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Related Concept Videos

Fast Fourier Transform01:10

Fast Fourier Transform

The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
The computational efficiency of the FFT becomes...
Transformations of Functions III01:20

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Transformations modify the graphical representation of a function without changing its fundamental form. One common transformation is reflection, which flips the graph across a designated axis. When the vertical coordinates of all points are multiplied by the negative one, the entire graph is mirrored over the horizontal axis. This transformation reverses the vertical orientation of peaks and troughs, akin to signal inversion in electrical systems, where a waveform is flipped, but the timing of...
Transformations of Functions II01:29

Transformations of Functions II

Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c, where c is a constant.
Continuous -time Fourier Transform01:11

Continuous -time Fourier Transform

The Fourier series is instrumental in representing periodic functions, offering a powerful method to decompose such functions into a sum of sinusoids. This technique, however, necessitates modification when applied to nonperiodic functions. Consider a pulse-train waveform consisting of a series of rectangular pulses. When these pulses have a finite period, they can be accurately represented by a Fourier series. Yet, as the period approaches infinity, resulting in a single, isolated pulse, the...
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Microbial communities are dynamic environments where cell lysis releases free DNA into the surroundings. Other cells can take up this extracellular DNA through a process known as transformation.When a cell incorporates this foreign DNA into its genome, resulting in genetic modification, the process is known as transformation. Cells capable of this process are termed competent. Competence can be natural, as observed in certain bacteria and archaea, or artificially induced in the...
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The Fourier Transform is a pivotal mathematical tool in signal processing, enabling the transformation of time-domain signals into their frequency-domain representations. Among the numerous elements within this domain, certain functions like the sinc function, delta function, and exponential signals hold significant importance due to their unique properties and implications.
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A Multimodal Wide-Field Fourier-Transform Raman Microscope
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Published on: December 30, 2025

A Fast Hermite Transform.

Gregory Leibon1, Daniel N Rockmore, Wooram Park

  • 1Department of Mathematics, Dartmouth College, Hanover, NH 03755.

Theoretical Computer Science
|December 23, 2009
PubMed
Summary
This summary is machine-generated.

We developed fast algorithms for approximating the Hermite transform of functions. This method enables efficient computation for applications like protein structure determination via tomographic reconstruction.

Related Experiment Videos

Last Updated: Jun 17, 2026

A Multimodal Wide-Field Fourier-Transform Raman Microscope
06:48

A Multimodal Wide-Field Fourier-Transform Raman Microscope

Published on: December 30, 2025

Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Computational Physics

Background:

  • The Hermite transform is crucial in various scientific fields.
  • Efficient computation of the Hermite transform is often limited by computational complexity.
  • Approximation methods are needed for practical applications.

Purpose of the Study:

  • To present novel algorithms for fast and stable approximation of the Hermite transform.
  • To address the trade-offs between bandlimit approximation and support region size.
  • To demonstrate the utility of the proposed algorithms in scientific applications.

Main Methods:

  • Development of fast algebraic algorithms for sums associated with a three-term relation.
  • Application of these algorithms to approximate the Hermite transform of compactly supported functions.
  • Analysis of approximation accuracy concerning bandlimit and support region.

Main Results:

  • Fast and stable approximation algorithms for the Hermite transform are presented.
  • Numerical experiments confirm the feasibility and utility of the developed approach.
  • The algorithms offer controllable trade-offs between approximation accuracy and computational cost.

Conclusions:

  • The proposed algorithms provide an efficient method for Hermite transform approximation.
  • The approach is generalizable to other orthogonal polynomial families.
  • Significant potential for applications in tomographic reconstruction and protein structure determination.