Short Course Description
What do Fourier series, Fourier transforms, orthogonal polynomials, and spherical harmonics have in common? Each arises from the spectral decomposition of a non-negative self-adjoint operator. This course reveals the deep unity behind these seemingly distinct tools of analysis by developing a general spectral framework that encompasses all of them. Through this lens, we explore powerful methods of function representation and approximation that are rooted in operator theory and functional calculus.
We begin with the spectral theorem for self-adjoint operators and develop a functional calculus specifically for non-negative self-adjoint operators. This enables the construction of highly localized kernel operators and leads naturally to the definition and analysis of spectral spaces ( spaces of functions with finite spectra). From there, we examine the role of test functions and distributions in the spectral setting and study function spaces associated with non-negative self-adjoint operators, such as Besov, Triebel-Lizorkin and Hardy.
By the end of the course, students will gain a unified understanding of many classical analytic constructions and the operator-theoretic structures that underpin them.
Topics Covered:
- Review of examples the general theory covers.
- The structure of spaces satisfying the "doubling condition"
- Spectral theory for self-adjoint operators
- Functional calculus for non-negative self-adjoint operators
- Localized kernel operators and spectral projectors
- Spectral spaces
- Distributions and test functions in the spectral setting
- Function spaces (e.g., Besov, Triebel?Lizorkin) tied to operator spectra
- Applications to harmonic analysis and PDEs
Full syllabus is to be published
