Reviews for Representation Theorems in Hardy Spaces

The average rating for Representation Theorems in Hardy Spaces based on 2 reviews is 3.5 stars.

Review # 1 was written on 2020-11-28 00:00:00 Loren Andrews The calculus, as typically taught in high school, is insufficient for the purposes of the research mathematician. It corresponds roughly to the level at which it was originally worked out during the seventeenth century by Newton and Leibnitz, which is to say, one satisfies oneself with merely heuristic derivations and the intuitive picture of the derivative as the tangent to a curve and the integral as the area under the curve. To tell the truth, neither Newton nor Leibnitz so much as defines the derivative (nor the integral, for that matter); to regard it as a difference quotient of infinitesimally small quantities leaves it indefinite as to what an infinitesimal actually is, and at the time nobody had an adequate notion (there was not to be an adequate notion of infinitesimals until Robinson's discovery of non-standard analysis in the 1950's). Berkeley famously skewered the concept of infinitesimal current among his contemporaries as the 'ghost of a departed quantity'. Not until the period from Cauchy in the early nineteenth century to Weierstrass late in the century did mathematicians seek to put the calculus on a firm, rigorous footing. For instance, the derivative can be defined as the limit of difference quotients (if it exists), where obviously one first needs to have a concept of what a limit is, which was Cauchy's initial breakthrough. Incidentally, we could observe that taking a limit is not so far from what Berkeley ironically suggested, i.e., seeing what remains after a quantity (the distance to the limiting value) has departed. For this reason, most mathematics majors have to take an honors course on calculus in college, in which they relearn everything they thought they knew, only this time supplied with rigorous definitions and proofs. There are three standard textbooks in English that cater to this audience: those by Courant and John, Apostol and Spivak. As this reviewer has worked his way completely through only the first of these, he will have little to say on their relative merits. Richard Courant was Hilbert's assistant in Göttingen, where he coauthored with him the fundamental reference work Methoden der mathematischen Physik, until his expatriation by the Nazis. He settled at NYU in New York, where he began to collaborate with Fritz John, an expert on partial differential equations (see the latter's celebrated textbook by that name in the Springer series in the applied mathematical sciences). In the 1960's, the two revised an earlier textbook on differential and integral calculus by Courant himself into the three-volume Introduction to calculus and analysis reviewed here. The additions forming this new work are material. The heart of their contribution is to be found in the 661-page first volume, which provides a thorough grounding in real analysis and calculus in one variable at the level of an advanced undergraduate. It starts out at the elementary level with the definition of the real number system (following Cauchy, as the limit of a nested sequence of intervals), the concepts of limit and continuity, functions (rational, algebraic, trigonometric, exponential and logarithmic), induction, convergence tests, infinite series and least upper bound and greatest lower bound. Beginning students of calculus will not necessarily appreciate the precise reasoning it takes to obtain a result and Courant and John lead them through numerous worked examples. As an illustration, they establish the lemma that (1+h)ⁿ is greater than or equal to 1+nh en route to proving that, for any p>0, the limit as n tends to infinity of ⁿ√p = 1, what is a standard result typical of introductory analysis. Not only are these concepts introduced and some non-trivial examples discussed, the authors make use of them to derive some interesting properties. For instance, there is a nice portrayal of Archimedes' method of exhaustion for determining π through the areas of successive polygons inscribed inside a circle. The second chapter gets down to the fundamental ideas of integral and differential calculus, illustrated with the aid of numerous worked examples and handsomely drawn figures. The logarithm is interpreted (rigorously now) as the integral of 1/x and the exponential function obtained as its inverse (there is much to appreciate in the economy of this approach, which differs perhaps from what most students are used to). A satisfactory proof of the existence of the definite integral of a continuous function rounds out the chapter. The techniques of calculus are further developed in chapter three. The avid student will find it salutary, for instance, to have to determine the derivatives of some sample functions directly from the definition as a limit of difference quotients, rather than to employ the convenient calculational rules. Rules follow for differentiation of rational and trigonometric functions, the derivative of an inverse function, the chain rule, hyperbolic functions, maxima and minima, orders of magnitude, substitution and integration by parts, integration of rational and some elliptic functions. In all, a fairly thorough review that goes into greater depth than is usual in a first course in calculus, such as is found in the discussion of the resolution of rational functions into partial fractions and their integration. Chapter four covers applications to physics and geometry, starting off with an extensive section on the theory of plane curves (cycloids, ellipses, lemniscates etc.) then going into physics proper, with vector analysis, Newton's laws of motion, constrained motion, elastic vibrations, the pendulum (ordinary and cycloidal), the universal law of gravitation and the solution of Kepler's problem of planetary motion and a little on work and energy. The presentation serves as a useful exemplification of the techniques of calculus but is perhaps too sketchy to be sufficient as a pedagogical account of the subject matter, for which one would want to consult the appropriate dedicated textbooks. The fifth chapter on Taylor expansions nicely derives the Cauchy and Lagrange formulae for the remainder and discusses exponential, trigonometric and binomial series, including attention to the question of convergence. There is a thoughtful appendix on interpolation. The sixth chapter on numerical methods could easily be skipped, although it contains useful derivations of Simpson's rule and Stirling's formula. The remainder of the text (chapters seven to nine) is filled out with a treatment of infinite sums and products. Here, the authors' pedagogical style is on display with good accounts of convergence properties (such as when one can differentiate or integrate a series term-by-term) and of Fourier analysis, including Dirichlet's proof of the main theorem on convergence of Fourier series for sectionally continuous periodic functions, many worked examples and a helpful discussion of the Gibbs phenomenon. Courant and John make it seem almost effortless (whereas Rudin in his textbook on introductory analysis shows convergence of the Fourier series only in the easier case of Lipschitz functions). There need not be as much detail given here about the second and third volumes, which treat multivariable calculus, linear algebra, improper integrals, differentiation under the integral sign, multiple integrals, differential forms, surfaces, parametrized families of curves, calculus of variations and a variety of more advanced topics in differential equations and complex analysis. The style continues to be what we are accustomed to: lucid, efficient, unpretentious. In this reviewer's opinion, however, these later two volumes are not nearly as strong as the first and there may be better texts from which to learn the material (or perhaps this merely means that the first volume is outstanding). The chapter on differential equations, for instance, leads one through a number of special cases, e.g. differential equations of Riccati form, but comes across more as a grab-bag of tricks than as a systematic account of the elements of the subject. One will not find a good proof of the existence theorem for solutions to ordinary differential equations or coverage of phase portraits, bifurcation theory, symmetries etc. The section on complex analysis is fine as far as it goes, but not comprehensive enough to serve as a full course. Speaking of the work in general, it can be highly recommended. The pedagogical style is excellent throughout. The reader may be taken unawares, as for the most part the authors prefer a continuous line of exposition rather than section into definitions, statement of theorems and proofs. But one should not be put off or misled by the slightly informal tone; everything is demonstrated that should be and when a recondite point that the student might miss arises, it is called out and explained well. From what little this reviewer has seen, the competitors Apostol and Spivak can be at once self-importantly pretentious and tedious. It must make a difference that Courant, at least, was a more talented and versatile scientist than either Apostol or Spivak, who are remembered today only as textbook authors. The numerous exercises for the student (almost nine hundred in all over the three volumes) are all good, thoughtful, and supplement the text nicely. Many of them can be quite challenging, but not beyond the reach of a determined effort. A handful, though, are very hard and seem to come from out of the blue. Moreover, a reasonable though not excessive number of the exercises call for quite intense computation, a page or more leading to very unwieldy formulae it can take some skill to reduce into a manageable form (for instance, one problem on confocal ellipsoidal coordinates). The authors date to an era before electronic computers with symbolic manipulation tools were widely available and the practicing applied mathematician probably did have to be adept at seeing such pencil-and-paper calculations through to the end, which does require stamina. Among the three leading textbooks in calculus at this level, Courant and John is reputed to have the hardest problems (which on the basis of this reviewer's experience could very well be true, although he has not worked enough of the problems in Apostol or Spivak to confirm this statement).

Review # 2 was written on 2014-03-24 00:00:00 Frances Drzewicki Best book to learn Calculus. Here's my note: goo.gl/FWjzU2