- Key People:
- Norbert Wiener
- Kodaira Kunihiko
- Antoni Zygmund
- Jean Bourgain
- Related Topics:
- analysis
- harmonic function
- harmonic analyzer
harmonic analysis
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What is harmonic analysis in mathematics?
Harmonic analysis is a mathematical technique used to describe and analyze periodically recurrent phenomena. It simplifies complex problems by breaking down complicated mathematical curves into sums of simpler components.
Many physical phenomena, such as sound waves, alternating electric currents, tides, and machine motions and vibrations, exhibit periodic behavior. These can be expressed as the sum of sine and cosine terms, known as a Fourier series. The determination of the coefficients of these terms is harmonic analysis.
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How are waves or signals related to harmonic analysis?
Harmonic analysis is a mathematical method used to describe and analyze periodically recurrent phenomena. It breaks down complex mathematical curves into sums of simpler components. Many physical phenomena, such as sound waves, can be periodic. These periodic functions can be expressed as a sum of sine and cosine terms called a Fourier series, where determining the coefficients of these terms is called harmonic analysis.
One term of a Fourier series has a period equal to that of the function f(x) and is called the fundamental. Other terms are harmonics with shorter periods that are integral submultiples of the fundamental. Harmonic analyzers are machines that measure the relative amplitudes of sinusoidal components of a periodically recurrent function.
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What is a Fourier series and why is it important in harmonic analysis?
A Fourier series is an infinite series of sine and cosine terms used to represent periodic functions. It's a tool to solve differential equations by breaking down complex mathematical curves into simpler components.
In harmonic analysis, the determination of the coefficients of these terms is called harmonic analysis. Many physical phenomena, like sound waves and electric currents, can be expressed as a sum of sine and cosine terms. The use of more terms increases the accuracy of the approximation.
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How does harmonic analysis help us break a function into simpler parts?
Harmonic analysis is a mathematical technique that simplifies complex problems by breaking down intricate mathematical curves into sums of simpler components. It expresses a periodic function as a sum of sines and cosines, specifically using a Fourier series. This method is used to analyze periodically recurrent phenomena like sound waves, electric currents, tides, and vibrations. The determination of the coefficients of these terms is called harmonic analysis.
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Where is harmonic analysis used in science or engineering?
Harmonic analysis is a mathematical procedure that describes and analyzes periodically recurrent phenomena. It is used in many complex problems to break down complicated mathematical curves into sums of simpler components.
Many physical phenomena, such as sound waves, alternating electric currents, tides, and machine motions and vibrations, may be periodic. With the periodic functions found in nature, the function can be expressed as the sum of a number of sine and cosine terms. This sum is known as a Fourier series, named after Joseph Fourier.
Harmonic analyzers, sometimes called spectrum analyzers, are used to measure the relative amplitudes of sinusoidal components of a periodically recurrent function.
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What is the difference between Fourier series and Fourier transform in harmonic analysis?
Both the Fourier series and the Fourier transform are tools in harmonic analysis that convert a signal from one domain to another. Harmonic analysis expresses a periodic function as a sum of sines and cosines using a Fourier series.
A Fourier series is an infinite series of sine and cosine functions used to solve differential equations and analyze periodic functions. A Fourier transform decomposes a signal in the time domain into its constituent frequencies, representing the signal in the frequency domain and giving each frequency component's phase and amplitude. It has applications in various fields, including electrical engineering, digital signal processing, and radio interferometry.
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harmonic analysis, mathematical procedure for describing and analyzing phenomena of a periodically recurrent nature. Many complex problems have been reduced to manageable terms by the technique of breaking complicated mathematical curves into sums of comparatively simple components.
Many physical phenomena, such as sound waves, alternating electric currents, tides, and machine motions and vibrations, may be periodic in character. Such motions can be measured at a number of successive values of the independent variable, usually the time, and these data or a curve plotted from them will represent a function of that independent variable. Generally, the mathematical expression for the function will be unknown. However, with the periodic functions found in nature, the function can be expressed as the sum of a number of sine and cosine terms. Such a sum is known as a Fourier series, after the French mathematician Joseph Fourier (1768–1830), and the determination of the coefficients of these terms is called harmonic analysis. One of the terms of a Fourier series has a period equal to that of the function, f(x), and is called the fundamental. Other terms have shortened periods that are integral submultiples of the fundamental; these are called harmonics. The terminology derives from one of the earliest applications, the study of the sound waves created by a violin (see analysis: Musical origins and Fourier analysis).
In 1822 Fourier stated that a function y = f(x) could be expressed between the limits x = 0 and x = 2π by the infinite series that is now given in the form 
provided the function is single-valued, finite, and continuous except for a finite number of discontinuities, and where 
and 
for k ≥ 0. With the further restriction that there be only a finite number of extremum (local maxima and minima), the theorem was proved by the German mathematician Peter Lejeune Dirichlet in 1829.
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The use of a larger number of terms will increase the accuracy of the approximation, and the large amounts of calculations needed are best done by machines called harmonic (or spectrum) analyzers; these measure the relative amplitudes of sinusoidal components of a periodically recurrent function. The first such instrument was invented by the British mathematician and physicist William Thomson (later Baron Kelvin) in 1873. This machine, used for the harmonic analysis of tidal observations, embodied 11 sets of mechanical integrators, one for each harmonic to be measured. A still more complicated machine, handling up to 80 coefficients, was designed in 1898 by the American physicists Albert Abraham Michelson and Samuel W. Stratton.
Early machines and methods made use of an experimentally determined curve or set of data. In the case of electric currents or voltages, an entirely different method is possible. Instead of making an oscillographic record of the voltage or current and analyzing it mathematically, the analysis is performed directly on the electric quantity by recording the response as the natural frequency of a tuned circuit is varied through a wide range. Thus, harmonic analyzers and synthesizers of the 20th century tended to be electromechanical rather than purely mechanical devices. Modern analyzers display the frequency-modulated signals visually by means of a cathode-ray tube, and digital or analog computer principles are used to carry out the Fourier analysis automatically, thereby achieving approximations of great accuracy.