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  2. Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Fourier_transform

    The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties. The Fourier transform ^ of any integrable function is uniformly continuous and [19] ‖ ^ ‖ ‖ ‖

  3. Fourier inversion theorem - Wikipedia

    en.wikipedia.org/wiki/Fourier_inversion_theorem

    More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an operator (see Fourier transform on function spaces). For example, the Fourier inversion theorem on f ∈ L 2 ( R n ) {\displaystyle f\in L^{2}(\mathbb {R} ^{n})} shows that the Fourier transform is a unitary operator on L 2 ( R n ) {\displaystyle L ...

  4. Convolution theorem - Wikipedia

    en.wikipedia.org/wiki/Convolution_theorem

    In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).

  5. Fourier analysis - Wikipedia

    en.wikipedia.org/wiki/Fourier_analysis

    The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. [8] In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum.

  6. Dirac delta function - Wikipedia

    en.wikipedia.org/wiki/Dirac_delta_function

    The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed as ∫ − ∞ ∞ 1 ⋅ e 2 π i x ξ d ξ = δ ( x ) {\displaystyle \int _{-\infty }^{\infty }1\cdot e^{2\pi ix\xi }\,d\xi =\delta (x)} and more rigorously, it follows since 1 , f ^ = f ( 0 ) = δ , f {\displaystyle \langle ...

  7. Sine and cosine transforms - Wikipedia

    en.wikipedia.org/wiki/Sine_and_cosine_transforms

    By applying Euler's formula (= ⁡ + ⁡), it can be shown (for real-valued functions) that the Fourier transform's real component is the cosine transform (representing the even component of the original function) and the Fourier transform's imaginary component is the negative of the sine transform (representing the odd component of the ...

  8. Discrete Fourier transform - Wikipedia

    en.wikipedia.org/wiki/Discrete_Fourier_transform

    In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...

  9. Riemann–Lebesgue lemma - Wikipedia

    en.wikipedia.org/wiki/Riemann–Lebesgue_lemma

    A version holds for Fourier series as well: if is an integrable function on a bounded interval, then the Fourier coefficients ^ of tend to 0 as . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line.