Harmonic analysis, a branch of mathematics.The main research function is expanded intoFourier seriesOr Fourier integral, andseriesandintegralThe various problems of.It originated fromphysicsGeneral onePeriodic oscillationThe problem of decomposing into the superposition of simple harmonic oscillations has developed into a widely applied discipline.[1]
Chinese name
harmonic analysis
Foreign name
harmonic analysis
Definition
The operation of function expansion into Fourier series
Mathematical tools
Fourier transform
Application
Mathematics, information processing, quantum mechanics, etc
Harmonic analysis is modernAnalytical mathematicsIts brilliant achievements have attracted generations of analysts to fall and struggle for it.According to Mr. Hua Luogeng, the operation of expanding a known function into Fourier series is called harmonic analysis.In fact, harmonic analysis has grown from the research of Fourier series and Fourier transform theory.From the physical point of view, harmonic analysis is to represent the signal as the superposition of the fundamental wave "irony and sub".[2]Over the centuries, harmonic analysis has formed a huge discipline system, and has important and profound applications in mathematics, information processing, quantum mechanics and other fields.
The development of harmonic analysis can be traced back toFourier analysis。Mathematical tools developed recently for harmonic analysis, such aswavelet transform And Gabor transformation are essentially optimal transformations in some situations (spaces with certain properties, such as Bosov spaces).Harmonic analysis has been successfully applied to the development of new forms of functional representation, which has proved the importance of harmonic analysis.Fourier transform and wavelet transform are two typical tools for function approximation.[3]
Related concepts
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Fourier series:Any periodic function can be usedSine functionandcosine functionIt is expressed by the infinite order number (sine function and cosine function are chosen as the basis function because they are orthogonal). Later generations called Fourier series as a special trigonometric series according toEuler formula, trigonometric function can be converted into exponential form, also known asFourier seriesIt is an exponential series.
Fourier transformFourier transform describes the total frequency distribution in the signal by using the complex sine basis function on the infinite interval and the inner product of the signal. It converts the research of the original time domain signal into the research of Fourier coefficients in the frequency domain. Fourier analysis is purely frequency domain analysis.Only applicable to deterministic stationary signals[3]
classification
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From the perspective of application, the operation to effectively determine the Fourier series problem is calledPractical harmonic analysis。Finite harmonic analysis is the main framework of practical harmonic analysis, that is, from the perspective of the most appropriate number of items that should be calculated for a limited number of data, the Fourier method of solving practical problems from a finite to finite way of thinking is where the application value of finite harmonic analysis lies.From the perspective of physics, people can find thatquantum mechanicsThe uncertainty relation in has the explanation of harmonic analysis version, that is, the Fourier transform of the generalized function with non-zero compact support described by Paley Wiener theorem has no compact support.
Abstract harmonic analysisIt is a more in-depth branch of modern mathematics of harmonic analysis, that is, researchTopological groupThe harmonic analysis theory, especially the Fourier transform theory.The Ponteyagin duality theory of Abel compact groups is a suitable portrayal of harmonic analysis characteristics in modern mathematical processing.For general non Abelian locally compact groups, harmonic analysis isUnitary groupClosely related to expressionism.The Fourier transformation of classical convolution is the product of Fourier transformation, which can be sublimated by the Peter Weyl theorem of compact groups.When a group is neither Abel nor compact, the general theory of abstract harmonic analysis is not perfect.For example, it is not known whether there is an analogue of the Ptancherel theorem at this time, but in many special cases, certain related problems can be analyzed through the infinite dimensional representation technology.[2]
