# Hilbert transform

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## 1—10 of 15 matching pages

##### 1: 1.14 Integral Transforms
###### §1.14(v) HilbertTransform
The Hilbert transform of a real-valued function $f(t)$ is defined in the following equivalent ways: …
###### Fourier Transform
A special case is the rule for Hilbert transforms1.14(v)):
3.5.46 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)=\frac{1}{\pi}\pvint_{-% \infty}^{\infty}\frac{f(t)}{t-x}\,\mathrm{d}t,$ $x\in\mathbb{R}$,
##### 3: Bibliography F
• B. D. Fried and S. D. Conte (1961) The Plasma Dispersion Function: The Hilbert Transform of the Gaussian. Academic Press, London-New York.
• ##### 4: Errata
• Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

• ##### 5: 1.15 Summability Methods
Moreover, $\lim_{y\to 0+}\Im\Phi(x+\mathrm{i}y)$ is the Hilbert transform of $f(x)$1.14(v)). …
##### 6: Bibliography S
• M. H. Stone (1990) Linear transformations in Hilbert space. American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.
• ##### 7: 18.38 Mathematical Applications
It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. …
##### 8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### §1.18(iii) Linear Operators on a Hilbert Space
In the following let $V$ be a Hilbert space. …