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. P

    Dirac delta; fourier representation

    Homework Statement I know that we can write ## \int_{-\infinity}^{\infinity}{e^{ikx}dx}= 2\pi \delta (k) ## But is there an equivalent if the interval which we are considering is finite? i.e. is there any meaning in ##\int_{-0}^{-L}{e^{i(k-a)x}dx} ## is a lies within 0 and L? Homework...
  2. J

    Solve Fourier Transform Homework: Wrong Answer?

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  3. S

    Is My Fourier Series Expansion of a Sawtooth Wave Correct?

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  4. T

    Show the Fourier transformation of a Gaussian is a Gaussian.

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  5. ElPimiento

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  6. J6204

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  7. J6204

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  8. J6204

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  9. A

    A Fourier Transform for 3rd kind of boundary conditions?

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  10. W

    How Long Does It Take for a Handle to Become Unbearably Hot?

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  11. B

    Fourier Series for |x|: Convergence & Answers

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  12. G

    Derivation of the Fourier series of a real signal

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  13. Allan McPherson

    Using Maxima to plot error in Fourier series

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  14. H

    Fourier series of a bandwidth limited periodic function

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  15. S

    How Does Increasing 'n' Affect the Cut-off Frequency 'k_cut' in DFT Analysis?

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  16. M

    I Measuring Thickness and Refractive Index with Fourier Domain OCT

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  17. S

    Python Discrete Fourier Transform in Python

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  18. L

    Understanding the discrete time Fourier transformation

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  19. M

    E&M separation of variables and Fourier

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  20. Jonski

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  21. redtree

    I Fourier conjugates and momentum

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  22. Svein

    Insights Further Sums Found Through Fourier Series - Comments

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  23. H

    Fourier Transfer of sawtooth function

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  24. baby_1

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  25. M

    Fourier Series of a function not centered at zero

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  26. M

    A Can the convolution operator be diagonalized using the Fourier transform?

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  27. F

    B Fourier Transform of Spacetime

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  28. A

    Odd and even in complex fourier series

    Homework Statement In Complex Fourier series, how to determine the function is odd or even or neither, as in the given equation $$ I(t)= \pi + \sum_{n=-\infty}^\infty \frac j n e^{jnt} $$Homework Equations ##Co=\pi## ##\frac {ao} 2 = \pi## ##Cn=\frac j n## ##C_{-n}= \frac {-j} n ## ##an=0##...
  29. Telemachus

    Fortran Discrete Fast Fourier transform with FFTW in FORTRAN77

    Hi, this thread is an extension of this one: https://www.physicsforums.com/posts/5829265/ As I've realized that the problem is that I don't know how to properly use FFTW, from http://www.fftw.org. I am trying to calculate a derivative using FFTW. I have ##u(x)=e^{\sin(x)}##, so...
  30. Telemachus

    Doubt about a discrete Fourier Transform

    Hi. I was checking the library for the discrete Fourier transform, fftw. So, I was using a functition ##f(x)=sin(kx)##, which when transformed must give a delta function in k. When I transform, and then transform back, I effectively recover the function, so I think I am doing something right...
  31. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

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  32. C

    What is the Definition of Period in Fourier Series?

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  33. davidge

    I Proving Fourier Method: Decompose Wave Forms with Sines & Cosines

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  34. R

    I Amplitudes of Fourier expansion of a vector as the generalized coordinates

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  35. tomdodd4598

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  36. D

    B Correctness of Equations in Electromagnetism Textbook

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  37. A

    Fourier Transform with inverse

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  38. A

    Understanding Fourier Transform: Solving a Homework Problem Step by Step

    Homework Statement Hello everyone, am trying to solve this Fourier Trans. problem, here is the original solution >> https://i.imgur.com/eJJ5FLF.pngQ/ How did he come up with this result and where is my mistake? Homework Equations All equation are in the above attached picture The Attempt at a...
  39. A

    Complex Fourier Series for cos(t/2)

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  40. B

    I Fourier transform and locality/uncertainty

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  41. A

    Supressed harmonics with certain initial conditions

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  42. John Jacke

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  43. R

    Matching Discrete Fourier Transform (DFT) Pairs

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  44. G

    Find Fourier coefficients - M. Chester text

    Homework Statement I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right ) and asked to find its Fourier...
  45. A

    Fourier transform of the ground state hydrogen wave function

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  46. R

    Discrete Fourier Transform (DFT) Matching

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  47. B

    I Fourier analysis and the sinusoidal plane wave

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  48. M

    Power signal calculation using Parseval's Theorem

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  49. DeathbyGreen

    A Fourier Transforming a HgTe 2D Hamiltonian

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  50. evinda

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