Fourier Series, Fourier Transform and Their Applications to Mathematical Physics (Instructor Solution Manual, Solutions)

Description

obtained thanks to https://t.me/HermitianSociety I Fourier Series and the Discrete Fourier Transform Introduction Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4 Formulation of Fourier Series Exercise 2.1 Exercise 2.2 Exercise 2.3 Exercise 2.4 Exercise 2.5 Exercise 2.6 Exercise 2.7 Fourier Coefficients and Their Properties Exercise 3.1 Exercise 3.2 Exercise 3.3 Exercise 3.4 Convolution and Parseval's Equality Exercise 4.1 Exercise 4.2 Exercise 4.3 Fejér Means of Fourier Series. Uniqueness of the Fourier Series. Exercise 5.1 Exercise 5.2 Exercise 5.3 The Riemann–Lebesgue Lemma Exercise 6.1 Exercise 6.2 Exercise 6.3 Exercise 6.4 Exercise 6.5 The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem. Exercise 7.1 Exercise 7.2 Exercise 7.3 Besov and Hölder Spaces Exercise 8.1 Exercise 8.2 Exercise 8.3 Exercise 8.4 Exercise 8.5 Exercise 8.6 Exercise 8.7 Absolute Convergence. Bernstein and Peetre Theorems. Exercise 9.1 Exercise 9.2 Exercise 9.3 Exercise 9.4 Exercise 9.5 Exercise 9.6 Exercise 9.7 Dirichlet Kernel. Pointwise and Uniform Convergence. Exercise 10.1 Exercise 10.2 Exercise 10.3 Exercise 10.4 Exercise 10.5 Exercise 10.6 Exercise 10.7 Exercise 10.8 Formulation of the Discrete Fourier Transform and Its Properties. Exercise 11.1 Exercise 11.2 Exercise 11.3 Connection Between the Discrete Fourier Transform and the Fourier Transform. Exercise 12.1 Exercise 12.2 Some Applications of the Discrete Fourier Transform. Exercise 13.1 Applications to Solving Some Model Equations The One-Dimensional Heat Equation Exercise 14.1 Exercise 14.2 Exercise 14.3 The One-Dimensional Wave Equation Exercise 14.4 Exercise 14.5 Exercise 14.6 Exercise 14.7 Exercise 14.8 The Laplace Equation in a Rectangle and in a Disk Exercise 14.9 Exercise 14.10 Exercise 14.11 Exercise 14.12 II Fourier Transform and Distributions The Fourier Transform in Schwartz Space Exercise 16.1 Exercise 16.2 Exercise 16.3 Exercise 16.4 Exercise 16.5 The Fourier Transform in Lp Exercise 17.1 Exercise 17.2 Exercise 17.3 Exercise 17.4 Exercise 17.5 Exercise 17.6 Exercise 17.7 Tempered Distributions Exercise 18.1 Exercise 18.2 Exercise 18.3 Exercise 18.4 Exercise 18.5 Exercise 18.6 Exercise 18.7 Exercise 18.8 Convolutions in S and Sprime Exercise 19.1 Sobolev Spaces Exercise 20.1 Exercise 20.2 Exercise 20.3 Exercise 20.4 Exercise 20.5 Exercise 20.6 Exercise 20.7 Sobolev spaces on bounded domains Exercise 20.8 Homogeneous Distributions Exercise 21.1 Exercise 21.2 Exercise 21.3 Exercise 21.4 Exercise 21.5 Exercise 21.6 Exercise 21.7 Fundamental Solution of the Helmholtz Operator Exercise 22.1 Exercise 22.2 Exercise 22.3 Exercise 22.4 Estimates for the Laplacian and Hamiltonian Exercise 23.1 Exercise 23.2 Exercise 23.3 Exercise 23.4 Exercise 23.5 III Operator Theory and Integral Equations Inner Product Spaces and Hilbert Spaces Exercise 25.1 Exercise 25.2 Exercise 25.3 Exercise 25.4 Exercise 25.5 Exercise 25.6 Exercise 25.7 Exercise 25.8 Symmetric Operators in Hilbert Spaces Exercise 26.1 Exercise 26.2 Exercise 26.3 Exercise 26.4 Exercise 26.5 Exercise 26.6 Exercise 26.7 Exercise 26.8 John von Neumann’s Spectral Theorem Exercise 27.1 Exercise 27.2 Exercise 27.3 Exercise 27.4 Exercise 27.5 Exercise 27.6 Exercise 27.7 Spectra of Self-Adjoint Operators Exercise 28.2 Exercise 28.3 Exercise 28.4 Exercise 28.5 Exercise 28.6 Exercise 28.7 Exercise 28.8 Exercise 28.9 Exercise 28.10 Exercise 28.11 Quadratic Forms. Friedrichs Extension. Exercise 29.1 Elliptic Differential Operators Exercise 30.1 Exercise 30.2 Exercise 30.3 Exercise 30.4 Exercise 30.5 Spectral Functions Exercise 31.1 Exercise 31.2 Exercise 31.3 The Schrödinger Operator Exercise 32.1 Exercise 32.2 Exercise 32.3 Exercise 32.4 Exercise 32.5 The Magnetic Schrödinger Operator Exercise 33.1 Exercise 33.2 Integral Operators with Weak Singularities. Integral Equations of the First and Second Kinds. Exercise 34.1 Exercise 34.2 Exercise 34.3 Volterra and Singular Integral Equations Exercise 35.1 Exercise 35.2 Approximate Methods Exercise 36.1 Exercise 36.2 Exercise 36.3 IV Partial Differential Equations Introduction Exercise 37.1 Exercise 37.2 Exercise 37.3 Exercise 37.4 Exercise 37.5 Exercise 37.6 Exercise 37.7 Exercise 37.8 Exercise 37.9 Exercise 37.10 Exercise 37.11 Local Existence Theory Exercise 38.1 Exercise 38.2 The Laplace Operator Exercise 39.1 Exercise 39.2 Exercise 39.3 Exercise 39.4 Exercise 39.5 Exercise 39.6 Exercise 39.7 Exercise 39.8 Exercise 39.9 Exercise 39.10 Exercise 39.11 Exercise 39.12 Exercise 39.13 Exercise 39.14 The Dirichlet and Neumann Problems Exercise 40.1 Exercise 40.2 Exercise 40.3 Exercise 40.4 Exercise 40.5 Exercise 40.6 Exercise 40.7 Exercise 40.8 Exercise 40.9 Layer Potentials Exercise 41.1 Exercise 41.2 Exercise 41.3 Exercise 41.4 Exercise 41.5 Exercise 41.6 Exercise 41.7 Exercise 41.8 Elliptic Boundary Value Problems Exercise 42.1 Exercise 42.2 Exercise 42.3 Exercise 42.4 Some Inverse Scattering Problems for the Schrödinger Operator Exercise 44.1 Exercise 44.2 The Heat Operator Exercise 45.1 Exercise 45.2 Exercise 45.3 Exercise 45.4