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Wiener–Hopf method

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The Wiener–Hopf method is a mathematical technique widely used in <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>. It was initially developed by <a href="Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> and <a href="Eberhard_Hopf" title="Eberhard Hopf">Eberhard Hopf</a> as a method to solve systems of <a href="Integral_equation" title="Integral equation">integral equations</a>, but has found wider use in solving two-dimensional <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a> with mixed <a href="Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</a> on the same boundary. In general, the method works by exploiting the <a href="Complex_analysis" title="Complex analysis">complex-analytical</a> properties of transformed functions. Typically, the standard <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> is used, but examples exist using other transforms, such as the <a href="Mellin_transform" title="Mellin transform">Mellin transform</a>.

In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively <a href="Analytic_function" title="Analytic function">analytic</a> in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the <a href="Complex_plane" title="Complex plane">complex plane</a>, typically, a thin strip containing the <a href="Real_line" class="mw-redirect" title="Real line">real line</a>. <a href="Analytic_continuation" title="Analytic continuation">Analytic continuation</a> guarantees that these two functions define a single function analytic in the entire complex plane, and <a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a> implies that this function is an unknown <a href="Polynomial" title="Polynomial">polynomial</a>, which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial.

<div class="mw-heading mw-heading2"><h2 id="Wiener–Hopf_decomposition">Wiener–Hopf decomposition</h2></div>

The fundamental equation that appears in the Wiener-Hopf method is of the form

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<annotation encoding="application/x-tex">{\displaystyle A(\alpha )\Xi _{+}(\alpha )+B(\alpha )\Psi _{-}(\alpha )+C(\alpha )=0,}</annotation>

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</math><img src="./663849cfe791805339b9b1ad28548d7e58aeb14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.727ex; height:2.843ex;" alt="{\displaystyle A(\alpha )\Xi _{+}(\alpha )+B(\alpha )\Psi _{-}(\alpha )+C(\alpha )=0,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> are known <a href="Holomorphic_function" title="Holomorphic function">holomorphic functions</a>, the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi _{+}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle \Xi _{+}(\alpha )}</annotation>

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</math><img src="./4e1d078735171d40348055a36bfaea755299d934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.358ex; height:2.843ex;" alt="{\displaystyle \Xi _{+}(\alpha )}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{-}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{-}(\alpha )}</annotation>

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</math><img src="./046a19abd8363497e875f86880e8a56baa301761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.616ex; height:2.843ex;" alt="{\displaystyle \Psi _{-}(\alpha )}" loading="lazy"> are unknown and the equation holds in a strip <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{-}<{\mathfrak {Im}}(\alpha )<\tau _{+}}">

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<annotation encoding="application/x-tex">{\displaystyle \tau _{-}<{\mathfrak {Im}}(\alpha )<\tau _{+}}</annotation>

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</math><img src="./063606708d4f6e8694528b6c799d75c8c2d6c458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.618ex; height:2.843ex;" alt="{\displaystyle \tau _{-}<{\mathfrak {Im}}(\alpha )<\tau _{+}}" loading="lazy"> in the <a href="Complex_plane" title="Complex plane">complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> plane</a>. Finding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Xi _{+}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle \Xi _{+}(\alpha )}</annotation>

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</math><img src="./4e1d078735171d40348055a36bfaea755299d934.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.358ex; height:2.843ex;" alt="{\displaystyle \Xi _{+}(\alpha )}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{-}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi _{-}(\alpha )}</annotation>

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</math><img src="./046a19abd8363497e875f86880e8a56baa301761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.616ex; height:2.843ex;" alt="{\displaystyle \Psi _{-}(\alpha )}" loading="lazy"> is what's called the Wiener-Hopf problem.<a href="#cite_note-FOOTNOTENoble1958§4.2-1">[1]</a>

The key step in many Wiener–Hopf problems is to decompose an arbitrary function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

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<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>

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</math><img src="./aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"> into two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\pm }}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{\pm }}</annotation>

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</math><img src="./f1054af76dcfe9d556bba6c194ebaae9aa292d11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.189ex; height:2.509ex;" alt="{\displaystyle \Phi _{\pm }}" loading="lazy"> with the desired properties outlined above. In general, this can be done by writing

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{+}(\alpha )={\frac {1}{2\pi i}}\int _{C_{1}}\Phi (z){\frac {dz}{z-\alpha }}}</annotation>

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</math><img src="./80640ae3dc88e00d24095a52c70eb474782d6ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.851ex; height:6.176ex;" alt="{\displaystyle \Phi _{+}(\alpha )={\frac {1}{2\pi i}}\int _{C_{1}}\Phi (z){\frac {dz}{z-\alpha }}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{-}(\alpha )=-{\frac {1}{2\pi i}}\int _{C_{2}}\Phi (z){\frac {dz}{z-\alpha }},}</annotation>

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</math><img src="./ac241f46e5f3dbc40abf684fd8e3238e68060539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.306ex; height:6.176ex;" alt="{\displaystyle \Phi _{-}(\alpha )=-{\frac {1}{2\pi i}}\int _{C_{2}}\Phi (z){\frac {dz}{z-\alpha }},}" loading="lazy"></dd></dl>

where the contours <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation>

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</math><img src="./babf569931f1a7b5182b9bec51873c2f5692fbb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation>

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</math><img src="./7ec545f7870665e1028b7492746848d149878808.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}" loading="lazy"> are parallel to the real line, but pass above and below the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\alpha }">

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<annotation encoding="application/x-tex">{\displaystyle z=\alpha }</annotation>

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</math><img src="./491986a485643c367377a46c6e074ba8646ef584.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.674ex; height:1.676ex;" alt="{\displaystyle z=\alpha }" loading="lazy">, respectively.<a href="#cite_note-FOOTNOTENoble1958Chapter_1-2">[2]</a>

Similarly, arbitrary scalar functions may be decomposed into a product of +/− functions, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(\alpha )=K_{+}(\alpha )K_{-}(\alpha )}">

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<annotation encoding="application/x-tex">{\displaystyle K(\alpha )=K_{+}(\alpha )K_{-}(\alpha )}</annotation>

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</math><img src="./441809aec8b5a138134be4a6dd85547c6a699bb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.023ex; height:2.843ex;" alt="{\displaystyle K(\alpha )=K_{+}(\alpha )K_{-}(\alpha )}" loading="lazy">, by first taking the logarithm, and then performing a sum decomposition. Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed.

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

Consider the linear <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {L}}_{xy}f(x,y)=0,}</annotation>

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</math><img src="./49fe3a8ef5bd158cfe0b8c3a89b8f727f360ea22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.261ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {L}}_{xy}f(x,y)=0,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {L}}_{xy}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {L}}_{xy}}</annotation>

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</math><img src="./7e5028873f4d1d2c47e024adbd8425657cdefaa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.747ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {L}}_{xy}}" loading="lazy"> is a linear operator which contains derivatives with respect to x and y, subject to the mixed conditions on y = 0, for some prescribed function g(x),

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<annotation encoding="application/x-tex">{\displaystyle f=g(x){\text{ for }}x\leq 0,\quad f_{y}=0{\text{ when }}x>0}</annotation>

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</math><img src="./ab681abb5401002b8879b8425293deee9f1dfb5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.023ex; height:3.009ex;" alt="{\displaystyle f=g(x){\text{ for }}x\leq 0,\quad f_{y}=0{\text{ when }}x>0}" loading="lazy"></dd></dl>

and decay at infinity i.e. f → 0 as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}\rightarrow \infty }">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}\rightarrow \infty }</annotation>

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</math><img src="./9e451d2ffb84e8650c958e35c1eca8d2b54cdd1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.47ex; height:1.843ex;" alt="{\displaystyle {\boldsymbol {x}}\rightarrow \infty }" loading="lazy">.

Taking a <a href="Fourier_transform" title="Fourier transform">Fourier transform</a> with respect to x results in the following <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {L}}_{y}{\widehat {f}}(k,y)-P(k,y){\widehat {f}}(k,y)=0,}</annotation>

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</math><img src="./2ae489a838a62440280688dcaa4fee9aa4fa24c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.472ex; height:3.676ex;" alt="{\displaystyle {\boldsymbol {L}}_{y}{\widehat {f}}(k,y)-P(k,y){\widehat {f}}(k,y)=0,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {L}}_{y}}">

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</math><img src="./298718b818ec29d86a924dfcd79d1165cfe2f7b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.806ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {L}}_{y}}" loading="lazy"> is a linear operator containing y derivatives only, P(k,y) is a known function of y and k and

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(k,y)=\int _{-\infty }^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x.}</annotation>

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</math><img src="./8021a63dd02e79f28b0548b54e796ccf2e332e72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.135ex; height:6.009ex;" alt="{\displaystyle {\widehat {f}}(k,y)=\int _{-\infty }^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x.}" loading="lazy"></dd></dl>

If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity is denoted F(k,y), a general solution can be written as

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(k,y)=C(k)F(k,y),}</annotation>

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</math><img src="./fb401e69983daff6735f6b18535b1c78f59c12b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.464ex; height:3.509ex;" alt="{\displaystyle {\widehat {f}}(k,y)=C(k)F(k,y),}" loading="lazy"></dd></dl>

where C(k) is an unknown function to be determined by the boundary conditions on y=0.

The key idea is to split <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}}</annotation>

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</math><img src="./153a4a4d50ef7099fc5f8804e34dccc539f08743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.019ex; width:1.79ex; height:3.343ex;" alt="{\displaystyle {\widehat {f}}}" loading="lazy"> into two separate functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}_{+}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}_{+}}</annotation>

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</math><img src="./a0b910147070fc18c60ef2328d0bcc6287a13639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.282ex; height:3.676ex;" alt="{\displaystyle {\widehat {f}}_{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}_{-}}">

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}_{-}}</annotation>

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</math><img src="./93042efc125cd1dc30d448b79e553dd08dcfab99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.282ex; height:3.676ex;" alt="{\displaystyle {\widehat {f}}_{-}}" loading="lazy"> which are analytic in the lower- and upper-halves of the complex plane, respectively,

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}_{+}(k,y)=\int _{0}^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x,}</annotation>

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</math><img src="./ecbd73933671cc8bec6b990f3a54221035eeffe3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.538ex; height:5.843ex;" alt="{\displaystyle {\widehat {f}}_{+}(k,y)=\int _{0}^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}_{-}(k,y)=\int _{-\infty }^{0}f(x,y)e^{-ikx}\,{\textrm {d}}x.}</annotation>

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</math><img src="./a9e385cb42c1c055d17826e0c78458a48ba4e99a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.645ex; height:6.343ex;" alt="{\displaystyle {\widehat {f}}_{-}(k,y)=\int _{-\infty }^{0}f(x,y)e^{-ikx}\,{\textrm {d}}x.}" loading="lazy"></dd></dl>

The boundary conditions then give

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}_{-}(k,0)+{\widehat {f}}_{+}(k,0)={\widehat {f}}(k,0)=C(k)F(k,0)}</annotation>

</semantics>

</math><img src="./43b9b48f6d3378e009a729d27324e523d841c460.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.729ex; height:3.676ex;" alt="{\displaystyle {\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}_{-}(k,0)+{\widehat {f}}_{+}(k,0)={\widehat {f}}(k,0)=C(k)F(k,0)}" loading="lazy"></dd></dl>

and, on taking derivatives with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">,

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

<msup>

<mover>

</mover>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}'_{-}(k,0)={\widehat {f}}'_{-}(k,0)+{\widehat {f}}'_{+}(k,0)={\widehat {f}}'(k,0)=C(k)F'(k,0).}</annotation>

</semantics>

</math><img src="./70c4f1c5778712fcbe6d2057a42b94d5b09e55ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:58.455ex; height:3.843ex;" alt="{\displaystyle {\widehat {f}}'_{-}(k,0)={\widehat {f}}'_{-}(k,0)+{\widehat {f}}'_{+}(k,0)={\widehat {f}}'(k,0)=C(k)F'(k,0).}" loading="lazy"></dd></dl>

Eliminating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(k)}</annotation>

</semantics>

</math><img src="./c13e999a7597207a934d20b8c3284b626379da51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.787ex; height:2.843ex;" alt="{\displaystyle C(k)}" loading="lazy"> yields

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K(k),}</annotation>

</semantics>

</math><img src="./f1d8405d94fb2851591a234521b9fd5579ea3e30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.356ex; height:3.843ex;" alt="{\displaystyle {\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K(k),}" loading="lazy"></dd></dl>

where

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(k)={\frac {F'(k,0)}{F(k,0)}}.}</annotation>

</semantics>

</math><img src="./070d83d668084ee1f8ace1a5c833b814b8b2e503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.384ex; height:6.509ex;" alt="{\displaystyle K(k)={\frac {F'(k,0)}{F(k,0)}}.}" loading="lazy"></dd></dl>

Now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(k)}</annotation>

</semantics>

</math><img src="./38499eed5b7d12f7789d9e2547145f71ec15c5bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.086ex; height:2.843ex;" alt="{\displaystyle K(k)}" loading="lazy"> can be decomposed into the product of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{-}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{-}}</annotation>

</semantics>

</math><img src="./1e7dc8673d1420e13d6aa1bfe58243b4598c986e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.509ex;" alt="{\displaystyle K^{-}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{+}}</annotation>

</semantics>

</math><img src="./a10baad86115451d9e9b8e6b0cb0a3ad67e2c502.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.509ex;" alt="{\displaystyle K^{+}}" loading="lazy"> which are analytic in the upper and lower half-planes respectively.

To be precise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k)=K^{+}(k)K^{-}(k),}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(k)=K^{+}(k)K^{-}(k),}</annotation>

</semantics>

</math><img src="./9650912872281611583324a1665eb826d757b570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.082ex; height:3.009ex;" alt="{\displaystyle K(k)=K^{+}(k)K^{-}(k),}" loading="lazy"> where

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log K^{-}={\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k>0,}</annotation>

</semantics>

</math><img src="./1c2853d6d701234891236adb884bf0ac2ce83ffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.402ex; height:6.343ex;" alt="{\displaystyle \log K^{-}={\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k>0,}" loading="lazy"></dd>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \log K^{+}=-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k<0.}</annotation>

</semantics>

</math><img src="./2017c166941073664b4a1129f9a0a4afc64be243.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.21ex; height:6.343ex;" alt="{\displaystyle \log K^{+}=-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k<0.}" loading="lazy"></dd></dl>

(Note that this sometimes involves scaling <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> so that it tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\rightarrow \infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\rightarrow \infty }</annotation>

</semantics>

</math><img src="./612a3ec99f1c9f12de1cfab011e306ae799858ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.149ex; height:2.176ex;" alt="{\displaystyle k\rightarrow \infty }" loading="lazy">.) We also decompose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{+}{\widehat {g\,}}}">

<msup>

</mrow>

</msup>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{+}{\widehat {g\,}}}</annotation>

</semantics>

</math><img src="./993732a7eb822ee57cb6ca8850b7f500bb75785e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.108ex; height:2.843ex;" alt="{\displaystyle K^{+}{\widehat {g\,}}}" loading="lazy"> into the sum of two functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G^{+}}</annotation>

</semantics>

</math><img src="./ea64c8eaee51f08eecca685b15f7fed130db38fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.338ex; height:2.509ex;" alt="{\displaystyle G^{+}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{-}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G^{-}}</annotation>

</semantics>

</math><img src="./b17f1aad85ad42f94562d4202ac0e9f1b924b631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.338ex; height:2.509ex;" alt="{\displaystyle G^{-}}" loading="lazy"> which are analytic in the lower and upper half-planes respectively, i.e.,

<msup>

</mrow>

</msup>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K^{+}(k){\widehat {g\,}}(k)=G^{+}(k)+G^{-}(k).}</annotation>

</semantics>

</math><img src="./016d94a2342bbd11980a8c96bfb3edbe6789705f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.451ex; height:3.009ex;" alt="{\displaystyle K^{+}(k){\widehat {g\,}}(k)=G^{+}(k)+G^{-}(k).}" loading="lazy"></dd></dl>

This can be done in the same way that we factorised <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K(k).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(k).}</annotation>

</semantics>

</math><img src="./18bd22052567c2a75f0fbd498d346719dc14ff3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.733ex; height:2.843ex;" alt="{\displaystyle K(k).}" loading="lazy">

Consequently,

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G^{+}(k)+K_{+}(k){\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K_{-}(k)-G^{-}(k).}</annotation>

</semantics>

</math><img src="./609e2613ec4bc569bce5738b9ed7511a5fb1fe88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:53.311ex; height:3.843ex;" alt="{\displaystyle G^{+}(k)+K_{+}(k){\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K_{-}(k)-G^{-}(k).}" loading="lazy"></dd></dl>

Now, as the left-hand side of the above equation is analytic in the lower half-plane, whilst the right-hand side is analytic in the upper half-plane, analytic continuation guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large k, an application of <a href="Liouville's_theorem_(complex_analysis)" title="Liouville's theorem (complex analysis)">Liouville's theorem</a> shows that this entire function is identically zero, therefore

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}_{+}(k,0)=-{\frac {G^{+}(k)}{K^{+}(k)}},}</annotation>

</semantics>

</math><img src="./c425ef4b96c36fa31ab507cfd388fcf4dae75fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.514ex; height:6.509ex;" alt="{\displaystyle {\widehat {f}}_{+}(k,0)=-{\frac {G^{+}(k)}{K^{+}(k)}},}" loading="lazy"></dd></dl>

and so

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(k)={\frac {K^{+}(k){\widehat {g\,}}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}}.}</annotation>

</semantics>

</math><img src="./f9fd3ec455cf3edef0640f90e690ab347bfaad84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.716ex; height:6.509ex;" alt="{\displaystyle C(k)={\frac {K^{+}(k){\widehat {g\,}}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}}.}" loading="lazy"></dd></dl>

<ul><li><a href="Wiener_filter" title="Wiener filter">Wiener filter</a></li>

<li><a href="Riemann%E2%80%93Hilbert_problem" title="Riemann–Hilbert problem">Riemann–Hilbert problem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-FOOTNOTENoble1958§4.2-1"><a href="#cite_ref-FOOTNOTENoble1958§4.2_1-0">^</a> <a href="#CITEREFNoble1958">Noble 1958</a>, §4.2.

</li>

<li id="cite_note-FOOTNOTENoble1958Chapter_1-2"><a href="#cite_ref-FOOTNOTENoble1958Chapter_1_2-0">^</a> <a href="#CITEREFNoble1958">Noble 1958</a>, Chapter 1.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://wikiwaves.org/Category:Wiener-Hopf">"Category:Wiener-Hopf - WikiWaves"</a>. wikiwaves.org. Retrieved 2020-05-19.</cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Wiener-Hopf_method">"Wiener-Hopf method"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite id="CITEREFFornberg,_Bengt2020" class="citation book cs1">Fornberg, Bengt (2020). Complex variables and analytic functions : an illustrated introduction. Piret, Cécile. Philadelphia. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61197-597-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1124781689">1124781689</a>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: location missing publisher (link)</li>

<li><cite id="CITEREFNoble1958" class="citation book cs1">Noble, Ben (1958). Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. New York, N.Y: Taylor & Francis US. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8284-0332-0</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-18" href="https://en.wikipedia.org/wiki/?title=Wiener%E2%80%93Hopf_method&oldid=1301258479">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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