Exercises in Integration - Brossura

George, Claude

Brossura

ISBN 10: 1468463225 ISBN 13: 9781468463224

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Having taught the theory of integration for several years at the University of Nancy I, then at the Ecole des Mines of the same city, I had followed the custom of the times of writing up de� tailed solutions of exercises and problems, which I used to dis� tribute to the students every week. Some colleagues who had had occasion to use these solutions have persuaded me that this work would be interesting to many students, teachers and researchers. The majority of these exercises are at the master's level; to them I have added a number directed to those who would wish to tackle greater difficulties or complete their knowledge on various points of the theory (third year students, diploma of education students, researchers, etc.). This book, I hope, will render to students the services that this kind of book brings them in general, with the reservation that can always be made in this case: that certain of them will be tempted to look at the solution to the exercises which are put to them without any personal effort. There is hardly any need to emphasize that such a use of this book would be no benefit. On the other hand, the student who after having worked seriously upon a problem, seeks some pointers from the solution, or compares it with his own, will be using this work in the optimal way.

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Contenuti

0: Outline of the Course.- 1: Measurable Sets.- (Exercises 1 � 1 1 � 21).- 2: ?-Algebras and Positive Measures.- Exercises 2 � 22 – 2 � 28).- 3: The Fundamental Theorems.- (Exercises 3 � 29 – 3 � 72).- 4: Asymptotic Evaluation of Integrals.- (Exercises 4 � 73 – 4 � 78).- 5: Fubini’s Theorem.- (Exercises 5 � 79 – 5 � 99).- 6: The LP Spaces.- (Exercises 6 � 100 – 6 � 125).- 7: The Space L2..- (Exercises 7 � 126 – 7 � 137).- 8: Convolution Products and Fourier Transforms.- (Exercises 8 � 138 – 8 � 162).- 9: Functions with Bounded Variation: Absolutely Continuous Functions: Differentiation and Integration (Exercises 9 � 163 – 9 � 173).- 10: Summation Processes: Trigonometric Polynomials.- (Exercises 10 � 174 – 10 � 184).- 11: Trigonometric Series.- (Exercises 11 � 185 – 11 � 230).- Erratum to Exercise 3 � 45.- Name Index.