Riesz ProjectionEdit
Riesz projection is a fundamental construction in harmonic analysis and complex function theory that isolates the analytic part of a function defined on the unit circle. Named after Marcel Riesz, it sits at the crossroads of Fourier analysis, Hardy spaces, and operator theory, and has important implications for both pure mathematics and applied signal processing. At its heart, the projection is a Fourier multiplier that selects nonnegative frequency components, thereby linking real-valued boundary data to holomorphic structure inside the disk.
In its most common form, the Riesz projection acts on functions defined on the unit circle and yields a function whose Fourier series contains only nonnegative frequencies. Concretely, if f(e^{it}) has Fourier series f(e^{it}) ~ sum_{n∈Z} \hat f(n) e^{int}, then the Riesz projection P f is given by Pf(e^{it}) ~ sum_{n≥0} \hat f(n) e^{int}. The range of P is the Hardy space on the boundary, Hardy space, denoted Hardy space, which consists of boundary values of holomorphic functions in the unit disk. When p=2, this projection is the orthogonal projection from L^2 on the unit circle onto H^2, often referred to in this context as the Szegő projection, i.e., the L^2 projection onto the analytic subspace.
Mathematical formulation
Definition and basic properties
- Domain and codomain: The Riesz projection P is a linear operator from L^p space to itself for 1
- Fourier multiplier viewpoint: P is the Fourier multiplier with symbol 1 for n≥0 and 0 for n<0. This multiplier viewpoint makes P a natural tool for decomposing signals into their analytic and anti-analytic parts.
- Orthogonality at p=2: When p=2, Pf is the orthogonal projection of f onto Hardy space within L^2(T). In this case, P is a contraction with operator norm 1.
- Boundedness on 1
- Domain and codomain: The Riesz projection P is a linear operator from L^p space to itself for 1
Relation to the Hilbert transform
- The Riesz projection can be related to the Hilbert transform Hilbert transform on the circle. Up to conventions, one has a representation of P in terms of I (identity) and H, for example P ≈ (I + iH)/2 on appropriate function spaces. The Hilbert transform itself is a singular integral operator with a well-known norm behavior on L^p space spaces.
Analytic versus anti-analytic decomposition
- For many purposes, f is written as f = Pf + (f − Pf), where Pf contains the analytic part (nonnegative frequencies) and f − Pf contains the anti-analytic part (negative frequencies). This decomposition underpins many spectral and factorization results, including links to Toeplitz operator theory.
Examples
- If f(e^{it}) = sum_{n∈Z} a_n e^{int}, then Pf(e^{it}) = sum_{n≥0} a_n e^{int}. In particular, if f is real-valued, Pf captures the analytic component of the signal, which is central in constructing analytic signals in signal processing contexts.
Connections and generalizations
- Szegő projection: On L^2(T), the Szegő projection is another name for the orthogonal projection onto H^2(T); the Riesz projection generalizes this idea to L^p spaces for 1<p<∞.
- Higher dimensions and other domains: Riesz-type projections extend to multi-dimensional tori T^d and to Hardy spaces on higher-dimensional domains, with corresponding Fourier multiplier descriptions and analytic-structure interpretations.
- Vector-valued and weighted settings: The projection framework extends to functions taking values in Banach spaces and to weighted L^p spaces, with adjustments to boundedness and norm estimates.
- Related operators: The Riesz projection sits alongside other projection operators in complex and harmonic analysis, such as the Szegő projection and various Fourier multiplier operators, and it interacts with Toeplitz operator theory in the study of analytic function spaces.
Historical notes
- The development of the Riesz projection is tied to early 20th-century work in harmonic analysis on the circle, with connections to Marcel Riesz and to the broader development of Hardy space theory and complex analysis on the disk. The concept also intersects with the theory of analytic signals and factorization results in operator theory.