| 单词 | fourier expansion |
| 释义 | > as lemmasFourier expansion Used attributively or in the possessive to designate certain principles enunciated by Fourier and many mathematical expressions and techniques arising out of his work, as Fourier analysis, the analysis of a periodic function into a number of simple harmonic functions or, more generally, into a series of functions from any orthonormal set; Fourier's law, that any non-sinusoidal periodic vibration can be regarded as the sum of a number of sinusoidal vibrations each having a frequency that is an integral multiple of some fundamental frequency; Fourier('s) series, a series of the form ½a0 + (a1 cos x + b1 sin x) + (a2 cos 2x + b2 sin 2x) + …, where the constants a0, a1, b1, etc. are defined in terms of a function f(x) to which the series may converge; Fourier's theorem, (a) that if a function f(x) satisfies certain conditions within the interval −π ≤ x ≤ π, it can be represented within that interval by a Fourier series; (b) (see quot. 1880); Fourier transform, a function f(x) related to a given function g(t) by the equation (2π)½f(x) = ∞−∞g(t)e±itxdt, used to represent a non-periodic function by a spectrum of sinusoidal functions. Also Fourier coefficient, Fourier expansion, Fourier integral, Fourier transformation, etc. ΘΚΠ the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > theorem > specific theorem > relating to series Fourier's theorem1834 the world > relative properties > number > mathematical number or quantity > numerical arrangement > [noun] > set > sequence > series > convergent convergency1791 convergence1858 Fourier('s) series1877 the world > relative properties > number > mathematics > [noun] > mathematical enquiry > proposition > theorem > specific theorem pons asinorum1718 Fermat's theorem1845 Bernoulli's theorem1865 Fermat's last theorem1865 Fourier's theorem1880 remainder theorem1886 Stokes' theorem1893 Jordan('s) (curve) theorem1900 Waring's theorem1920 Gödel's theorem1933 maximin1953 incompleteness theorem1955 Schwarz inequality1955 the world > relative properties > number > algebra > [noun] > expression > function function1758 exponential1784 potential function1828 syzygy1850 permutant1852 Green function1863 theta-function1871 Greenian1876 Gudermannian1876 discriminoid1877 Weierstrassian function1878 gradient1887 beta function1888 distribution function1889 Riemann zeta function1899 Airy integral1903 Poisson bracket1904 Stirling approximation1908 functional1915 metric1921 Fourier transform1923 recursive function1934 utility function1934 Airy function1939 transfer function1948 objective function1949 restriction1949 multifunction1954 restriction mapping1956 scalar function1956 Langevin function1960 mass function1961 the world > relative properties > number > algebra > [noun] > expression > method of calculation or analysis extrapolation1872 functional analysis1876 inversion1880 Fourier analysis1929 formalism1940 linear programming1949 quadratic programming1951 simplex method1951 convex programming1963 deconvolution1967 1834 Rep. Brit. Assoc. 1833 343 If the interval of the roots be determined, by the application of Fourier's theorem of the succession of signs of the original function X and its derivatives. 1842 A. De Morgan Differential & Integral Calculus xx. 641 In applying Fourier's theorem..to discontinuous functions, we find that at the point where the discontinuity takes place, and a function which generally can have but one value might be expected to have two, it takes neither, and gives only the mean between them. 1877 Ld. Rayleigh Theory of Sound I. ii. 24 The pre~eminent importance of Fourier's series in Acoustics. 1880 G. S. Carr Synopsis Elem. Results Math. I. 134 Fourier's Theorem.—Fourier's functions are..f(x), f′(x), f″(x)…fn(x)…As x increases, Fourier's functions lose one change of sign for each root of the equation f(x) = 0, through which x passes, and r changes of sign for r repeated roots. 1884 A. Daniell Text-bk. Princ. Physics v. 127 Longitudinal vibrations of a string or rod..whose ends are held fixed obey the same principles as transverse vibrations. Fourier's law holds good. 1902 E. T. Whittaker Course Mod. Anal. vii. 152 The question arises..whether the Fourier expansion is unique. 1911 Proc. Royal Soc. A. 85 14 We can also sum the series of the products of the Fourier coefficients of two such functions. 1912 London, Edinb., & Dublin Philos. Mag. 6th Ser. 24 866 Fourier's integral. 1923 Proc. Cambr. Philos. Soc. 21 463 The notion of Fourier transforms arises from Fourier's integral formula,..which gives..reciprocal relations..connecting the two functions f(x) and F(x). 1929 V. Bush Operational Circuit Anal. x. 186 Direct operational methods may be regarded as shorthand processes of evaluating and tabulating the results of Fourier analysis. 1936 Discovery Apr. 114/2 All sound-waves (except those from a flute, closed organ-pipe, etc.) are composed of many combined vibrations whose composition follows Fourier's law. 1957 R. S. Longhurst Geom. & Physical Optics xi. 226 The Fraunhofer pattern is the Fourier transform of the amplitude across the diffracting aperture and vice versa. 1963 R. W. Ditchburn Light (ed. 2) iv. 89 The ‘top-hat curve’ shown in fig. 4.6 can be represented by an appropriate Fourier series..for all values of x0 because the curve to be represented is periodic. 1964 Oceanogr. & Marine Biol. 2 14 The correlation coefficient fu(τ) is related by a Fourier transformation to the spectrum function Fu(n). 1965 S. I. Pearson & G. J. Maler Introd. Circuit Anal. ix. 439 The Fourier transform..is useful in analyzing pulses from a frequency standpoint. 1967 E. U. Condon & H. Odishaw Handbk. Physics (ed. 2) ii. iii. 26/2 The part of F(t) effective in exciting the oscillator is the component in its Fourier integral representation associated with the natural frequency of the oscillator. < as lemmas |
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