“Fourier analysis”的概念、定义、翻译、参考文献-科学参考

Fourier analysis

Physics

The representation of a function f(x), which is periodic in x, as an infinite series of sine and cosine functions,

$f (x) = a_{0} / 2 + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x) .$

A series of this type is called a Fourier series. If the function is periodic with a period 2π‎, the coefficients a₀, a_n, b_n are:

$\begin{array}{l} a_{0} = \int_{- π}^{π} f (x) d x, \\ a_{n} = \int_{- π}^{π} f (x) cos n x d x, (n = 1, 2, 3, \dots), \\ b_{n} = 1 / π \int_{- π}^{π} f (x) sin n x d x, (n = 1, 2, 3, \dots) . \end{array}$

Fourier analysis and Fourier series are named after the French mathematician and engineer Joseph Fourier (1768–1830). Fourier series have many important applications in mathematics, science, and engineering, having been invented by Fourier in the first quarter of the 19th century in his analysis of the problem of heat conduction.

Mathematics

The use of Fourier series and the Fourier transform in analysis, in many applications such including signal processing.

Astronomy

A technique used to find the frequencies present in a complex signal; also known as frequency analysis. A series of measurements made at different times is broken down mathematically into the sum of simple oscillations of various frequencies. The lowest frequency present is called the fundamental frequency, and the higher frequencies are then harmonics (whole multiples) of the fundamental frequency. The brightness fluctuations of some variable stars, for example, can be broken down into two or three simple sinusoidal variations by the use of Fourier analysis. The technique was invented by the French mathematician ( Jean-Baptiste) Joseph Fourier (1768–1830).

Chemistry

The representation of a function f(x), which is periodic in x, as an infinite series of sine and cosine functions:
$f (x) = a_{0} / 2 + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x)$
A series of this type is called a Fourier series. If the function is periodic with a period 2π‎, the coefficients a₀, a_n, b_n are:
$\begin{array}{l} a_{0} = \int_{- π}^{+ π} f (x) d x, \\ a_{n} = \int_{- π}^{+ π} f (x) cos n x d x, (n = 1, 2, 3, …), \\ b_{n} = \int_{- π}^{+ π} 1 / π f (x) sin n x d x, (n = 1, 2, 3, .…) . \end{array}$
Fourier analysis and Fourier series are named after the French mathematician and engineer Joseph Fourier (1768–1830). Fourier series have many important applications in mathematics, science, and engineering, including X-ray crystallography; they were invented by Fourier in the first quarter of the 19th century in his analysis of the problem of heat conduction.

Computer

The analysis of an arbitrary waveform into its constituent sinusoids (of different frequencies and amplitudes). See also Fourier transform, orthonormal basis.

Electronics and Electrical Engineering

A mathematical method of analysing complex waveshapes or signals into a series of simple harmonic functions, the frequencies of which are integer (1,2,3,…) multiples of the fundamental frequency. An arbitrary periodic phenomenon u, of period T, may be represented by the Fourier series provided that certain conditions – Dirichlet conditions – are satisfied. The Fourier series is then given by
$u = F (t)$
where
$F (t) = Σ_{n = - \infty}^{n = + \infty} a_{n} e^{j n ω t}$
where ω is equal to 2π/T, j is the square root of –1, and n is an integer; a_n is the nth coefficient and is given by
$a_{n} = (\frac{1}{T}) \int_{0}^{T} F (t) e^{- j n ω t} d t$
The Fourier series may alternatively be written as a series of sines and cosines.
As the period, T, becomes infinitely large so that 1/T tends to zero, the Fourier series in its limiting form becomes an integral – the Fourier integral. The values of the Fourier series or the Fourier integral are determined by the physical conditions of the phenomenon under consideration.
Fourier analysis is widely used in electronics, where a slightly different representation is commonly used in which the Fourier integral is written as
$F (t) = \int_{- \infty}^{+ \infty} g (ω) e^{j ω t} dω$
where the function
$g (ω) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} F (t) e^{- j ω t} d t$
is called the Fourier transform of the function F(t). Similarly F(t) is also the Fourier transform of the function g(ω). See also discrete Fourier transform.
http://hyperphysics.phy-astr.gsu.edu/hbase/Audio/fourier.html An introduction to Fourier analysis

Geology and Earth Sciences

The method whereby any periodic function can be broken down into a covergent trigonometric series of the form f(x) = a₀/2 + Σ‎_∞ⁿ⁼¹(a_ncos nx + b_nsin nx), where a_n and b_n are constant coefficients. Fourier analysis is the process of determining the frequency-domain function from a time function (e.g. a seismic-trace wave-form). See also fourier transform.

Economics

An expansion of a periodic function into an infinite sum of sines and cosines, each at a different frequency, also known as harmonic analysis or spectral analysis. Techniques based on Fourier analysis are applied primarily in time series econometrics. See also frequency domain analysis.

单词	Fourier analysis
释义	Fourier analysis Physics The representation of a function f(x), which is periodic in x, as an infinite series of sine and cosine functions, $f (x) = a_{0} / 2 + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x) .$ A series of this type is called a Fourier series. If the function is periodic with a period 2π‎, the coefficients a₀, a_n, b_n are: $\begin{array}{l} a_{0} = \int_{- π}^{π} f (x) d x, \\ a_{n} = \int_{- π}^{π} f (x) cos n x d x, (n = 1, 2, 3, \dots), \\ b_{n} = 1 / π \int_{- π}^{π} f (x) sin n x d x, (n = 1, 2, 3, \dots) . \end{array}$ Fourier analysis and Fourier series are named after the French mathematician and engineer Joseph Fourier (1768–1830). Fourier series have many important applications in mathematics, science, and engineering, having been invented by Fourier in the first quarter of the 19th century in his analysis of the problem of heat conduction. Mathematics The use of Fourier series and the Fourier transform in analysis, in many applications such including signal processing. Astronomy A technique used to find the frequencies present in a complex signal; also known as frequency analysis. A series of measurements made at different times is broken down mathematically into the sum of simple oscillations of various frequencies. The lowest frequency present is called the fundamental frequency, and the higher frequencies are then harmonics (whole multiples) of the fundamental frequency. The brightness fluctuations of some variable stars, for example, can be broken down into two or three simple sinusoidal variations by the use of Fourier analysis. The technique was invented by the French mathematician ( Jean-Baptiste) Joseph Fourier (1768–1830). Chemistry The representation of a function f(x), which is periodic in x, as an infinite series of sine and cosine functions: $f (x) = a_{0} / 2 + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x)$ A series of this type is called a Fourier series. If the function is periodic with a period 2π‎, the coefficients a₀, a_n, b_n are: $\begin{array}{l} a_{0} = \int_{- π}^{+ π} f (x) d x, \\ a_{n} = \int_{- π}^{+ π} f (x) cos n x d x, (n = 1, 2, 3, …), \\ b_{n} = \int_{- π}^{+ π} 1 / π f (x) sin n x d x, (n = 1, 2, 3, .…) . \end{array}$ Fourier analysis and Fourier series are named after the French mathematician and engineer Joseph Fourier (1768–1830). Fourier series have many important applications in mathematics, science, and engineering, including X-ray crystallography; they were invented by Fourier in the first quarter of the 19th century in his analysis of the problem of heat conduction. Computer The analysis of an arbitrary waveform into its constituent sinusoids (of different frequencies and amplitudes). See also Fourier transform, orthonormal basis. Electronics and Electrical Engineering A mathematical method of analysing complex waveshapes or signals into a series of simple harmonic functions, the frequencies of which are integer (1,2,3,…) multiples of the fundamental frequency. An arbitrary periodic phenomenon u, of period T, may be represented by the Fourier series provided that certain conditions – Dirichlet conditions – are satisfied. The Fourier series is then given by $u = F (t)$ where $F (t) = Σ_{n = - \infty}^{n = + \infty} a_{n} e^{j n ω t}$ where ω is equal to 2π/T, j is the square root of –1, and n is an integer; a_n is the nth coefficient and is given by $a_{n} = (\frac{1}{T}) \int_{0}^{T} F (t) e^{- j n ω t} d t$ The Fourier series may alternatively be written as a series of sines and cosines. As the period, T, becomes infinitely large so that 1/T tends to zero, the Fourier series in its limiting form becomes an integral – the Fourier integral. The values of the Fourier series or the Fourier integral are determined by the physical conditions of the phenomenon under consideration. Fourier analysis is widely used in electronics, where a slightly different representation is commonly used in which the Fourier integral is written as $F (t) = \int_{- \infty}^{+ \infty} g (ω) e^{j ω t} dω$ where the function $g (ω) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} F (t) e^{- j ω t} d t$ is called the Fourier transform of the function F(t). Similarly F(t) is also the Fourier transform of the function g(ω). See also discrete Fourier transform. http://hyperphysics.phy-astr.gsu.edu/hbase/Audio/fourier.html An introduction to Fourier analysis Geology and Earth Sciences The method whereby any periodic function can be broken down into a covergent trigonometric series of the form f(x) = a₀/2 + Σ‎_∞ⁿ⁼¹(a_ncos nx + b_nsin nx), where a_n and b_n are constant coefficients. Fourier analysis is the process of determining the frequency-domain function from a time function (e.g. a seismic-trace wave-form). See also fourier transform. Economics An expansion of a periodic function into an infinite sum of sines and cosines, each at a different frequency, also known as harmonic analysis or spectral analysis. Techniques based on Fourier analysis are applied primarily in time series econometrics. See also frequency domain analysis.
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