BOOK REVIEWS 403

long section on generalizations includes anisotropic Sobolev spaces, systems of equa-tions and a brief description of Sobolev-Orlicz spaces. The regularity problem isdiscussed.

In the second half of the book we come to modern existence theory for weaksolutions. Chapter IV presents variational methods and the role of convexity,reflexivity and monotone and coercive operators. Functional analysis results areexplicitly restated in terms of the given differential equations so that one can comparethese methods with those introduced later. A section on minimal surfaces illustrateshow one can proceed in the nonreflexive space W1’1, and the finite element method isintroduced in terms of some examples.

Chapter V surveys topological methods, including degree arguments, Schauder’sfixed point theorem and the Browder and Leray-Lions theorems on operators whichare monotone or monotone in the principal part. Proofs for some of the deeper resultsare sketched; the sketch for the Browder theorem seems intended for mathematicallyexperienced readers. Examples and explicit interpretation in terms of differentialequations are again provided.

Chapter VI deals with noncoercive problems, mostly in terms of the example

-u"(x)- g(u(x)) f(x) (0 < x < r), u(O) u(r) O.

The final chapter discusses variational inequalities and their interpretation as boundaryvalue problems with constraints or free boundaries.

Through judicious choice of material and clear, careful, lively writing, the authorshave made their subject accessible to a wide audience. This is not a reference book forexperts, but it should help others find their way in the field.

The reviewer noticed few misprints, most of which are easily corrected. Threepoints were more confusing. In the proof of the key Theorem 26.11, the term {f, u} isnot accounted for in the estimates; this can be fixed. In example 27.6, arccos should bearccosh. Finally, the hypotheses of Theorems 30.5 and 30.6 should be interchanged.

KENNETH B. HANNSGENVirginia Polytechnic Institute and State University

Padd-Type Approximation and General Orthogonal Polynomials. By CLAUDEBREZINSKI. Birkhauser-Verlag, Boston. 250 pp., $34.00.This volume contains, among other material, an account of the author’s recent

researches on the construction of generalized rational approximations.It has been known for some time (cf. the work of J. L. Fields and Y. L. Luke in the

1960’s) that the problem of constructing rational approximations to, say, analyticfunctions can be approached through the concept of Toeplitz summability. One formsthe weighted sum of the partial sums for the power series of the given function, f(z),using weights which are themselves polynomials, and then divides by the sum of theweights. In certain instances, e.g., when f is a generalized hypergeometric function andthe weights are judiciously chosen hypergeometric coefficients, these rational approx-imations converge. For a select class of functions, it is known how to choose the weightsto produce what is called the Pad6 approximation to the function, the unique rationalapproximation of given numerator and denominator degrees which interpolatesmaximally to f at the point of development of its Taylor series (i.e., derivatives of f and

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404 BOOK REVIEWS

its approximation agree as far as possible at that point). This select class of functionsconsists of the Gaussian hypergeometric function F(1, b; c; z) and its limiting cases.

In addition, Pad6 approximations for functions defined by Stieltjes integrals,

dO(t)f(z)= 1-zt’

can be constructed in principle by employing in the weights the coefficients of thepolynomials orthogonal with respect to the distribution d&. Such approximationscannot be said to be in closed form, since the coefficients themselves are not known.(They can, however, be computed recursively.)

Concerning the construction of rational approximations in other than these specialcases, not nearly enough is known.

For analytic functions Luke has made an intriguing conjecture about how theToeplitz weights should be chosen to produce efficient rational approximations, andthere is an abundance of numerical evidence to support his conjecture. Yet nothingspecific has been demonstrated about the approximations, and the conjectured weightswill certainly not yield the Pad6 for f(z).

To this somewhat muddled state of affairs Claude Brezinski, whose field issummability theory, has brought some fresh and ingenious ideas. He has seen that asalient feature of the problem is one of selection: can one focus on these functions withrepresentations which lend themselves both to the actual construction of rationalapproximations and to reasonably uncluttered error analyses? How the correct selec-tion of functions is made is not, perhaps, unexpected, and is very simple to explain.

Let c be a linear functional acting on some normed linear space of functions whosedomain @ includes polynomials (in t) and the functions (1- zt) -1, z in some region ofthe complex plane. Clearly, @ contains the functions ti/(1 zt), 0, 1, 2,.. . Denotethe moments of c by

mi=c(ti),and define

f(z)=c[(1-zt)-a].

Let v be a polynomial of degree k. Then

w(z)= c[ v(t)- z

is a polynomial of degree k- 1 at most. Defining 5(z)= zk-lw(z-1), (Z)= ZkV(Z -1)and doing some straightforward manipulations give

[ l(*) f(z)- R (z) -c[1 ’ zJ’where

R(z)-(z)"

R is called the Padg-type approximant to f and denoted (k-1/k)r. By writingf(z) mo + maz + + m.z + zn+IF(z) and applying the above construction to F(z),rational approximations of the form (n + k/k)r can be generated.

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BOOK REVIEWS 405

The advantage of the representation (*) is that it provides a ready-made error termwhich renders some of the ingredients of an effective rational approximation immedi-ately apparent. For instance, the distribution of the zeros of the polynomial t(z) mustclearly be a major consideration in the construction. (For a wide class of problems, itturns out to be the consideration.)

In the first chapter of this book, the author develops the formal (algebraic)properties of these approximations and shows how the construction can be interpretedas a formal summability process. He gives a number of applications, including general-ized rational approximations to e-’, and shows how these are related to the search forA-stable methods for integrating stiff differential equations. He has included a novelapproach to the inversion of Laplace transforms which I want to reproduce here.

Let

and make the decomposition

Then

f(t) Io g(x) e -x’ dx,

() kk-1. (t)= E A,(1-x,t)-k f i=1

ky AixTa(t-xT)-.i=1

w(x,)a,=v’(x)’

and inverting gives the approximation

g(x) Ai x/x,

i=1 Xi

Another interesting application is to the generation of p-adic codes for the computerrepresentation of rational numbers.

The reader will notice that when the sequence {mi} is totally monotone or totallypositive, c can be represented as an integral with respect to a distribution function withsupport in [0, 1] or [0, c) respectively, and so the problem reduces to the oneconsidered previously. Choosing v to be a polynomial from the set orthogonal withrespect to dO then produces a Pad6 approximant to f.

Even when {rni} is not one of these special sequences, taking v from a system ofpolynomials orthogonal in a wider sense produces a generalized kind of Pad6 approxi-mant. In fact, this is the main idea of Chapter 2. Under certain conditions a family {Pn}of polynomials can be constructed satisfying

c[XPk(x)] O, O <- <-_ k -1.

(Such polynomials have been discussed by previous authors also.) Choosing v P in(*) gives an error term on the right-hand side,

2k

(z)C1- ztJ’

and so it is natural to call R a generalized Padd approximant. It is not a Paddapproximant, because f(z) need not be analytic at 0 (or anywhere else). Like the

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406 BOOK REVIEWS

traditional orthogonal polynomials, the {Pk satisfy a three-term recursion relationshipand a Christoffel-Darboux formula. Furthermore, Pk and Pk/l have no common zerowhen {mi} is definite. The author gives a number of applications of such generalizedorthogonal polynomials outside of the context of rational approximations, mainly tosome popular projection methods in the theory of linear operators, including theLanczos biorthogonalization method and the Galerkin method.

It is well known that recursion relations exist between the various elements of thePad0 table for a given function, one of the most famous going by the name of thee-algorithm (due to P. Wynn). The usual proof of the e-algorithm identity is hideouslymessy and requires the use of so-called Schweinsian determinantal identities. InChapter 3, e-algorithm identities are shown to hold also for general Pad6 approxima-tions and, just as importantly, are derived solely from the already established propertiesof generalized orthogonal polynomials. An intimate connection obtains betweencontinued fractions and orthogonal polynomials, and the author shows that this is truealso for the general polynomials.

In Chapter 4 the author treats generalizations of the previous material to abstractspaces. It is rather surprising that the e-algorithm can be defined in such a way that itmakes sense in a topological vector space. It turns out that the need for reciprocals ofvectors can be avoided by defining a quasi-inverse with the aid of elements from thedual space. The procedure is ingenious but straightforward, and it may be useful to givean example of how such a generalization is accomplished for a famous formula ofnumerical analysis. Let {sn} be a sequence in a topological vector space and & anyelement from the dual of . Then

S"n Sn --((, A2Sn ASh

provides a topological generalization of the Aitken 2 method which reduces to theclassical method when R and & is the identity map. The above can be used toaccelerate the convergence of sequences of vectors. The e-algorithm is extended inmuch the same way.

The book closes with a discussion of Pad6-type approximations to double powerseries. An appendix contains a "conversational" program in Fortran to computerecursively sequences of Pad6 approximations.

I found this an exciting book. In a delightful way it was a time-consuming book toread; not because of the presentation" it is carefully organized and clearly written.Rather the material continually goaded me into the pursuit of small research projects.In fact the author is scrupulous about defining major unsolved problems and developinghis subject in a way that indicates directions for future research. Anyone interested inrational approximations will want to have this book on his bookshelf, and anyone doingresearch will find it a generous source of ideas.

.lET WIMPDrexel University

The Algebra of Econometrics. By D. S. G. POLLOCK. John Wiley, New York, 1979.xv + 360 pp., $45.00.The author’s preface provides an accurate description of the nature and contents of

this highly linear algebraic treatment of theoretical econometrics" "By emphasizing the

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