOTT, S. W.

GUTKNECHT, M. H.

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IMA Journal of Numerical Analysis (1983) 3, 221-227The Caratheodory-Fejer Extensionof a Finite Geometric SeriesS. W. ELLACOTT*Brighton Polytechnic, SussexANDM . H . GUTKNECHTETH Zurich, Switzerland[Received 5 May 1982]It is shown that the Caratheodory-Fejer extension of a finite geometric series can begiven explicitly up to a simple polynomial equation in an auxiliary variable. Thisresult allows us to analyse the Caratheodory-Fejer approximation method in thecase where the quotients of successive Maclaurin coefficients of the given functiontend to a limit.0. IntroductionTHE THEOREM of Caratheodory & Fejer (1911) (or, briefly, CF theorem) states: Givencomplex numbers c0,..., cK, there is among all functions g that are analytic in the unitdisc D and satisfyg{w) = co + clw + c1w1 + ... + cKwK + 0(wK+1) (0.1)as w -• 0 a uniquely determined one for which the supremum normattains a minimum. The optimal function g = g*, which is called the CF extension ofco+ ... +c^wx, is a scalar multiple of a finite Blaschke product with at most degreeK. It is the only function of this type which satisfies (0.1).(Note that one may assume c0 =A 0; otherwise the problem can be reduced to onewith smaller K.) As is well known (Takagi, 1924/25)is obtained by solving the singular-value problem Cu = cru, wherec r c x _! . . . c o \\ / °\2ind o- must be chosen to be the greatest singular value of C.• Research carried out while this author was at ETH Zurich partially supported by a Royal SocietyEuropean Visiting Fellowship.2210272-4979/83/020221 +07 S03.00/0 © "83 Academic Prea Inc. (London) Limited222 s- w - ELLACOTT AND M. H. GUTKNECHTThis singular-value problem is of course equivalent to a characteristic equationplus a linear system. However, in this paper we show that in the case c} = c* (c # 0fixed), it can be reduced to a particularly simple polynomial equation in an auxiliaryvariable; the singular value a and the corresponding singular vector u are simplefunctions of the solution of this equation.This result allows us to analyse a particular case of the Caratheodory-Fejerapproximation method (briefly called CF method), which was proposed byTrefethen (1981a) and has since been generalized in various directions, (For moredetails and references see Ellacott & Gutknecht, 1983; Gutknecht, 1983; Trefethen,1981b.)The smaller singular values of C yield examples for Takagi's generalization of theCF theorem (Takagi, 1924/25; Gutknecht, 1983). There one tries to determine afunction g satisfying (0.1) that is meromorphic in D, has at most a fixed number v(< K) of poles in D, is bounded near the unit circle, and has minimal norm on D. Ingeneral, there exist irregular cases where the singular-value problem leads to afunction g that does not match all the given derivatives at zero, but this can beexcluded in our example.1. The CF Extension of a Finite Geometric SeriesThe basic result of this paper is:THEOREM 1.1 (i) The finite geometric series 1 + cw + c2w2 + ... c*w* (c>0,K^0)has the CF extensionV0(x0)where Ut denotes the Ith Chebyshev polynomial of the second kind (which is of degree/), x0 is the (algebraically) largest zero of UK+1(x) —cU^x\ andCTO:= U^x0) S* max {1, c*}.(ii) If c > 1, then as K -*oo9*M c2 c-wuniformly on D.Note that the general case of a finite geometric series with complex ratio c ¥= 0can be reduced to the case c> 0 treated here by replacing wbywe*"".The proof of Theorem 1 is deferred to Section 3. It will be seen there that thealgebraically smaller roots Xj< x0 of UK+l(x) = cUg(x) belong to singular values<jj = Ug(xj) < a0 of C (in the same order), that all these singular values are simple(for every K > 0), and that the singular vectors too are obtained by formallyreplacing x0 by xr Hence, it is clear that Takagi's irregular case, which requiresmultiple singular values of the matrix obtained by deleting the first row and the lastcolumn of C, cannot occur here (cf. Takagi, 1924/25; Gutknecht, 1983). Therefore,Takagi's "regular" generalization of the CF theorem holds here and yields anextension of part (i) of Theorem 1.1:CARATHEODORY-FEJER EXTENSION 223THEOREM 1.2 The unique function g* that(i) is meromorphic in D,(ii) satisfies (0.1),(iii) has at most v poles in D (0 < v < K),(iv) is bounded in some annulus {w e C; 1 < \w\ < pig)},and minimizes \\g\\ among all Junctions g satisfying (i)-(iv) is)}, (1-3)where x0 > xx > ... > xK denote the K + l distinct zeros of Ut+1(x) — cUg{x), andLikewise the function g* is the solution of the maximum problem related to atheorem of Landau cited by Takagi (1924/25, p. 90) and Gutknecht (1983,Th. 1.1 (v)).Notable formulae following from our proof in Section 3 arev = 0,.. , K, (1.4)cf. Equation (3.2bX and the inequalitiesxy < {« , v = l , . . . ,X, (1.5b)where ij'i' > & > ... > $° denote the zeros of U,. By solving (1.4) for a* and using(1.5) one obtains simple lower and (except for v = 0) upper bounds for arA useful observation for testing programs is that if c is chosen to bec = {K + 2)/{K +1), then x0 = 1 and the coefficients U£l) in (1.1) equal /+1.2. Application to the CF MethodLet 9m denote the space of complex polynomials p of degree m. 9m is a subspaceof the space &m of functions p that are analytic i n f l :={weC; 1 < |w| <oo}, arebounded in every bounded subset of fl, and have a pole of order at most mat oo.Let/be a given function that is analytic in the unit disc D, and let/ A » : = E <J*W* (weD) (11)t-odenote the Mth partial sum of its Maclaurin series. The polynomial CF method ofTrefethen (1981a) for approximating/by a polynomial of degree m < M is based onextendingbackwards to a function q e &u according to the CF theorem, Le. wh-ty^qil/w) isthe CF extension of 0^+ ... +am+lwM~m~l. It is easy to see that p:=fM-q e &mis the best approximation of/m out of ^_. By deleting the terms with negative index2 2 4 s- w - ELLACOTT AND M. H. GUTKNECHTin the series for p, one ends up with the CF approximation p*1 e &m that is normallyvery close to the best approximation oifu, and hence of/if M was big enough.Our results of Section 1 allow us to analyse the model problem with at = a1, Le.The best approximation p* e 9m to / i s known explicitiy (Al'per, 1959; Rivlin, 1967),p*(w) = l+aw+...+(aw)m~1 + }r-fr- (2.3)l - | a |The error curveo"+ 1 w—d/(w)-p*(w) = — — j w- 1 (2.4)1 — \a\ 1 —awis exactly circular. (This is just an example where the generalization of the CFtheorem due to Adamjan, Arov & Krein (1971) yields p e 9m Le. p = p*.) For finiteM, p $ &m and the error function q =fu~P looks far less simple. But since q isobtained by extending am+1wm+l +... + aMwM backwards according to the CFtheorem, we can just apply Theorem 1.1 after having divided by {aw)" and madetwo simple substitutions (w -• 1/w, c: = I/a). Assuming 0 < a < 1 for simplicityagain, we obtain:THEOREM 2.1 Letfu be given by (22) with 0 < a < 1. Let K:= M-m-1 2* 0, andlet x0 be the (algebraically) largest root of aUK+1(x)— Ug(x) = 0. Then the bestapproximation p tofM out of&m is given by •'* * - ' - < • > - • " - t t f r ^where ao:= Ug{x0) > 0. As m, M -> 00 with M — m fixed, p converges uniformly on theunit circle 9D to p* given by (23).As mentioned, a0 is a simple singular value of a real symmetric matrix andu: = (U0(x0\..., Ux(x0))T is a corresponding singular vector. Hence aQ and u arestable under perturbations of the matrix, and even for non-hennitian perturbations,cf. Stewart (1973). In particular, <r0 and u are the limits of the correspondingquantities in the singular value problem belonging to any more general / whosecoefficients a} satisfylim 2*±± = a (26);~» ajwith 0 < a < 1, an assumption under which the CF method is known to work wellin practice. In view of aM-j ~ aMa~J we obtain:THEOREM 2.2 Let {ay}JL0 satisfy (2.6) with 0 < a < 1, letfM be defined by (2IX andlet p denote the best approximation to fM out of'&„. Ifm, M ->oo in such a way thatM — m is fixed, thenK ]uniformly on 3D, where x0 and a0 are the same as in Theorem 21 .CARATHEODORY-FEJER EXTENSION 225In particular, this theorem implies that as m, M -»oo, M—m = K + l fixed, the polesof all Blaschke products q/\\q\\ = (Ju—p)/ll/M—p\\ he in a disc \w\ < £ with fixedi < 1. (In the error analyses of Hollenhorst (1976) and Trefethen (1981a), whorequire weaker assumptions than (2.6), this is only guaranteed if \a\ < 0-43426 . . . or-&, respectively.) By applying Lemma 1.2 of Ellacott & Gutknecht (1983), which isbased on Cauchy's coefficient estimate, it follows that the relative truncation error\\{ff—PW/Wfit—P\\ tends t0 0 geometrically as m,M->oo {M—m fixed): for anyR £ «, 1)where ||/M—p\\ ~ aoau according to (2.7). (Note that X = M - m - 1 , <j~ = p "^—p,and a = ||./^—pll in the lemma we refer to.) For further comments on this andrelated error estimates see Section 1 of Ellacott & Gutknecht (1983).3. Proof of Tbeorem 1.1The matrix C defined in (0.3) is real and symmetric since c} = c1 with c > 0. Hence,the singular-value problem Cu = au can be reduced to an eigenvalue problemCv = Xv, where a = |I| and u = v if 1 > 0, u = iv if 1 < 0. The formulae simplify if wesubstitute a: = 1/c, X: = I/c*. Then Cv = Xv is equivalent toa'Y^^M, 1 = 0,..^K.Defining p x + x : = 0, p_ x: = co/a we obtain^ , - r , + 1 ) = ax + 1^_ / , / = 0,._A:, (3.1)whenceor, after reordering,»l+1-2x(a,A)i;, + o ,_ l =0, l = 0,..^K, (3.2a)where1 / 1 a2K+l\( ^ J (3.2b)It is well known that the Chebyshev polynomials^ ((/+!) arccos x)T,(x) = cos(/arccosx) and sin(arccosx)(/ = 0 ,1 , . . -) form together a fundamental set of solutions of (32a). Since (3.2a) issymmetric in r,+1 and p,-!, this is also true for the reverse recursion starting at vt,i.e. (32a) has the general solution vK.} = a7}(x) + /?l/,(x). Moreover, since U0(x) = 1,U^x) = 2x, the recursion for U, holds for / = 0 if we set l/_ x(x): = 0. Hence, in viewof vK+1 = 0,. (3.3)is the solution sought [Note that a constant factor would cancel in (0.2).] Since226 s- w - ELLACOTT AND M. H. GUTKNECHTD_ 1 = vja, x and a are directly related bywrk=a=\- (3-4)The rational function UK/UZi.l has exactly K + l simple poles and K interlacingzeros in (— 1,1) and is strictly decreasing on the whole real line except at these K + lpoles. For x > l ( x < — 1) it is positive (negative). Hence it is easy to see from thegraph of this function that (3.4) (with a > 0) has exactly K + l solutionsx0 > x t > . . . > xK > — 1, which satisfy (1.5). Obviously, at most x0 > 1; in factx0 > 1 iff a < (K + l)/K + 2) and x0 = 1 iff a = (K + l)/{K + 2). If we denote theeigenvalue corresponding to x, by X» then by (3.2b), \X0\ > \Xt\ > ... > \XZ\.Moreover, from (3.1) with / = K we conclude that XvK = O^PQ, hence by (3.3)A, = a* UK(xyl X\ = U,,(xJ, y = 0,..^K. (3.5)Since the zeros of Ur interlace with the points x« the eigenvalues JL, alternate in sign,XQ being positive. Hence a, = (—iyX\ and u, = p, if v is even, u, = ivt if v is odd.Finally, applying Cauchy's coefficient estimate to g*{w) yieldsa0 = \\g*\\ > max \c,\ > max (1, c*}.This establishes the first part of Theorem 1.1 and the ingredients mentioned inSection 1 for the proof of Theorem 1.2.Of course, Vj can also be written in terms of the powers of the reciprocal roots r\and l/tj of the auxiliary equation z2 — 2x(a, X)z+l = 0 of recursion (32a) since thesepowers form another fundamental set of solutions. If x > 1, r\ and l/rj are easily seento be real and positive: say, ^ e (0,1). Moreover, it is well known [and in view of therelation x = # j +1/17) easy to check] that U,(x) = {rjl + 1 -»7"'~1)/(>7-^"1) if |x| * 1.Thus,»j = - n— 0" = - ! . • • • , K+l). (3.6)1 — 1In the case c > 1,on 8D as K-*co. Since X2, = c~KcrQ > 1, it follows from (3.2b) thatx0 -> i [a+( l / a ) ] > 1. Consequently,and UJ/P0 -> a^  (for _/ fixed). In fact,hence, there exist y > 0 and Xo > 0 such that for all K > Ko\rj-a\^ya2K+1 and 0 < rj < b:=CARATHEODORY-FEJER EXTENSION 227Thus,t(3-8)2J-oand this tends to 0 as K -»co. In view of (3.6),i_ -j+ir+i•-a*1 - i fSo, by using (3.8) we get for all w el-b2K+r (vj/vo)w>- £ M J j -o J»THIt follows that ]£(D/I>O)W ' tCQds uniformly to 1/(1 —aw) as JC->oo. Likewise,E ( V o ) w " ; -> w/(w-a). Hence by (0.2) and (3.7)a*(w) 1 1 — aw c2 c — w~*l-a2 w-a = c 2 - l c w - lunifonnly on D. This concludes the proof .of Theorem 1.1. •REFERENCESADAMJAN, V., AROV, D. & KRHN, M. 1971 Analytic properties of Schmidt pairs for a Hankeloperator and the generalized Schur-Takagi problem. Math. USSR Sb. 15, 31-73.AL'PER, S. J. 1959 Asymptotic values of best approximation of analytic functions in acomplex domain. Usp. mat. Nauk. 14, 131-134.CARATHEODORY, G. & FEJER, L. 1911 Ober den Zusammenhang der Extremen vonharmonischen Funktionen mit ihren KoefSzienten und fiber den Picard-LandauschcnSatz. Re. Ore. mat. Palermo 32, 218-239.ELLACOTT, S. W. & GUTKNECHT, M. H. 1983 The polynomial Caratheodory-Fejerapproximation method for Jordan regions. IMA J. num. Analysis 3, 207-220.GuncNECHT, M. H. 1983 Rational Caratheodory-Fejer approximation on a disk, a circle,and an interval. / . Approx. Theory (in press).HoiXENHORST, M. 1976 Nichtlineare Verfahren bei der Polynomapproximation. Diss.Universitat Erlangen-Nurnberg.RrvLiN, T. J. 1967 Some explicit polynomial approximations in the complex plane. Bull. Am.math. Soc. 73, 467-469.STEWART, G. W. 1973 Error and perturbation bounds for subspaces associated with certaineigenvalue problems. SIAM Rev. 15, 727-764.TAKAGI, T. 1924/25 On an algebraic problem related to an analytic theorem of Caratheodoryand Fejer and on an allied theorem of Landau. Jap. J. Math. 1, 83-91 (1924); 2, 13-17(1925).TREFETHEN, L. N. 1981a Near-circularity of the error curve in complex Chebyshevapproximation. J. Approx. Theory 31, 344-367.TREFETHEN, L. N. 19816 Rational Chebyshev approximation on the unit disk. Num. Math.37, 297-320.

The Carathéodory—Fejér Extension of a Finite Geometric Series

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