Credits: 7.5

Type: bridging

Objectives:

To introduce infinite vector spaces, scalar products and continuous linear applications. The Fourier transform is the most important example, and the concepts are taught by addressing the treatments of continuous and discrete signals and their use in solving differential equations with constant coefficients.

Knowledge:

• Hilbert spaces: scalar product and projection theorem.
• Signals: L2 and I2 spaces. Continuous and discrete signals. Time-invariant systems. Introduction to filters.
• Fourier transform: Fourier series and Fourier integrals, L2 theory.
• Applications: orthonormal bases. Haar's basis and the idea of the wavelet. Solving linear differential equations with constant coefficients.
• Elementary spectrum theory: Hilbert-Schmidt operators. Compactness. Spectrum theory. Applications to Sturm-Liouville problems.

Subject-specific competences:

• To understand the concept of the vector space of functions and the operator between spaces of this sort.
• To use infinite linear spaces readily.
• To understand the role of continuity.
• To understand the various variants of the Fourier transform and their link to time-invariant operators.
• To demonstrate capability in the use of the course unit concepts to solve problems using signal theory and differential equations.