Distributions, Fourier Transforms and Some of Their Applications to Physics
In this book, distributions are introduced via sequences of functions. This approach due to Temple has two virtues: It only presupposes standard calculus.It allows to justify manipulations necessary in physical applications. The Fourier transform is defined for functions and generalized to distributions, while the Green function is defined as the outstanding application of distributions. Using Fourier transforms, the Green functions of the important linear differential equations in physics are computed. Linear algebra is reviewed with emphasis on Hilbert spaces. The author explains how linear differential operators and Fourier transforms naturally fit into this frame, a point of view that leads straight to generalized fourier transforms and systems of special functions like spherical harmonics, Hermite, Laguerre, and Bessel functions.
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algebraic ansatz Bessel functions boundary condition Calculate Cauchy's theorem causal Green function circle closed curve compact support complex conjugation complex function complex numbers complex plane converges weakly convolution product defined denote derivative differential equation dimensions Dirac distribution D(x domain of definition eigenvalues eigenvectors example Exercises fast decrease fn(x Fourier coefficients Fourier series Fourier transform function f(x functions of fast Green function harmonic oscillator harmonic saw Hilbert space inequality infinity inversion formula isolated singularity Laplace operator Laplace transform limit magic formula orthogonal orthonormal basis Parseval's theorem periodic distribution periodic function piecewise differentiable function point spectrum polar coordinates positive Pre-Hilbert space propagator Prove quantum rational numbers real numbers residue theorem respect scalar product self-adjoint operator sense of distributions sequence of functions Show solution solve spectral decomposition sphere step function subset tempered distribution test function vanishes vector space velocity wave equation weakly convergent weakly convergent sequence