Learning objectives
The aim of the course consists in introducing the students to the theory of distributions, showing also how this theory can be applied to solve PDEs.
Prerequisites
Analisi Superiore 1
Course unit content
<strong>1. Some results from Functional Analysis.</strong><br />
Topological vector spaces. L^p-spaces. The Schwarz space of rapidly decreasing functions.<br />
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<strong>2. Elements of the theory of distributions.</strong><br />
Test functions and regularization. The support of a function. The set D'. Distributions<br />
viewed upon as a generalization of locally integrable functions. Extensions of distributions. Differentiation of distributions. Multiplication of a distribution by a smooth function. The support of a distribution. The set of all distributions with bounded support. Substitution in distributions. Primitives of a distribution. Distributions viewed upon as derivatives of continuous functions. Fixing a variable in a distribution. Tensor product of distributions. Convolution of two distributions.<br />
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<strong>3. Fourier transform.</strong><br />
Fourier transform in the set of rapidly decreasing functions. Main properties of the Fourier transform. Fourier trasform in L^2. The space S' of tempered distributions. Convolution of two tempered distributions. Fixing a variable in a tempered distribution. Fourier transform in S'. Fourier transform in the set of distributions with bounded support. Fourier transform of the convolution of two distributions and the <br />
tensor product of two distributions. The Fourier transform of<br />
some important distributions. The role of Fourier transform in the study of <br />
differential equations with constant coefficients.<br />
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<strong>4. Linear differential equations.</strong><br />
Connection between classical and distributional solutions. Fundamental solutions. The method of descent. Transformation of a differential equation by substitution. ODE with a parameter. Distributional initial value problems. The wave equation.
Full programme
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Bibliography
Z. Szmydt: Fourier transformation and linear differential equations, D. Reidel Publishing company Dordrecht-Holland 1977.<br />
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Lecture notes provided by the teacher.
Teaching methods
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Assessment methods and criteria
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Other information
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