Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior. The methods of harmonic analysis decompose functions and related objects, such as measures, into components based on symmetries, scales, spectra, or oscillation. It is also concerned with the analytic estimates for operators arising from such decompositions. Basic examples include Fourier series and the Fourier transform, while modern harmonic analysis also studies maximal functions, singular integrals, oscillatory integrals, Fourier multipliers, Littlewood–Paley theory, and spectral decompositions. A related tradition is abstract harmonic analysis where the emphasis is on functions and representations on topological groups, including Pontryagin duality, the Peter–Weyl theorem, and Plancherel-type theorems. Harmonic analysis overlaps substantially with Fourier analysis, real analysis, functional analysis, partial differential equations, potential theory, ergodic theory, representation theory, and number theory.
Overview Harmonic analysis shares many methods with Fourier analysis, which is also concerned with the decomposition of functions into frequencies or harmonics. It differs from Fourier analysis chiefly in the kinds of functions considered and the types of questions addressed. Fourier analysis has a basic form in Hilbert space, where orthogonality and Plancherel's theorem are central, and studies objects close to the orthogonal frequency decomposition, such as multipliers and convolutions and linear constant-coefficient partial differential equations. In contrast, harmonic analysis often studies Fourier-like decompositions in situations where orthogonality alone is not enough. As a result, it often must look for finer kinds of decompositions and properties of functions than those of Fourier theory. One source of harmonic analysis, and especially of its real-variable tradition, is the study of harmonic functions and classical potential theory. In particular, the Poisson integral formula represents a harmonic function in a disk or half-space in terms of boundary data. But the boundary behavior of harmonic functions, and of the Poisson kernel, is often more delicate than Fourier methods alone can resolve. Thus questions about the convergence of Poisson integrals, the existence of boundary values, and the behavior of conjugate harmonic functions lead to maximal estimates and singular integral operators. Modern harmonic analysis is usually concerned with real variable methods first encountered the study of harmonic functions. The Poisson integral sits between real and complex methods: complex analysis gives powerful tools for holomorphic and harmonic functions, while boundary convergence and estimates in Lp, weak L1, and related spaces often require real-variable methods. The real variable theory is also apparent in the related Hilbert transform, which arises in the theory of conjugate harmonic functions and boundary values of holomorphic functions. These transforms became a prototype for more general singular integral operators, which the real variable methods of harmonic analysis are more suited for. In higher dimensions, analogous operators include the Riesz transforms, which are connected with the derivatives of harmonic and Newtonian potentials. One ingredient is Hardy–Littlewood maximal function. Maximal functions are used to control pointwise convergence, differentiation of integrals, and boundary limits of harmonic or subharmonic functions. They also provide a model for many later estimates in real-variable analysis, including weak-type inequalities, interpolation arguments, and weighted norm inequalities. The theory of Calderón–Zygmund operators is one of the main developments in the analytic treatment of singular integral operators. This theory gives conditions under which singular integral operators are bounded on spaces such as Lp and related function spaces. It handles the Hilbert transform, Riesz transforms, many convolution operators, and singular integral operators arising in elliptic and parabolic partial differential equations. Littlewood–Paley theory is another component of harmonic analysis. It seeks to decompose functions by scale or frequency and estimating square functions built from the resulting pieces. This technique is used together with Fourier theory, but replaces the orthogonality arguments of Fourier theory with almost-orthogonality methods that are suited to situations where orthogonality is no longer available, such as L p {\displaystyle L^{p}} where p ≠ 2 {\displaystyle p\neq 2} . The kind of decompositions involved are often finer than the Hilbert-space methods that suffice for many basic questions in Fourier analysis. The Fourier transform remains a fundamental tool in harmonic analysis. But much of modern real-variable harmonic analysis is concerned less with explicit Fourier inversion than with estimates for operators. This distinguishes it from classical Fourier analysis and also from abstract harmonic analysis, where the emphasis is on symmetry, locally compact groups, and representation theory.
Examples
Decomposition and singular integrals A real-variable method in harmonic analysis is to decompose a function into a controlled part and a collection of localized exceptional parts. One example is the Calderón–Zygmund decomposition. Given an integrable function f {\displaystyle f} and a threshold λ > 0 {\displaystyle \lambda >0} , one selects intervals or cubes on which the average size of f {\displaystyle f} is larger than λ {\displaystyle \lambda } . The function is then written schematically as
f = g + ∑ j b j . {\displaystyle f=g+\sum _{j}b_{j}.}
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