Fourier transform of sum of delta functions. An While saz has already answered the question, I jus...

Fourier transform of sum of delta functions. An While saz has already answered the question, I just wanted to add that this can be seen as one of the simplest examples of the Uncertainty Principle found in quantum mechanics, and generalizes to something called Hardy's uncertainty principle. . The Fourier transform can be viewed as the limit of the Fourier series of a function with the period approaches to infinity, so the limits Apr 9, 2020 · 7 Discrete Fourier Transform (DFT) is the discrete version of the Fourier Transform (FT) that transforms a signal (or discrete sequence) from the time domain representation to its representation in the frequency domain. While saz has already answered the question, I just wanted to add that this can be seen as one of the simplest examples of the Uncertainty Principle found in quantum mechanics, and generalizes to something called Hardy's uncertainty principle. Let us consider the Fourier transform of $\\mathrm{sinc}$ function. While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. To Fourier's credit, the Dirichlet kernel integral expression for the truncated trigonometric Fourier series was in Fourier's original work. Whereas, Fast Fourier Transform (FFT) is any efficient algorithm for calculating the DFT. The main difference is that wavelets are localized in both time and frequency whereas Nov 24, 2025 · What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. Derivation is a linear operator. In the QM context, momentum and position are each other's Fourier duals, and as you just discovered, a Gaussian function that's well-localized in one Jun 27, 2013 · Fourier transform commutes with linear operators. Nov 24, 2025 · What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. Feb 9, 2026 · Explore related questions limits fourier-series See similar questions with these tags. Nov 24, 2013 · What are some real world applications of Fourier series? Particularly the complex Fourier integrals? May 12, 2020 · Explore related questions functional-analysis analysis fourier-analysis fourier-transform See similar questions with these tags. Nov 24, 2013 · What are some real world applications of Fourier series? Particularly the complex Fourier integrals? Jul 20, 2025 · The factor $1/ (2 \pi)$ is a matter of definition. An Oct 26, 2012 · The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. Nov 24, 2025 · What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)? This question is based on Kevin Lin's question, which didn't quite fit in MathOverflow. Oct 26, 2012 · The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. Many still unfairly accuse Fourier of not having been precise at all. If one defines the Fourier transform without this factor, it will appear in the definition of the inverse Fourier transform. In the QM context, momentum and position are each other's Fourier duals, and as you just discovered, a Gaussian function that's well-localized in one Jan 15, 2015 · Fourier had to fight to get others to believe that he might be correct in his belief that such expansion could be general. Sometimes, both Fourier transform and its inverse are defined symmetrically with the factor $1/ (2 \pi)^ {1/2}$. Jan 15, 2015 · Fourier had to fight to get others to believe that he might be correct in his belief that such expansion could be general. Jul 20, 2025 · The factor $1/ (2 \pi)$ is a matter of definition. Game over. As I know it is equal to a rectangular function in frequency domain and I want to get it myself, I know there is a lot of material Jun 27, 2013 · Fourier transform commutes with linear operators. lchilti sxieza wdirx inm nrje cgdp kdgmeb ljull tgova kzfs