Contents

Hilbert transform

The Hilbert transform of u can be thought of as the convolutionofu(t) with the function h(t) = ⁠1/πt⁠, known as the Cauchy kernel. Because 1/t is not integrable across t = 0, the integral defining the convolution does not always converge. Instead, the Hilbert transform is defined using the Cauchy principal value (denoted here by p.v.). Explicitly, the Hilbert transform of a function (or signal) u(t) is given by

$${\displaystyle \operatorname {H} ($$u)(t)={\frac {1}{\pi }}\,\operatorname {p.v.} \int _{-\infty }^{+\infty }{\frac {u(\tau )}{t-\tau }}\,\mathrm {d} \tau ,}

provided this integral exists as a principal value. This is precisely the convolution of u with the tempered distribution p.v. ⁠1/πt⁠.^[1] Alternatively, by changing variables, the principal-value integral can be written explicitly^[2]as

$${\displaystyle \operatorname {H} ($$u)(t)={\frac {2}{\pi }}\,\lim _{\varepsilon \to 0}\int _{\varepsilon }^{\infty }{\frac {u(t-\tau )-u(t+\tau )}{2\tau }}\,\mathrm {d} \tau .}

When the Hilbert transform is applied twice in succession to a function u, the result is

$${\displaystyle \operatorname {H} {\bigl (}\operatorname {H} ($$u){\bigr )}(t)=-u(t),}

provided the integrals defining both iterations converge in a suitable sense. In particular, the inverse transform is ${\displaystyle -\operatorname {H} }$. This fact can most easily be seen by considering the effect of the Hilbert transform on the Fourier transformofu(t) (see § Relationship with the Fourier transform below).

For an analytic function in the upper half-plane, the Hilbert transform describes the relationship between the real part and the imaginary part of the boundary values. That is, if f(z) is analytic in the upper half complex plane {z : Im{z} > 0}, and u(t) = Re{f (t + 0·i)}, then Im{f(t + 0·i)} = H(u)(t) up to an additive constant, provided this Hilbert transform exists.

Insignal processing the Hilbert transform of u(t) is commonly denoted by ${\displaystyle {\hat {u}}($t)} .^[3] However, in mathematics, this notation is already extensively used to denote the Fourier transform of u(t).^[4] Occasionally, the Hilbert transform may be denoted by ${\displaystyle {\tilde {u}}($t)} . Furthermore, many sources define the Hilbert transform as the negative of the one defined here.^[5]

The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions,^[6][7] which has come to be known as the Riemann–Hilbert problem. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle.^[8][9] Some of his earlier work related to the Discrete Hilbert Transform dates back to lectures he gave in Göttingen. The results were later published by Hermann Weyl in his dissertation.^[10] Schur improved Hilbert's results about the discrete Hilbert transform and extended them to the integral case.^[11] These results were restricted to the spaces L² and ℓ². In 1928, Marcel Riesz proved that the Hilbert transform can be defined for uin${\displaystyle L^{p}(\mathbb {R} )}$ (L^p space) for 1 < p < ∞, that the Hilbert transform is a bounded operatoron${\displaystyle L^{p}(\mathbb {R} )}$ for 1 < p < ∞, and that similar results hold for the Hilbert transform on the circle as well as the discrete Hilbert transform.^[12] The Hilbert transform was a motivating example for Antoni Zygmund and Alberto Calderón during their study of singular integrals.^[13] Their investigations have played a fundamental role in modern harmonic analysis. Various generalizations of the Hilbert transform, such as the bilinear and trilinear Hilbert transforms are still active areas of research today.

The Hilbert transform is a multiplier operator.^[14] The multiplier of Hisσ_H(ω) = −i sgn(ω), where sgn is the signum function. Therefore:

$${\displaystyle {\mathcal {F}}{\bigl (}\operatorname {H} ($$u){\bigr )}(\omega )=-i\operatorname {sgn}(\omega )\cdot {\mathcal {F}}(u)(\omega ),}

where ${\displaystyle {\mathcal {F}}}$ denotes the Fourier transform. Since sgn(x) = sgn(2πx), it follows that this result applies to the three common definitions of ${\displaystyle {\mathcal {F}}}$.

ByEuler's formula, $${\displaystyle \sigma _{\operatorname {H} }(\omega )={\begin{cases}~~i=e^{+i\pi /2},&{\text{for }}\omega <0,\\~~0,&{\text{for }}\omega =0,\\-i=e^{-i\pi /2},&{\text{for }}\omega >0.\end{cases}}}$$

Therefore, H(u)(t) has the effect of shifting the phase of the negative frequency components of u(t) by +90° (π⁄2 radians) and the phase of the positive frequency components by −90°, and i·H(u)(t) has the effect of restoring the positive frequency components while shifting the negative frequency ones an additional +90°, resulting in their negation (i.e., a multiplication by −1).

When the Hilbert transform is applied twice, the phase of the negative and positive frequency components of u(t) are respectively shifted by +180° and −180°, which are equivalent amounts. The signal is negated; i.e., H(H(u)) = −u, because

$${\displaystyle \left(\sigma _{\operatorname {H} }(\omega )\right)^{2}=e^{\pm i\pi }=-1\quad {\text{for }}\omega \neq 0.}$$

In the following table, the frequency parameter ${\displaystyle \omega }$ is real.

Signal ${\displaystyle u($t)}	Hilbert transform[fn 1] ${\displaystyle \operatorname {H} ($u)(t)}
${\displaystyle \sin(\omega t+\varphi )}$ [fn 2]	${\displaystyle {\begin{array}{lll}\sin \left(\omega t+\varphi -{\tfrac {\pi }{2}}\right)=-\cos \left(\omega t+\varphi \right),\quad \omega >0\\\sin \left(\omega t+\varphi +{\tfrac {\pi }{2}}\right)=\cos \left(\omega t+\varphi \right),\quad \omega <0\end{array}}}$
${\displaystyle \cos(\omega t+\varphi )}$ [fn 2]	${\displaystyle {\begin{array}{lll}\cos \left(\omega t+\varphi -{\tfrac {\pi }{2}}\right)=\sin \left(\omega t+\varphi \right),\quad \omega >0\\\cos \left(\omega t+\varphi +{\tfrac {\pi }{2}}\right)=-\sin \left(\omega t+\varphi \right),\quad \omega <0\end{array}}}$
${\displaystyle e^{i\omega t}}$	${\displaystyle {\begin{array}{lll}e^{i\left(\omega t-{\tfrac {\pi }{2}}\right)},\quad \omega >0\\e^{i\left(\omega t+{\tfrac {\pi }{2}}\right)},\quad \omega <0\end{array}}}$
${\displaystyle e^{-i\omega t}}$	${\displaystyle {\begin{array}{lll}e^{-i\left(\omega t-{\tfrac {\pi }{2}}\right)},\quad \omega >0\\e^{-i\left(\omega t+{\tfrac {\pi }{2}}\right)},\quad \omega <0\end{array}}}$
${\displaystyle 1 \over t^{2}+1}$	${\displaystyle t \over t^{2}+1}$
${\displaystyle e^{-t^{2}}}$	${\displaystyle {\frac {2}{\sqrt {\pi \,}}}F($t)} (see Dawson function)
Sinc function ${\displaystyle \sin($t) \over t}	${\displaystyle 1-\cos($t) \over t}
Dirac delta function ${\displaystyle \delta ($t)}	${\displaystyle {1 \over \pi t}}$
Characteristic function ${\displaystyle \chi _{[a,b]}($t)}	${\displaystyle {{\frac {1}{\,\pi \,}}\ln \left\vert {\frac {t-a}{t-b}}\right\vert }}$

Notes

An extensive table of Hilbert transforms is available.^[15] Note that the Hilbert transform of a constant is zero.

It is by no means obvious that the Hilbert transform is well-defined at all, as the improper integral defining it must converge in a suitable sense. However, the Hilbert transform is well-defined for a broad class of functions, namely those in ${\displaystyle L^{p}(\mathbb {R} )}$ for 1 < p < ∞.

More precisely, if u is in ${\displaystyle L^{p}(\mathbb {R} )}$ for 1 < p < ∞, then the limit defining the improper integral

$${\displaystyle \operatorname {H} ($$u)(t)={\frac {2}{\pi }}\lim _{\varepsilon \to 0}\int _{\varepsilon }^{\infty }{\frac {u(t-\tau )-u(t+\tau )}{2\tau }}\,d\tau }

exists for almost every t. The limit function is also in ${\displaystyle L^{p}(\mathbb {R} )}$ and is in fact the limit in the mean of the improper integral as well. That is,

$${\displaystyle {\frac {2}{\pi }}\int _{\varepsilon }^{\infty }{\frac {u(t-\tau )-u(t+\tau )}{2\tau }}\,\mathrm {d} \tau \to \operatorname {H} ($$u)(t)}

asε → 0 in the L^p norm, as well as pointwise almost everywhere, by the Titchmarsh theorem.^[16]

In the case p = 1, the Hilbert transform still converges pointwise almost everywhere, but may itself fail to be integrable, even locally.^[17] In particular, convergence in the mean does not in general happen in this case. The Hilbert transform of an L¹ function does converge, however, in L¹-weak, and the Hilbert transform is a bounded operator from L¹toL^1,w.^[18] (In particular, since the Hilbert transform is also a multiplier operator on L², Marcinkiewicz interpolation and a duality argument furnishes an alternative proof that H is bounded on L^p.)

If1 < p < ∞, then the Hilbert transform on ${\displaystyle L^{p}(\mathbb {R} )}$ is a bounded linear operator, meaning that there exists a constant C_p such that

$${\displaystyle \left\|\operatorname {H} u\right\|_{p}\leq C_{p}\left\|u\right\|_{p}}$$

for all ${\displaystyle u\in L^{p}(\mathbb {R} )}$.^[19]

The best constant ${\displaystyle C_{p}}$ is given by^[20] $${\displaystyle C_{p}={\begin{cases}\tan {\frac {\pi }{2p}}&{\text{for}}~1<p\leq 2,\\[4pt]\cot {\frac {\pi }{2p}}&{\text{for}}~2<p<\infty .\end{cases}}}$$

An easy way to find the best ${\displaystyle C_{p}}$ for ${\displaystyle p}$ being a power of 2 is through the so-called Cotlar's identity that ${\displaystyle (\operatorname {H} f)^{2}=f^{2}+2\operatorname {H} (f\operatorname {H} f)}$ for all real valued f. The same best constants hold for the periodic Hilbert transform.

The boundedness of the Hilbert transform implies the ${\displaystyle L^{p}(\mathbb {R} )}$ convergence of the symmetric partial sum operator $${\displaystyle S_{R}f=\int _{-R}^{R}{\hat {f}}(\xi )e^{2\pi ix\xi }\,\mathrm {d} \xi }$$

tofin${\displaystyle L^{p}(\mathbb {R} )}$.^[21]

The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between ${\displaystyle L^{p}(\mathbb {R} )}$ and the dual space ${\displaystyle L^{q}(\mathbb {R} )}$, where p and q are Hölder conjugates and 1 < p, q < ∞. Symbolically,

$${\displaystyle \langle \operatorname {H} u,v\rangle =\langle u,-\operatorname {H} v\rangle }$$

for ${\displaystyle u\in L^{p}(\mathbb {R} )}$ and ${\displaystyle v\in L^{q}(\mathbb {R} )}$.^[22]

The Hilbert transform is an anti-involution,^[23] meaning that

$${\displaystyle \operatorname {H} {\bigl (}\operatorname {H} \left(u\right){\bigr )}=-u}$$

provided each transform is well-defined. Since H preserves the space ${\displaystyle L^{p}(\mathbb {R} )}$, this implies in particular that the Hilbert transform is invertible on ${\displaystyle L^{p}(\mathbb {R} )}$, and that

$${\displaystyle \operatorname {H} ^{-1}=-\operatorname {H} }$$

Because H² = −I ("I" is the identity operator) on the real Banach spaceofreal-valued functions in ${\displaystyle L^{p}(\mathbb {R} )}$, the Hilbert transform defines a linear complex structure on this Banach space. In particular, when p = 2, the Hilbert transform gives the Hilbert space of real-valued functions in ${\displaystyle L^{2}(\mathbb {R} )}$ the structure of a complex Hilbert space.

The (complex) eigenstates of the Hilbert transform admit representations as holomorphic functions in the upper and lower half-planes in the Hardy space H² by the Paley–Wiener theorem.

Formally, the derivative of the Hilbert transform is the Hilbert transform of the derivative, i.e. these two linear operators commute:

$${\displaystyle \operatorname {H} \left({\frac {\mathrm {d} u}{\mathrm {d} t}}\right)={\frac {\mathrm {d} }{\mathrm {d} t}}\operatorname {H} ($$u)}

Iterating this identity,

$${\displaystyle \operatorname {H} \left({\frac {\mathrm {d} ^{k}u}{\mathrm {d} t^{k}}}\right)={\frac {\mathrm {d} ^{k}}{\mathrm {d} t^{k}}}\operatorname {H} ($$u)}

This is rigorously true as stated provided u and its first k derivatives belong to ${\displaystyle L^{p}(\mathbb {R} )}$.^[24] One can check this easily in the frequency domain, where differentiation becomes multiplication by ω.

The Hilbert transform can formally be realized as a convolution with the tempered distribution^[25]

$${\displaystyle h($$t)=\operatorname {p.v.} {\frac {1}{\pi \,t}}}

Thus formally,

$${\displaystyle \operatorname {H} ($$u)=h*u}

However, a priori this may only be defined for u a distribution of compact support. It is possible to work somewhat rigorously with this since compactly supported functions (which are distributions a fortiori) are denseinL^p. Alternatively, one may use the fact that h(t) is the distributional derivative of the function log|t|/π; to wit

$${\displaystyle \operatorname {H} ($$u)(t)={\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {1}{\pi }}\left(u*\log {\bigl |}\cdot {\bigr |}\right)(t)\right)}

For most operational purposes the Hilbert transform can be treated as a convolution. For example, in a formal sense, the Hilbert transform of a convolution is the convolution of the Hilbert transform applied on only one of either of the factors:

$${\displaystyle \operatorname {H} (u*v)=\operatorname {H} ($$u)*v=u*\operatorname {H} (v)}

This is rigorously true if u and v are compactly supported distributions since, in that case,

$${\displaystyle h*(u*v)=(h*u)*v=u*(h*v)}$$

By passing to an appropriate limit, it is thus also true if u ∈ L^p and v ∈ L^q provided that

$${\displaystyle 1<{\frac {1}{p}}+{\frac {1}{q}}}$$

from a theorem due to Titchmarsh.^[26]

The Hilbert transform has the following invariance properties on ${\displaystyle L^{2}(\mathbb {R} )}$.

• It commutes with translations. That is, it commutes with the operators T_a f(x) = f(x + a) for all ain${\displaystyle \mathbb {R} .}$
• It commutes with positive dilations. That is it commutes with the operators M_λf (x) = f (λ x) for all λ > 0.
• Itanticommutes with the reflection R f (x) = f (−x).

Up to a multiplicative constant, the Hilbert transform is the only bounded operator on L² with these properties.^[27]

In fact there is a wider set of operators that commute with the Hilbert transform. The group ${\displaystyle {\text{SL}}(2,\mathbb {R} )}$ acts by unitary operators U_g on the space ${\displaystyle L^{2}(\mathbb {R} )}$ by the formula

$${\displaystyle \operatorname {U} _{g}^{-1}f($$x)={\frac {1}{cx+d}}\,f\left({\frac {ax+b}{cx+d}}\right)\,,\qquad g={\begin{bmatrix}a&b\\c&d\end{bmatrix}}~,\qquad {\text{ for }}~ad-bc=\pm 1.}

This unitary representation is an example of a principal series representationof${\displaystyle ~{\text{SL}}(2,\mathbb {R} )~.}$ In this case it is reducible, splitting as the orthogonal sum of two invariant subspaces, Hardy space ${\displaystyle H^{2}(\mathbb {R} )}$ and its conjugate. These are the spaces of L² boundary values of holomorphic functions on the upper and lower halfplanes. ${\displaystyle H^{2}(\mathbb {R} )}$ and its conjugate consist of exactly those L² functions with Fourier transforms vanishing on the negative and positive parts of the real axis respectively. Since the Hilbert transform is equal to H = −i (2P − I), with P being the orthogonal projection from ${\displaystyle L^{2}(\mathbb {R} )}$ onto ${\displaystyle \operatorname {H} ^{2}(\mathbb {R} ),}$ and I the identity operator, it follows that ${\displaystyle \operatorname {H} ^{2}(\mathbb {R} )}$ and its orthogonal complement are eigenspaces of H for the eigenvalues ±i. In other words, H commutes with the operators U_g. The restrictions of the operators U_gto${\displaystyle \operatorname {H} ^{2}(\mathbb {R} )}$ and its conjugate give irreducible representations of ${\displaystyle {\text{SL}}(2,\mathbb {R} )}$ – the so-called limit of discrete series representations.^[28]

It is further possible to extend the Hilbert transform to certain spaces of distributions (Pandey 1996, Chapter 3). Since the Hilbert transform commutes with differentiation, and is a bounded operator on L^p, H restricts to give a continuous transform on the inverse limitofSobolev spaces:

$${\displaystyle {\mathcal {D}}_{L^{p}}={\underset {n\to \infty }{\underset {\longleftarrow }{\lim }}}W^{n,p}(\mathbb {R} )}$$

The Hilbert transform can then be defined on the dual space of ${\displaystyle {\mathcal {D}}_{L^{p}}}$, denoted ${\displaystyle {\mathcal {D}}_{L^{p}}'}$, consisting of L^p distributions. This is accomplished by the duality pairing:
For ${\displaystyle u\in {\mathcal {D}}'_{L^{p}}}$, define:

$${\displaystyle \operatorname {H} ($$u)\in {\mathcal {D}}'_{L^{p}}=\langle \operatorname {H} u,v\rangle \ \triangleq \ \langle u,-\operatorname {H} v\rangle ,\ {\text{for all}}\ v\in {\mathcal {D}}_{L^{p}}.}

It is possible to define the Hilbert transform on the space of tempered distributions as well by an approach due to Gel'fand and Shilov,^[29] but considerably more care is needed because of the singularity in the integral.

The Hilbert transform can be defined for functions in ${\displaystyle L^{\infty }(\mathbb {R} )}$ as well, but it requires some modifications and caveats. Properly understood, the Hilbert transform maps ${\displaystyle L^{\infty }(\mathbb {R} )}$ to the Banach spaceofbounded mean oscillation (BMO) classes.

Interpreted naïvely, the Hilbert transform of a bounded function is clearly ill-defined. For instance, with u = sgn(x), the integral defining H(u) diverges almost everywhere to ±∞. To alleviate such difficulties, the Hilbert transform of an L^∞ function is therefore defined by the following regularized form of the integral

$${\displaystyle \operatorname {H} ($$u)(t)=\operatorname {p.v.} \int _{-\infty }^{\infty }u(\tau )\left\{h(t-\tau )-h_{0}(-\tau )\right\}\,\mathrm {d} \tau }

where as above h(x) = ⁠1/πx⁠ and

$${\displaystyle h_{0}($$x)={\begin{cases}0&{\text{for}}~|x|<1\\{\frac {1}{\pi \,x}}&{\text{for}}~|x|\geq 1\end{cases}}}

The modified transform H agrees with the original transform up to an additive constant on functions of compact support from a general result by Calderón and Zygmund.^[30] Furthermore, the resulting integral converges pointwise almost everywhere, and with respect to the BMO norm, to a function of bounded mean oscillation.

Adeep result of Fefferman's work^[31] is that a function is of bounded mean oscillation if and only if it has the form f + H(g) for some ${\displaystyle f,g\in L^{\infty }(\mathbb {R} )}$.

The Hilbert transform can be understood in terms of a pair of functions f(x) and g(x) such that the function $${\displaystyle F($$x)=f(x)+i\,g(x)} is the boundary value of a holomorphic function F(z) in the upper half-plane.^[32] Under these circumstances, if f and g are sufficiently integrable, then one is the Hilbert transform of the other.

Suppose that ${\displaystyle f\in L^{p}(\mathbb {R} ).}$ Then, by the theory of the Poisson integral, f admits a unique harmonic extension into the upper half-plane, and this extension is given by

$${\displaystyle u(x+iy)=u(x,y)={\frac {1}{\pi }}\int _{-\infty }^{\infty }f($$s)\;{\frac {y}{(x-s)^{2}+y^{2}}}\;\mathrm {d} s}

which is the convolution of f with the Poisson kernel

$${\displaystyle P(x,y)={\frac {y}{\pi \,\left(x^{2}+y^{2}\right)}}}$$

Furthermore, there is a unique harmonic function v defined in the upper half-plane such that F(z) = u(z) + i v(z) is holomorphic and $${\displaystyle \lim _{y\to \infty }v\,(x+i\,y)=0}$$

This harmonic function is obtained from f by taking a convolution with the conjugate Poisson kernel

$${\displaystyle Q(x,y)={\frac {x}{\pi \,\left(x^{2}+y^{2}\right)}}.}$$

Thus $${\displaystyle v(x,y)={\frac {1}{\pi }}\int _{-\infty }^{\infty }f($$s)\;{\frac {x-s}{\,(x-s)^{2}+y^{2}\,}}\;\mathrm {d} s.}

Indeed, the real and imaginary parts of the Cauchy kernel are $${\displaystyle {\frac {i}{\pi \,z}}=P(x,y)+i\,Q(x,y)}$$

so that F = u + i v is holomorphic by Cauchy's integral formula.

The function v obtained from u in this way is called the harmonic conjugateofu. The (non-tangential) boundary limit of v(x,y)asy → 0 is the Hilbert transform of f. Thus, succinctly, $${\displaystyle \operatorname {H} ($$f)=\lim _{y\to 0}Q(-,y)\star f}

Titchmarsh's theorem (named for E. C. Titchmarsh who included it in his 1937 work) makes precise the relationship between the boundary values of holomorphic functions in the upper half-plane and the Hilbert transform.^[33] It gives necessary and sufficient conditions for a complex-valued square-integrable function F(x) on the real line to be the boundary value of a function in the Hardy space H²(U) of holomorphic functions in the upper half-plane U.

The theorem states that the following conditions for a complex-valued square-integrable function ${\displaystyle F:\mathbb {R} \to \mathbb {C} }$ are equivalent:

• F(x) is the limit as z → x of a holomorphic function F(z) in the upper half-plane such that $${\displaystyle \int _{-\infty }^{\infty }|F(x+i\,y)|^{2}\;\mathrm {d} x<K}$$
• The real and imaginary parts of F(x) are Hilbert transforms of each other.
• The Fourier transform ${\displaystyle {\mathcal {F}}($F)(x)} vanishes for x < 0.

A weaker result is true for functions of class L^p for p >1.^[34] Specifically, if F(z) is a holomorphic function such that

$${\displaystyle \int _{-\infty }^{\infty }|F(x+i\,y)|^{p}\;\mathrm {d} x<K}$$

for all y, then there is a complex-valued function F(x)in${\displaystyle L^{p}(\mathbb {R} )}$ such that F(x + i y) → F(x) in the L^p norm as y → 0 (as well as holding pointwise almost everywhere). Furthermore,

$${\displaystyle F($$x)=f(x)-i\,g(x)}

where f is a real-valued function in ${\displaystyle L^{p}(\mathbb {R} )}$ and g is the Hilbert transform (of class L^p) of f.

This is not true in the case p = 1. In fact, the Hilbert transform of an L¹ function f need not converge in the mean to another L¹ function. Nevertheless,^[35] the Hilbert transform of f does converge almost everywhere to a finite function g such that

$${\displaystyle \int _{-\infty }^{\infty }{\frac {|g($$x)|^{p}}{1+x^{2}}}\;\mathrm {d} x<\infty }

This result is directly analogous to one by Andrey Kolmogorov for Hardy functions in the disc.^[36] Although usually called Titchmarsh's theorem, the result aggregates much work of others, including Hardy, Paley and Wiener (see Paley–Wiener theorem), as well as work by Riesz, Hille, and Tamarkin^[37]

One form of the Riemann–Hilbert problem seeks to identify pairs of functions F₊ and F₋ such that F₊isholomorphic on the upper half-plane and F₋ is holomorphic on the lower half-plane, such that for x along the real axis, $${\displaystyle F_{+}($$x)-F_{-}(x)=f(x)}

where f(x) is some given real-valued function of ${\displaystyle x\in \mathbb {R} }$. The left-hand side of this equation may be understood either as the difference of the limits of F_± from the appropriate half-planes, or as a hyperfunction distribution. Two functions of this form are a solution of the Riemann–Hilbert problem.

Formally, if F_± solve the Riemann–Hilbert problem $${\displaystyle f($$x)=F_{+}(x)-F_{-}(x)}

then the Hilbert transform of f(x) is given by^[38] $${\displaystyle H($$f)(x)=-i{\bigl (}F_{+}(x)+F_{-}(x){\bigr )}.}