Harmonic Analysis

Convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. Learners will understand its applications in filtering, signal processing, and differential equations.

Distributions, or generalized functions, extend the concept of derivative to all continuous functions. Learners will understand how to formulate and solve weak solutions for partial differential equations that lack classical solutions.

Fourier series represent periodic functions as infinite sums of sine and cosine waves. Learners will understand how to decompose complex periodic signals into their frequency components and solve partial differential equations.

Fourier transforms extend Fourier series to non-periodic functions, mapping signals from the time domain to the frequency domain. Learners will understand how to analyze continuous signals, solve differential equations, and process digital data.

Hardy spaces are mathematical spaces of holomorphic functions on the unit disk or upper half-plane. Learners will understand their boundary behavior, norm properties, and applications in operator theory and control engineering.

Harmonic functions are twice-differentiable real-valued functions that satisfy Laplace's equation. Learners will understand their connection to analytic complex functions, explore the mean value property, and solve boundary value problems in physics.

Maximal functions measure the local behavior of a function by taking averages over shrinking neighborhoods. Learners will understand the Hardy-Littlewood maximal operator and its role in proving pointwise convergence and bounding singular integrals.

Singular integrals are integral transforms with kernels that have singularities. Learners will understand the Hilbert and Riesz transforms, Calderón-Zygmund theory, and how these operators behave on various function spaces.

Sobolev spaces are vector spaces of functions equipped with norms involving both the functions and their weak derivatives. Learners will understand how to apply these spaces to establish the existence and regularity of solutions to partial differential equations.

Wavelets are mathematical functions used to divide data into different frequency components. Learners will understand how to perform multi-resolution analysis for signal compression, noise reduction, and image processing.