ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Positive-definite function</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>positive-definite function</b> is, depending on the context, either of two types of <a href="Function_(mathematics)" title="Function (mathematics)">function</a>.

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<div class="mw-heading mw-heading2"><h2 id="Definition_1">Definition 1</h2></div>

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

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</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.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 \mathbb {R} }" loading="lazy"></span> be the set of <a href="Real_number" title="Real number">real numbers</a> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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

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</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.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 \mathbb {C} }" loading="lazy"></span> be the set of <a href="Complex_number" title="Complex number">complex numbers</a>.

</p><p>A function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {C} }</annotation>

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</math></span><img src="./56fd4f6d5889adc68cfb7e6043cfc3cf8d0dd258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.186ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {C} }" loading="lazy"></span> is called <i>positive semi-definite</i> if for all real numbers <i>x</i><sub>1</sub>, …, <i>x</i><sub><i>n</i></sub> the <i>n</i> × <i>n</i> <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>

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<annotation encoding="application/x-tex">{\displaystyle A=\left(a_{ij}\right)_{i,j=1}^{n}~,\quad a_{ij}=f(x_{i}-x_{j})}</annotation>

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</math></span><img src="./28f63fd6ae633b01d96902c82a17d9239e43bbe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:33.432ex; height:3.509ex;" alt="{\displaystyle A=\left(a_{ij}\right)_{i,j=1}^{n}~,\quad a_{ij}=f(x_{i}-x_{j})}" loading="lazy"></span></dd></dl>

<p>is a <a href="Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive <i>semi-</i>definite matrix</a>.

</p><p>By definition, a positive semi-definite matrix, such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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"></span>, is <a href="Hermitian_matrix" title="Hermitian matrix">Hermitian</a>; therefore <i>f</i>(−<i>x</i>) is the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <i>f</i>(<i>x</i>)).

</p><p>In particular, it is necessary (but not sufficient) that

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<annotation encoding="application/x-tex">{\displaystyle f(0)\geq 0~,\quad |f(x)|\leq f(0)}</annotation>

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</math></span><img src="./58ba25e6236ad76344d333402785632b246bf7c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.508ex; height:2.843ex;" alt="{\displaystyle f(0)\geq 0~,\quad |f(x)|\leq f(0)}" loading="lazy"></span></dd></dl>

<p>(these inequalities follow from the condition for <i>n</i> = 1, 2.)

</p><p>A function is <i>negative semi-definite</i> if the inequality is reversed. A function is <i>definite</i> if the weak inequality is replaced with a strong (<, > 0).

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<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle (X,\langle \cdot ,\cdot \rangle )}</annotation>

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</math></span><img src="./47fa76665db6afe71aaa5b62a292cabdd5848bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.96ex; height:2.843ex;" alt="{\displaystyle (X,\langle \cdot ,\cdot \rangle )}" loading="lazy"></span> is a real <a href="Inner_product_space" title="Inner product space">inner product space</a>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{y}\colon X\to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle g_{y}\colon X\to \mathbb {C} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle x\mapsto \exp(i\langle y,x\rangle )}</annotation>

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</math></span><img src="./914f54a50e367266f73fc567507476bb21bea3de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.436ex; height:2.843ex;" alt="{\displaystyle x\mapsto \exp(i\langle y,x\rangle )}" loading="lazy"></span> is positive definite for every <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in X}">

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</math></span><img src="./6015f751d0278b3aa3b5e4c33740456f08e888b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.976ex; height:2.509ex;" alt="{\displaystyle y\in X}" loading="lazy"></span>: for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle u\in \mathbb {C} ^{n}}</annotation>

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</math></span><img src="./ab078b3ad5344a703ab086260334c4b583a13c3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle u\in \mathbb {C} ^{n}}" loading="lazy"></span> and all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}">

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

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</math></span><img src="./737e02a5fbf8bc31d443c91025339f9fd1de1065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{1},\ldots ,x_{n}}" loading="lazy"></span> we have

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<annotation encoding="application/x-tex">{\displaystyle u^{*}A^{(g_{y})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}e^{i\langle y,x_{k}-x_{j}\rangle }=\sum _{k=1}^{n}{\overline {u_{k}}}e^{i\langle y,x_{k}\rangle }\sum _{j=1}^{n}u_{j}e^{-i\langle y,x_{j}\rangle }=\left|\sum _{j=1}^{n}{\overline {u_{j}}}e^{i\langle y,x_{j}\rangle }\right|^{2}\geq 0.}</annotation>

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</math></span><img src="./b0f409a16ad7e129f9687bcb7f22bfb51a455c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:81.991ex; height:8.343ex;" alt="{\displaystyle u^{*}A^{(g_{y})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}e^{i\langle y,x_{k}-x_{j}\rangle }=\sum _{k=1}^{n}{\overline {u_{k}}}e^{i\langle y,x_{k}\rangle }\sum _{j=1}^{n}u_{j}e^{-i\langle y,x_{j}\rangle }=\left|\sum _{j=1}^{n}{\overline {u_{j}}}e^{i\langle y,x_{j}\rangle }\right|^{2}\geq 0.}" loading="lazy"></span></dd></dl>

<p>As nonnegative linear combinations of positive definite functions are again positive definite, the <a href="Cosine_function" class="mw-redirect" title="Cosine function">cosine function</a> is positive definite as a nonnegative linear combination of the above functions:

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<annotation encoding="application/x-tex">{\displaystyle \cos(x)={\frac {1}{2}}(e^{ix}+e^{-ix})={\frac {1}{2}}(g_{1}+g_{-1}).}</annotation>

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<p>One can create a positive definite function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \phi \colon X\to \mathbb {R} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f^{*}:=f\circ \phi }</annotation>

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</math></span><img src="./660ff33e5f7b16fa870374900b37db13efd804d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.979ex; height:2.676ex;" alt="{\displaystyle f^{*}:=f\circ \phi }" loading="lazy"></span>.

Then

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<annotation encoding="application/x-tex">{\displaystyle u^{*}A^{(f^{*})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f^{*}(x_{k}-x_{j})=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f(\phi (x_{k})-\phi (x_{j}))=u^{*}{\tilde {A}}^{(f)}u\geq 0,}</annotation>

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</math></span><img src="./88e2eb244431263f9253a1dc8f793aea9064e369.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:78.003ex; height:7.176ex;" alt="{\displaystyle u^{*}A^{(f^{*})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f^{*}(x_{k}-x_{j})=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f(\phi (x_{k})-\phi (x_{j}))=u^{*}{\tilde {A}}^{(f)}u\geq 0,}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {A}}^{(f)}={\big (}f(\phi (x_{i})-\phi (x_{j}))=f({\tilde {x}}_{i}-{\tilde {x}}_{j}){\big )}_{i,j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}^{(f)}={\big (}f(\phi (x_{i})-\phi (x_{j}))=f({\tilde {x}}_{i}-{\tilde {x}}_{j}){\big )}_{i,j}}</annotation>

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</math></span><img src="./d52745c0378d8a8b6c13a791164c0c26c21891b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:41.436ex; height:4.509ex;" alt="{\displaystyle {\tilde {A}}^{(f)}={\big (}f(\phi (x_{i})-\phi (x_{j}))=f({\tilde {x}}_{i}-{\tilde {x}}_{j}){\big )}_{i,j}}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {x}}_{k}:=\phi (x_{k})}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}_{k}:=\phi (x_{k})}</annotation>

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</math></span><img src="./426434c9ede0f0ca5181178792ef0f9c9b343d0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.777ex; height:2.843ex;" alt="{\displaystyle {\tilde {x}}_{k}:=\phi (x_{k})}" loading="lazy"></span> are distinct as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> is <a href="Linear" class="mw-redirect" title="Linear">linear</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Bochner's_theorem">Bochner's theorem</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bochner's_theorem" title="Bochner's theorem">Bochner's theorem</a></div>

<p>Positive-definiteness arises naturally in the theory of the <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>; it can be seen directly that to be positive-definite it is sufficient for <i>f</i> to be the Fourier transform of a function <i>g</i> on the real line with <i>g</i>(<i>y</i>) ≥ 0.

</p><p>The converse result is <i><a href="Bochner's_theorem" title="Bochner's theorem">Bochner's theorem</a></i>, stating that any <a href="Continuous_function" title="Continuous function">continuous</a> positive-definite function on the real line is the Fourier transform of a (positive) <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading4"><h4 id="Applications">Applications</h4></div>

<p>In <a href="Statistics" title="Statistics">statistics</a>, and especially <a href="Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a>, the theorem is usually applied to real functions. Typically, <i>n</i> scalar measurements of some scalar value at points in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{d}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5c2fc383f28fee81277353c81fb7ede49303ebc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.676ex;" alt="{\displaystyle R^{d}}" loading="lazy"></span> are taken and points that are mutually close are required to have measurements that are highly correlated. In practice, one must be careful to ensure that the resulting covariance matrix (an <span class="nowrap"><i>n</i> × <i>n</i></span> matrix) is always positive-definite. One strategy is to define a correlation matrix <i>A</i> which is then multiplied by a scalar to give a <a href="Covariance_matrix" title="Covariance matrix">covariance matrix</a>: this must be positive-definite. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function <i>f</i>), then function <i>f</i> must be positive-definite to ensure the covariance matrix <i>A</i> is positive-definite. See <a href="Kriging" title="Kriging">Kriging</a>.

</p><p>In this context, Fourier terminology is not normally used and instead it is stated that <i>f</i>(<i>x</i>) is the <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> of a <a href="Symmetric" class="mw-redirect" title="Symmetric">symmetric</a> <a href="Probability_density_function" title="Probability density function">probability density function (PDF)</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Generalization">Generalization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Positive-definite_function_on_a_group" title="Positive-definite function on a group">Positive-definite function on a group</a></div>

<p>One can define positive-definite functions on any <a href="Locally_compact_abelian_topological_group" class="mw-redirect" title="Locally compact abelian topological group">locally compact abelian topological group</a>; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the <a href="Representation_theory" title="Representation theory">representation theory</a> of groups on <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a> (i.e. the theory of <a href="Unitary_representation" title="Unitary representation">unitary representations</a>).

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition_2">Definition 2</h2></div>

<p>Alternatively, a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./306c097f43c91dce633d12cde024948d39e73752.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }" loading="lazy"></span> is called <i>positive-definite</i> on a <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> <i>D</i> of the origin if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation>

</semantics>

</math></span><img src="./8d308c32c9894b88115262081194321ae7d9bbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)>0}</annotation>

</semantics>

</math></span><img src="./af29e26aaac6c969d32c8afac1f33ae703f442c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)>0}" loading="lazy"></span> for every non-zero <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in D}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./aaf3e2c3607ccf1c1d7ee6620b44ef9d9e2e1f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.095ex; height:2.176ex;" alt="{\displaystyle x\in D}" loading="lazy"></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>Note that this definition conflicts with definition 1, given above.

</p><p>In physics, the requirement that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(0)=0}</annotation>

</semantics>

</math></span><img src="./8d308c32c9894b88115262081194321ae7d9bbf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=0}" loading="lazy"></span> is sometimes dropped (see, e.g., Corney and Olsen<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>).

</p>

<ul><li><a href="Positive_definiteness" title="Positive definiteness">Positive definiteness</a></li>

<li><a href="Positive-definite_kernel" title="Positive-definite kernel">Positive-definite kernel</a></li></ul>

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

<ul><li>Christian Berg, Christensen, Paul Ressel. <i>Harmonic Analysis on Semigroups</i>, GTM, Springer Verlag.</li>

<li>Z. Sasvári, <i>Positive Definite and Definitizable Functions</i>, Akademie Verlag, 1994</li>

<li>Wells, J. H.; Williams, L. R. <i>Embeddings and extensions in analysis</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 84. Springer-Verlag, New York-Heidelberg, 1975. vii+108 pp.</li></ul>

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

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</style><cite id="CITEREFCheney2009" class="citation book cs1">Cheney, Elliot Ward (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=II6DAwAAQBAJ"><i>A course in Approximation Theory</i></a>. American Mathematical Society. pp. <span class="nowrap">77–</span>78. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780821847985</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">3 February</span> 2022</span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFBochner1959" class="citation book cs1"><a href="Salomon_Bochner" title="Salomon Bochner">Bochner, Salomon</a> (1959). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/lecturesonfourie0000boch"><i>Lectures on Fourier integrals</i></a></span>. Princeton University Press.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFVerhulst1996" class="citation book cs1">Verhulst, Ferdinand (1996). <i>Nonlinear Differential Equations and Dynamical Systems</i> (2nd ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-60934-2</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHahn1967" class="citation book cs1">Hahn, Wolfgang (1967). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/stabilityofmotio0000hahn"><i>Stability of Motion</i></a></span>. Springer.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFCorneyOlsen2015" class="citation journal cs1">Corney, J. F.; Olsen, M. K. (19 February 2015). "Non-Gaussian pure states and positive Wigner functions". <i>Physical Review A</i>. <b>91</b> (2): 023824. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1412.4868">1412.4868</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PhRvA..91b3824C">2015PhRvA..91b3824C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.91.023824">10.1103/PhysRevA.91.023824</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1050-2947">1050-2947</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119293595">119293595</a>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Positive-definite_function">"Positive-definite function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul></div><div><div class="zim-footer">

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