Tables of Bessel Transforms. by Fritz Oberhettinger; Fourier Expansions: A Collection ofFormulas. by Fritz Oberhettinger; Fourier Transforms of Distributions and their Inverses--ACollection of Tables by Fritz Oberhettinger; Tables of Laplace Transforms. by F.Oberhettinger; L. BadiiReview by: Richard AskeySIAM Review, Vol. 17, No. 4 (Oct., 1975), pp. 707-712Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2029281 .

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BOOK REVIEWS 707

characteristic functions. Special mention should be made of the new material on estimation of param-

eters in a linear model when the observations have a possibly singular covariance matrix. The existing theories and methods, going back as far as Gauss (1809), are applicable only when the covariance matrix is known to be nonsingular. Rao presents a new unified approach in Chapter 4, valid for all situations, whether the covariance matrix is singular or not.

About the only criticism that I have of this book is the large number of misspelled names. Also, the subject index should have been arranged better than to include most entries under very general titles, such as density, or estimation, or testing of hypotheses, etc.

At the ends of each chapter are very useful references to the principal research papers in statistics and probability discussed in the chapter, and at the end of the book is an extensive (and impressive) nine-page bibliography of all of Rao's many books and research papers, from 1941 to 1973. There are a fair number of numerical examples scattered through Chapters 4, 6, 7 and 8 to illustrate the theory. I should mention that Rao has assumed on the part of the reader a previous knowledge of statistics at an undergraduate level.

My favorite quotation from this book (which I have been quoting to my students in statistical inference ever since the publication of the first edition) is (p. 346), "There is some amount of confusion in statistical literature regarding the concept of efficiency of an estimator." Rao has an excellent discussion of the problems involved, and then presents his new definition of asymptotic efficiency on pages 348-349.

I recommend this book without reservation to all mathematical statisticians as an absolute necessity for their personal libraries and to those research workers with a good mathematical background who want to learn more about modern statistical techniques.

M ORTON KUPPERMAN University of Leicester, England

Tables of Bessel Transforms. By FRITZ OBERHETTINGER. Springer-Verlag, New York, 1972. ix + 289 pp., $8.80.

Fourier Expansions. A Collection of Formulas. By FRITZ OBERHETTINGER. Academic Press, New York, 1973. xi + 64 pp., $11.00.

Fourier Transforms of Distributions and their Inverses A Collection of Tables. By FRITZ OBERHETTINGER. Academic Press, New York, 1973. ix + 167 pp., $18.00.

Tables of Laplace Transforms. By F. OBERHETTINGER and L. BADII. Springer- Verlag, New York, 1973. vii + 428 pp., $16.00. Tables of useful integral transforms are hard to compile. and it is nearly

impossible to eliminate all misprints. If such tables are well done, then the mathe- matical and scientific community owes the author a vote of thanks, and the author can see this when looking at all the references to his work in Science Citation Index. On the other hand, if the author does a poor job of choosing useful integrals, or if a large number of misprints occur, then a number of people are going to curse the author and publisher for the bad job they did. Often if takes years before it

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708 BOOK REVIEWS

it is clear which class a book falls into. I do not know how I will ultimately feel about these books, since they contain some useful integrals and sums, but some other useful facts are missing. Most of the formulas seem to be correct, but not all, so I would recommend checking a formula before using it.

To get more specific, consider the book Fourier Expansions. There are fewer collections of Fourier series than there are of Fourier transforms, and specific Fourier series can be very useful, so this book fills a need. However, there are minor errors which should not be there. On page 2, line 2, limhO should be limh,O+. On page 5, line 2, If(x)I should bef(Ixl) and in line 4, sgn xf(x) should be sgn xf(Ixl). On the same page, line 6 from the bottom, "for a < t < b" should be "off a ? t < b", and the conditions on m1 and m2 in the last line are given incorrectly. In the tables, these conditions are given correctly. In the tables, the conditions are used only in the symmetric case, so the author could describe them by use of the greatest-integer functions. In the nonsymmetric case, Iverson's notation Fal and [aJ could have been used to state the conditions concisely.

The table itself is broken into five parts. The first gives Fourier series with elementary coefficients representing elementary functions. By and large, the formulas which are given are useful, but a number of formulas which I have used in the last few years are not given here. Among these are the following:

sin (n +1)0 n

(1) sin(n - Z cos(n-2k)0, Sin 0 k =O

sin (n + 1)0 sin (m + 1)0 min (m, n) (2) siO- Z sin (n + m? 1I- 2k)O,

(3) 1-z)sin -+ 1) sin (n + 1)0, (1 - 2zcos0 + z2)2 n=O

(1- z) sin -[(1 + r)2 + 2r(1 + cos 0)] (4) 2Z Z(n + ) sin (n + 2)0 2[(1 A- Z)2 - 4z COS2 012

0 Cos - n 2 '

(5) (1 - cos n0) = 2 sin kO + sin n1O

sin - 2

cs 0 \1/2

(6) ' c~c cos 2_ ico 2 0<Cos , n =O -,s inO \2 sin /

0 0 1/2 oc O2- CO2

(7) E c sin ni0 = 2) = 0 < O < 70 ,

n~~~~~~~~~~~~~~~~~~~~~~~~1 ~~~~~~~~~~~~~~~~~~~~~,sinO0 sin/

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BOOK REVIEWS 709

where c2, = c2n + 1 (2) /nn!,

?? sii (n + 1)O sin (n +? p

n=0 n + 1 (8) '

= cos p log 2 + 1 sin 2 [0- 7 g 2( (p)

The interest in these formulas is the following. (3) is the Poisson kernel on the three-dimensional sphere, and (4) is the Poisson kernel on the real three-dimensional projective space. (2) defines a generalized translation for the 11 coefficient algebra associated with zonal spherical harmonics on the 3-sphere. (1) gives a projection operator from the 3-sphere to the circle, and when connected with positive definite functions is tied up with the fact that the circle can be isometrically imbedded in the 3-sphere. (5) is one of many ways of recapturing the nonnegativity of

T sin t dt = 1 - cos x

from the sum itl

Z sin kx = sin (nx/2) sin ((n + 1)x/2)/sin (x/2), k = 1

which is not nonnegative. (6) and (7) are formulas of Vietoris [3] which he used in his proof that these two series have positive partial sums for 0 < x < 7[. The positivity of these partial sums is quite useful, and is related to the question of the positivity of certain quadrature methods; see [1]. (8) gives an explicit formula for a kernel of Turan; see Hylten-Cavallius [2]. The most recent of these formulas was published in 1958, yet none of them is included in this book, and most. are not given in any book in this form. These are mentioned here, not to belittle the author for having omitted them, but to give the reader some indication why it is hard to write a really first-rate set of tables. They are all special cases of results about Jacobi polynomials, and only in this context can they really be understood and fully appreciated. The author is not an expert on Jacobi polynomials, so it is not surprising that he did not include these important formulas. No one is an expert in all the areas in which Fourier series have been used, so it is likely that every one would omit some useful formulas.

Mathematics of Computation has a section of table errata which should be consulted by all authors who contemplate writing a new set of tables. The errors from one book which have been corrected should not be passed on in future books. It is also important that new facts which have been discovered, or old ones which have been rediscovered, are included in the new books. So I would like to suggest that some journal start a section of useful facts which should be included in future handbooks. Since most tables of integrals and series are written by taking formulas from previous books and adding a few that the author has

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710 BOOK REVIEWS

encountered in his work, this would be a useful way of aiding authors in the hard and important task of compiling future tables.

The second part deals with Fourier series with elementary coefficients which represent higher transcendental functions. Most of the series given represent elliptic functions or theta functions. There are also many sums which represent hypergeometric functions, and only a few of these are given. Heine's sum,

Pn(cos x) E ak n sin (n + 1 + 2k)x, k = O

is given, but Laplace's sum, n

(21X)(21)n - k Pn(COSX) = k!(n - k)!cos(n - 2k)x,

is not given. Formula 2.44 gives one extension of Heine's sum 2.42 to Legendre functions. In this form. the parmieters r and n are restricted to be integers. If the formula is rewritten in terms of Gegenbauer polynomials or hypergeometric functions, then this restriction can be relaxed. Also, when written as a series for Gegenbauer polynomials, this is only one of two known extensions of Heine's formula, one given by Szeg6, the other given implicitly by Gegenbauer and explicitly by Askey. I do not understand why some of these formulas were given and not others. The basic formula is Gegenbauer's connection formula,

[n/2]

C (X) = , ak,nCnl-2k(X), k = O

with ak, n a product of gamma functions. The special cases mentioned above are all almost equally useful. They all have applications to harmonic analysis and most of them have been used in probability theory and in studying mixed boundary value problems.

Mention of these applications suggests another point. The author should know applications of most of the formulas which are included in the tables, and it would be useful if references to applications could be included. Also references to actual derivations should be given if they are known. There are a number of reasons for this. The most obvious is that misprints occur all too often in sets of formulas. I became aware of another reason when trying to use the Bateman Project as a student. Often I would find a formula which looked close to the one I needed, but I did not know how to derive this formula and so was unable to derive the formula I really needed. My experience was not unique, and so, since we no longer teach our students how to evaluate complicated integrals and sums, we must start to provide them with enough information so they can start to educate themselves. Boas told me Hardy complained that students had no technical ability in the 1930's, and I am sure this complaint can be traced back to Euler. So we might as well come to grips with the fact that not everyone can compute everything they need to use and take this into consideration when writing books.

Part 3 deals with elementary functions expanded in Fourier series whose coefficients are related to higher transcendental functions. The coefficients are

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BOOK REVIEWS 711

usually Bessel functions of some sort or Legendre functions. Legendre functions are 2F1's which have a quadratic transformation times an elementary function. The general 2F1 is even more interesting and useful, and a few formulas should have been given which involve them in both parts 3 and 4. Part 4 deals with "higher" functions whose Fourier expansion coefficients are also "higher". Again, Bessel functions and Legendre functions predominate. The last part is a small set of Fourier series written in exponential form and nine Fourier-Bessel series.

I did not systematically check the formulas, but noticed the following errata. The last factor in 3.9 should be cos (z cos x sin t). In 3.81, cos nx is omitted on the left-hand side. In 4.3,

(4 2 ) 4 2)

should be

IF + 2) -2).

Similar comments can be made about the other books. Except for Fourier Transforms of Distributions and their Inverses, they are quite adequately described by their titles. This book on Fourier transforms deals with Fourier transforms of positive functions of total mass one, but many of the formulas, especially those dealing with Bessel functions, really hold without the restrictions which are necessary to make the functions positive. Because of this, I have doubts that it was necessary to write a set of tables with this restriction. One misprint in this book is the function on the left in (15) in the Appendix (written by Laha). It should be (2/7c)X2(1 + x2)-2.

In the book on Laplace transforms, a more detailed acknowledgment should have been made to Roberts and Kaufmann, Tables of Laplace Transforms, W. B. Saunders, Philadelphia, 1966. To see this, compare the first sets of tables in both books. There is a humorous misprint in 1.23. See Roberts and Kaufmann for the correct statement. There is an unfortunate tradition of not giving adequate acknowledgment in books of tables which future authors should not contirue. There is no harm in telling the reader where much of the book came from. A diligent reader can trace formulas from one book to another (especially those with misprints). It is clear that much of each new book comes from older books. The previous authors should receive the credit which is due to them.

There is a nice feature in the book of Fourier series, a small set of formulas dealing with generalized functions. Hopefully, future authors will expand the list of formulas of this type.

Formula (7.63) in the book on Laplace transforms,

f sin 2nteP dt = p- (p2 + 4)- l(p2 + 16)-' ... (p2 + 4112)- 1(2n)!(1 -e l rp

is a very nice extension of Hamburger's special case m = co. Other readers will find other formulas which interest them in these books which could only be

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712 BOOK REVIEWS

found elsewhere with a lot of work, or possibly not at all. So everyone who uses integrals should become familiar with the contents of these books.

RICHARD ASKEY University of Wisconsin-Madison

REFERENCES

[1] R. ASKEY AND J. STEINIG, Some positive trigonometric sums, Trans. Amer. Math. Soc., 187 (1974), pp. 295-307.

[2] C. HYLTEN-CAVALLIUS, A positive trigonometrical kernel, Tolfte Skand. Mathematiker Kongressen 1953, Lund, 1954.

[3] L. VIETORIS, Cber das Vorzeichen gewisser trigonometrischer Suinmen, Sitzungsber. Ost. Akad. Wiss., 167 (1958), pp. 125-135.

Characterization Problems in Mathematical Statistics. By A. M. KAGAN, Yu. V. LINNIK and C. RADHAKRISHNA RAO. John Wiley, New York, 1973. xii + 499 pp. Characterization problems are of fairly recent origin; the first results are

probably due to R. C. Geary [1] and to J. Marcinkiewicz [2]; later several other authors worked too on these problems. Some monographs such as E. Lukacs and R. G. Laha [3] or B. Ramachandran [4] already contain some chapters dealing with characterization theorems. The volume reviewed here is the first book devoted exclusively to characterization problems. Its size and its bibliography of almost 200 items indicate the rapid growth of this field.

Chapter 1 presents the necessary analytical tools; these include a variety of mathematical results. As examples we mention: properties of characteristic functions, some results from the theory of entire functions, of differential and of functional equations. Chapter 2 deals with Linnik's result on identically distributed linear forms [5] and a closely related characterization of the normal law. Linnik's proof was simplified by A. A. Zinger in his (unpublished) dissertation in 1969 and his approach is presented in this chapter. Even the simplified proof is difficult. This seems to be inherent in the nature of the problem, and this reviewer doubts whether further simplifications would be possible. Independently distributed linear statistics are studied in Chapter 3. The Skitovic-Darmois theorem and its multivariate analogues are derived, and. also the generalization of the Skitovic- Darmois theorem to linear forms in denumerably many independent random variables, as well as other extensions are treated. Chapter 4 starts with the classical result that the independence of the sample mean and the sample variance implies the normality of the population and continues with well-known extensions of this result. Quasi-polynomial statistics are introduced, and the independence of a quasipolynomial statistic and a linear form is studied. This chapter also contains interesting results due to A. A. Zinger and Yu. V. Linnik [6] concerning linear forms whose coefficients are random variables, and also the study of the indepen- dence of polynomial statistics and the sample mean. Tube statistics are defined and Anosov's theorem [7] is derived. This theorem shows that the independence of the sample mean and an (n - 1)-dimensional tube statistic implies the normality of the population, provided that the population distribution function has a continuous density.

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