Fourier Definition and 1000 Threads

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.
The Fourier transform of a function of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.

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  1. J

    I understanding the Fourier components of a square wave

    In my physics book there is an example of making a square wave by "simply" summing up a few cosine waves. The book says these first three waves are the first three Fourier components of a square wave, yet when I sum the three wave functions up, I get something way off; as does my calculator...
  2. K

    Fourier Transform : Analysis of 2 different signals

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  3. B

    Complex exponetial form of Fourier series

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  4. R

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  5. S

    What is the correct Fourier Series for f(x) = sinx on the interval 0 < x < ∏?

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  6. U

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  7. T

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  8. M

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  9. D

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  10. A

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  11. M

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  12. U

    Fixed Rope Fourier: Complete Expression of Motion

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  13. D

    Fourier series - DC component, integration problem

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  14. M

    Calculate Fourier transform for the characteristic function of a rv

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  15. S

    Complex exponential and sine-cosine Fourier series

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  16. K

    Convolution integral and fourier transform in linear response theory

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  17. H

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  18. X

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  19. S

    Fourier Transform - Scaling Property

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  20. M

    Discrete Time Fourier Transform

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  21. G

    Orthogonality Problem (From Fourier Analysis Text)

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  22. P

    MHB How Can Fourier Coefficients Help Solve Infinite Series Problems?

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  23. S

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  25. J

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  26. C

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  27. K

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  28. B

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  29. T

    Hilbert Space Interpretation of Fourier Transform

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  30. cocopops12

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  31. M

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  32. R

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  33. F

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  34. S

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  35. E

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  36. L

    Fourier Transform: Solve Homework Equations for fd

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  37. aNxello

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  38. P

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  39. L

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  40. I

    Is this Fourier Series exam question solved correctly?

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  41. H

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  42. J

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  43. J

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  44. W

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  45. E

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  46. L

    Help with computing/understanding Fourier Sine Expansion.

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  47. L

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  48. D

    MHB Fourier series damped driven oscillator ODE

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  49. B

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  50. I

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