Discourse on Fourier Series by Cornelius Lanczos (PDF)

Description

Originally published in 1966, this well-written and still-cited text covers Fourier analysis, a foundation of science and engineering. Many modern textbooks are filled with specialized terms and equations that may be confusing, but this book uses a friendly, conversational tone to clarify the material and engage the reader. The author meticulously develops the topic and uses 161 problems integrated into the text to walk the student down the simplest path to a solution.Audience: Intended for students of engineering, physics, and mathematics at both advanced undergraduate and graduate levels.Contents: Foreword to the Classics Edition; Preface; Chapter 1: The Fourier Series; Chapter 2: The Fourier Series in Approximation Problems; Chapter 3: The Fourier Integral; Bibliography; Index.

User’s Reviews

Editorial Reviews: Review This is a radically different approach from modern mathematics texts, which tend to hide behind vast arrays of symbols and formalism. Lanczos, like Feynman, was so brilliant that even very complicated mathematics and physics seemed simple to him. His goal was to help the reader see how simple it all was, too. –From the Foreword by John Boyd Book Description First published in 1966, this still-cited text covers Fourier analysis. The author meticulously develops the topic in a conversational tone to engage readers. 161 problems are integrated into the text to walk undergraduate and graduate students of engineering, physics, and mathematics down the simplest path to a solution. About the Author Cornelius Lanczos (1893-1974) held positions at Purdue University, the U.S. National Bureau of Standards, University of Washington, Boeing, and the Dublin Institute for Advanced Studies. He wrote eight books and won the Chauvenet Prize for Excellence in Expository Mathematical Writing in 1960. The six-volume Collected Published Papers with Commentaries appeared in 1998. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐What an amazing book! Lanczos excels at an intuitive discussion of the Fourier series and weighted summation techniques (Cesaro summation) for the spectrum of distributions (delta functions etc.). This is a must read! Some background in complex analysis recommended. While this is a masterful book it is geared as an introduction to students so it doesn’t cover a variety of advanced topics, so other resources on Fourier analysis are recommended as well.

⭐What’s not to like about Lanczos. He wrote so that students could understand and wrote because he disliked how other texts were written and talked down to an audience. Here he opens up how Fourier series is all about and starts with the definition of functions and takes it all the way. Like a Symphony. The buildup is gradual. Lanczos considered Fourier Series as the only Mathematical feature worth saving if all of the mathematics came to being discarded. So you need only to know that to know the passion that went into writing this averagely sized volume. One of the mathematical treasures for generations and on par with Hilbert’s works

⭐Of Cornelius Lanczos: if familiar with his monograph on Variational Principles of Mechanics, chances are that you will greatly enjoy this offering. If that is not reason enough to pique your interest, recall that Lanczos published a paper (February 26,1926) where he “demonstrated how the operations of matrix mechanics can be formulated in terms of integral equations.” (Lanczos Centenary Conference, 1994). With that background, we turn to this remarkable book. Lanczos falls between two textbook extremes: it is less detailed and expansive than Korner’s tome on Fourier Analysis (which, I like) yet, not as complete as the wonderful text of Folland, Fourier Analysis And its Applications (which, I really like). Lanczos can be studied as an adjunct to those two textbooks–those two textbook extremes. In contrast to those two textbooks, Lanczos’ book appears to be much less widely known. Neither Korner nor Folland refers to it !That is a pity, as it makes perfect accompaniment to both.Read Lanczos: “…it is necessary to develop the subject from its early beginnings and this explains the fact that even so-called ‘elementary’ concepts, such as the idea of a function, the meaning of a limit, uniform convergence and similar were included in the discussion.” (preface) Therefore you do meet those concepts: function, limit, uniform convergence, in the initial twenty-five pages. What a delight ! Lanczos proceeds in an historical vein. You will meet mathematics of Fourier, Riemann, Bonnet, Dirichlet and Fejer, among others ( If so inclined, I can think of no better than Bressoud, Radical Approach to Real Analysis). One ascertains that this field abounds with excellent tomes !Read Lanczos: “…the primary aim of the author was to convey something of the excitement and enthusiasm which imbued the hundreds of mathematicians who have contributed to this remarkable chapter of analysis.” This aim is admirably achieved ! A Question and Answer dialogue permeates the prose. That is, Lanczos presents somewhat of a Socratic approach, excellent pedagogy ! A random sample: “Is it always advocated to use the Laplace transform method in the solution of differential equations ? ” Following which, an Answer (page 233). Problems are scattered throughout the prose (excellent pedagogy). The answers are supplied (excellent pedagogy).Three Chapters:(1) The Fourier Series (110 pages). Progressing from elementary concepts to more advanced concepts, such as orthogonal functions and eigenvalues. Read: “we will now pretend that we have no geometrical knowledge of the sine and cosine functions and develop all their properties.” (page 89).(2) The Fourier Series in Approximation Problems (40 pages). From simple curve fitting to weighting by convergence.We read: “Is the use of the sine functions preferable to the use of the cosine functions ? ” Lanczos pontificates upon such (page 125). Excellent pedagogic strategy !(3) The Fourier Integral (90 pages). I highlight the excellent discussion of applying calculus of residues to physical problems. We read: “the method of residues cannot be considered an automatic procedure which will work in all cases.” (page 195). Think about that !(4) Concluding my review: Each page of this discourse is stamped with Lanczos’ inimitable imprint. This discourse is a pleasure to read and to study. It deserves wide readership. It deserves to be much better known. As a stepping stone (elementary to advanced) this monograph is unlike any other.Highly recommended to all (both students and their professors).

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