Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations
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A01=Kristian P Evans
A01=Niels Jacob
Author_Kristian P Evans
Author_Niels Jacob
Boundary and Eigenvalue Problems
Category=PBK
eq_isMigrated=1
eq_isMigrated=2
eq_nobargain
Euler-Lagrange Equations
EulerAcAEURA"Lagrange Equations
Euler–Lagrange Equations
Fourier Series
Fourier Transform
Hamiliton-Jacobi Theory
HamilitonAcAEURA"Jacobi Theory
Hamiliton–Jacobi Theory
Hardy Spaces
Hypergeometric Differential Equations and Special Functions of Mathematical Physics
Legendre Conditions
Linear Systems
Orthonormal Expansions
Picard-Lindelof and Peano Theorems
PicardAcAEURA"LindelAfA?f and Peano Theorems
Picard–Lindelöf and Peano Theorems
Stability in the Sense of Ljapunov
Sturm-Liouville Problems
SturmAcAEURA"Liouville Problems
Sturm–Liouville Problems
Product details
- ISBN 9789813273511
- Publication Date: 05 Sep 2018
- Publisher: World Scientific Publishing Co Pte Ltd
- Publication City/Country: SG
- Product Form: Hardback
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In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures.Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard-Lindelöf and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm-Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows.Our introduction to the calculus of variations includes a discussion of the Euler-Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail.The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.
