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Shift operator

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about shift operators in mathematics. For operators in computer programming languages, see <a href="Bit_shift" class="mw-redirect" title="Bit shift">Bit shift</a>. For the shift operator of group schemes, see <a href="Verschiebung_operator" title="Verschiebung operator">Verschiebung operator</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, and in particular <a href="Functional_analysis" title="Functional analysis">functional analysis</a>, the shift operator, also known as the translation operator, is an <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a> that takes a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> x ↦ f(x)

to its translation x ↦ f(x + a).<a href="#cite_note-1">[1]</a> In <a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">time series analysis</a>, the shift operator is called the <a href="Lag_operator" title="Lag operator">lag operator</a>.

Shift operators are examples of <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a>, important for their simplicity and natural occurrence. The shift operator action on <a href="Function_of_a_real_variable" title="Function of a real variable">functions of a real variable</a> plays an important role in <a href="Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>, for example, it appears in the definitions of <a href="Almost_periodic_function#Uniform_or_Bohr_or_Bochner_almost_periodic_functions" title="Almost periodic function">almost periodic functions</a>, <a href="Positive-definite_function" title="Positive-definite function">positive-definite functions</a>, <a href="Derivative" title="Derivative">derivatives</a>, and <a href="Convolution" title="Convolution">convolution</a>.<a href="#cite_note-mar-2">[2]</a> Shifts of sequences (functions of an <a href="Integer" title="Integer">integer</a> variable) appear in diverse areas such as <a href="Hardy_space" title="Hardy space">Hardy spaces</a>, the theory of <a href="Abelian_variety" title="Abelian variety">abelian varieties</a>, and the theory of <a href="Symbolic_dynamics" title="Symbolic dynamics">symbolic dynamics</a>, for which the <a href="Baker's_map" title="Baker's map">baker's map</a> is an explicit representation. The notion of <a href="Triangulated_category" title="Triangulated category">triangulated category</a> is a <a href="Categorification" title="Categorification"> categorified</a> analogue of the shift operator.

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

<div class="mw-heading mw-heading3"><h3 id="Functions_of_a_real_variable">Functions of a real variable</h3></div>

The shift operator T t (where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {R} }">

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

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</math><img src="./592bced0c39b10fc90e74c6a66223abfbfb029de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {R} }" loading="lazy">⁠) takes a function f on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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</math><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">⁠ to its translation ft,

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<annotation encoding="application/x-tex">{\displaystyle T^{t}f(x)=f_{t}(x)=f(x+t)~.}</annotation>

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</math><img src="./7fd00b00ba481bd0b90694beabac9e2f4a5c5867.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.59ex; height:3.009ex;" alt="{\displaystyle T^{t}f(x)=f_{t}(x)=f(x+t)~.}" loading="lazy"></dd></dl>

A practical <a href="Operational_calculus" title="Operational calculus">operational calculus</a> representation of the linear operator T t in terms of the plain derivative ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d}{dx}}}">

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</math><img src="./9b9695f938c603b3b40404808946e3c25c6b35b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.636ex; height:3.843ex;" alt="{\displaystyle {\tfrac {d}{dx}}}" loading="lazy">⁠ was introduced by <a href="Lagrange" class="mw-redirect" title="Lagrange">Lagrange</a>,

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<annotation encoding="application/x-tex">{\displaystyle T^{t}=e^{t{\frac {d}{dx}}}~,}</annotation>

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</math><img src="./376da3618a3fef24ebd0771fc4b8ec4ee12f331c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.079ex; height:4.009ex;" alt="{\displaystyle T^{t}=e^{t{\frac {d}{dx}}}~,}" loading="lazy">

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which may be interpreted operationally through its formal <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> in t; and whose action on the monomial xn is evident by the <a href="Binomial_theorem" title="Binomial theorem">binomial theorem</a>, and hence on all series in x, and so all functions f(x) as above.<a href="#cite_note-3">[3]</a> This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.

The operator thus provides the prototype<a href="#cite_note-4">[4]</a>

for Lie's celebrated <a href="Iterated_function#Lie's_data_transport_equation" title="Iterated function">advective flow for Abelian groups</a>,

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(t\beta (x){\frac {d}{dx}}\right)f(x)=\exp \left(t{\frac {d}{dh}}\right)F(h)=F(h+t)=f\left(h^{-1}(h(x)+t)\right),}</annotation>

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</math><img src="./64297b7af0ab7839f6bcf42a6aadd2cc4ce8aa57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.51ex; height:6.176ex;" alt="{\displaystyle \exp \left(t\beta (x){\frac {d}{dx}}\right)f(x)=\exp \left(t{\frac {d}{dh}}\right)F(h)=F(h+t)=f\left(h^{-1}(h(x)+t)\right),}" loading="lazy"></dd></dl>

where the canonical coordinates h (<a href="Abel_equation" title="Abel equation">Abel functions</a>) are defined such that

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<annotation encoding="application/x-tex">{\displaystyle h'(x)\equiv {\frac {1}{\beta (x)}}~,\qquad f(x)\equiv F(h(x)).}</annotation>

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</math><img src="./fd4ef37677c3526eed5e2706a0d82e12a0765045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.019ex; height:6.009ex;" alt="{\displaystyle h'(x)\equiv {\frac {1}{\beta (x)}}~,\qquad f(x)\equiv F(h(x)).}" loading="lazy"></dd></dl>

For example, it easily follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta (x)=x}">

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(tx{\frac {d}{dx}}\right)f(x)=f(e^{t}x),}</annotation>

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</math><img src="./4e706db6d8fa8454ff5921c5ecb8264bad90f5e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.014ex; height:6.176ex;" alt="{\displaystyle \exp \left(tx{\frac {d}{dx}}\right)f(x)=f(e^{t}x),}" loading="lazy"></dd></dl>

hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left(i\pi x{\tfrac {d}{dx}}\right)f(x)=f(-x)}">

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(i\pi x{\tfrac {d}{dx}}\right)f(x)=f(-x)}</annotation>

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</math><img src="./44484e54b70b0f7deb301d84d85936a3051ec9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.17ex; height:4.843ex;" alt="{\displaystyle \exp \left(i\pi x{\tfrac {d}{dx}}\right)f(x)=f(-x)}" loading="lazy"> (parity); likewise,

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<annotation encoding="application/x-tex">{\displaystyle \beta (x)=x^{2}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(tx^{2}{\frac {d}{dx}}\right)f(x)=f\left({\frac {x}{1-tx}}\right),}</annotation>

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</math><img src="./af518c7465b7131609cd8f8764cb522bfcea730c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.224ex; height:6.176ex;" alt="{\displaystyle \exp \left(tx^{2}{\frac {d}{dx}}\right)f(x)=f\left({\frac {x}{1-tx}}\right),}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle \exp \left({\frac {t}{x}}{\frac {d}{dx}}\right)f(x)=f\left({\sqrt {x^{2}+2t}}\right),}</annotation>

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</math><img src="./67eb9956221918c6fa42e66d4a111832316563e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.062ex; height:6.176ex;" alt="{\displaystyle \exp \left({\frac {t}{x}}{\frac {d}{dx}}\right)f(x)=f\left({\sqrt {x^{2}+2t}}\right),}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \beta (x)=e^{x}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(te^{x}{\frac {d}{dx}}\right)f(x)=f\left(\ln \left({\frac {1}{e^{-x}-t}}\right)\right),}</annotation>

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</math><img src="./d5f2374193fff1ef4f2e10282fbf02f4cf6525dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.499ex; height:6.176ex;" alt="{\displaystyle \exp \left(te^{x}{\frac {d}{dx}}\right)f(x)=f\left(\ln \left({\frac {1}{e^{-x}-t}}\right)\right),}" loading="lazy"></dd></dl>

etc.

The <a href="Initial_condition" title="Initial condition">initial condition</a> of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation<a href="#cite_note-acz-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle f_{t}(f_{\tau }(x))=f_{t+\tau }(x).}</annotation>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Shift_space" title="Shift space">Shift space</a></div>

The left shift operator acts on one-sided <a href="Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a> of numbers by

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<annotation encoding="application/x-tex">{\displaystyle S^{*}:(a_{1},a_{2},a_{3},\ldots )\mapsto (a_{2},a_{3},a_{4},\ldots )}</annotation>

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</math><img src="./980686885e1e3cf3cb87ecf77837786060bc2d56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.1ex; height:2.843ex;" alt="{\displaystyle S^{*}:(a_{1},a_{2},a_{3},\ldots )\mapsto (a_{2},a_{3},a_{4},\ldots )}" loading="lazy"></dd></dl>

and on two-sided infinite sequences by

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<annotation encoding="application/x-tex">{\displaystyle T:(a_{k})_{k\,=\,-\infty }^{\infty }\mapsto (a_{k+1})_{k\,=\,-\infty }^{\infty }.}</annotation>

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</math><img src="./f148d719fe5337cb6c6a45637d27ad018d45bf53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.317ex; height:3.176ex;" alt="{\displaystyle T:(a_{k})_{k\,=\,-\infty }^{\infty }\mapsto (a_{k+1})_{k\,=\,-\infty }^{\infty }.}" loading="lazy"></dd></dl>

The right shift operator acts on one-sided <a href="Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a> of numbers by

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<annotation encoding="application/x-tex">{\displaystyle S:(a_{1},a_{2},a_{3},\ldots )\mapsto (0,a_{1},a_{2},\ldots )}</annotation>

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</math><img src="./6be8cb5c30aa045ce53fdb86fa8144f35c92710a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.902ex; height:2.843ex;" alt="{\displaystyle S:(a_{1},a_{2},a_{3},\ldots )\mapsto (0,a_{1},a_{2},\ldots )}" loading="lazy"></dd></dl>

and on two-sided infinite sequences by

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<annotation encoding="application/x-tex">{\displaystyle T^{-1}:(a_{k})_{k\,=\,-\infty }^{\infty }\mapsto (a_{k-1})_{k\,=\,-\infty }^{\infty }.}</annotation>

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</math><img src="./71605f5617ddaf1d7a79cdb18ea68d823038a597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:32.733ex; height:3.509ex;" alt="{\displaystyle T^{-1}:(a_{k})_{k\,=\,-\infty }^{\infty }\mapsto (a_{k-1})_{k\,=\,-\infty }^{\infty }.}" loading="lazy"></dd></dl>

The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.

<div class="mw-heading mw-heading3"><h3 id="Abelian_groups">Abelian groups</h3></div>

In general, as illustrated above, if F is a function on an <a href="Abelian_group" title="Abelian group">abelian group</a> G, and h is an element of G, the shift operator T g maps F to<a href="#cite_note-acz-6">[6]</a><a href="#cite_note-7">[7]</a>

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<annotation encoding="application/x-tex">{\displaystyle F_{g}(h)=F(h+g).}</annotation>

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</math><img src="./c79fb664b3bcd68e5301028fea8108bf5754829c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.255ex; height:3.009ex;" alt="{\displaystyle F_{g}(h)=F(h+g).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Properties_of_the_shift_operator">Properties of the shift operator</h2></div>

The shift operator acting on real- or complex-valued functions or sequences is a linear operator which preserves most of the standard <a href="Norm_(mathematics)" title="Norm (mathematics)">norms</a> which appear in functional analysis. Therefore, it is usually a <a href="Continuous_operator" class="mw-redirect" title="Continuous operator">continuous operator</a> with norm one.

<div class="mw-heading mw-heading3"><h3 id="Action_on_Hilbert_spaces">Action on Hilbert spaces</h3></div>

The shift operator acting on two-sided sequences is a <a href="Unitary_operator" title="Unitary operator">unitary operator</a> on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}(\mathbb {Z} ).}">

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<annotation encoding="application/x-tex">{\displaystyle \ell _{2}(\mathbb {Z} ).}</annotation>

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</math><img src="./777932c7c374d4e45ff88f613886769c13be39f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.03ex; height:2.843ex;" alt="{\displaystyle \ell _{2}(\mathbb {Z} ).}" loading="lazy">⁠ The shift operator acting on functions of a real variable is a unitary operator on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}(\mathbb {R} ).}">

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<annotation encoding="application/x-tex">{\displaystyle L_{2}(\mathbb {R} ).}</annotation>

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In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}T^{t}=M^{t}{\mathcal {F}},}</annotation>

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where M t is the <a href="Multiplication_operator" title="Multiplication operator">multiplication operator</a> by exp(itx). Therefore, the spectrum of T t is the <a href="Unit_circle" title="Unit circle">unit circle</a>.

The one-sided shift S acting on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}(\mathbb {N} )}">

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<annotation encoding="application/x-tex">{\displaystyle \ell _{2}(\mathbb {N} )}</annotation>

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</math><img src="./324955dd550f67ecf969f1cb1bf4db99fce8f541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.511ex; height:2.843ex;" alt="{\displaystyle \ell _{2}(\mathbb {N} )}" loading="lazy">⁠ is a proper <a href="Isometry" title="Isometry">isometry</a> with <a href="Range_of_a_function" title="Range of a function">range</a> equal to all <a href="Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a> which vanish in the first <a href="Coordinate" class="mw-redirect" title="Coordinate">coordinate</a>. The operator S is a <a href="Compression_(functional_analysis)" title="Compression (functional analysis)">compression</a> of T−1, in the sense that

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<annotation encoding="application/x-tex">{\displaystyle T^{-1}y=Sx{\text{ for each }}x\in \ell ^{2}(\mathbb {N} ),}</annotation>

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</math>

where y is the vector in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell _{2}(\mathbb {Z} )}">

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<annotation encoding="application/x-tex">{\displaystyle \ell _{2}(\mathbb {Z} )}</annotation>

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</math><img src="./12623095fbb4634fdc8046fb2f99622e6a2dc636.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.384ex; height:2.843ex;" alt="{\displaystyle \ell _{2}(\mathbb {Z} )}" loading="lazy">⁠ with yi = xi for i ≥ 0 and yi = 0 for i < 0. This observation is at the heart of the construction of many <a href="Unitary_dilation" class="mw-redirect" title="Unitary dilation">unitary dilations</a> of isometries.

The <a href="Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectrum</a> of S is the <a href="Unit_disk" title="Unit disk">unit disk</a>. The shift S is one example of a <a href="Fredholm_operator" title="Fredholm operator">Fredholm operator</a>; it has Fredholm index −1.

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

<a href="Jean_Delsarte" title="Jean Delsarte">Jean Delsarte</a> introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by <a href="Boris_Levitan" title="Boris Levitan">Boris Levitan</a>.<a href="#cite_note-mar-2">[2]</a><a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a>

A family of operators ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{L^{x}\}_{x\in X}}">

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<annotation encoding="application/x-tex">{\displaystyle \{L^{x}\}_{x\in X}}</annotation>

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</math><img src="./5374f4ff53bea35cdc53c1b13ce6ea0ef54339ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.749ex; height:2.843ex;" alt="{\displaystyle \{L^{x}\}_{x\in X}}" loading="lazy">⁠ acting on a space Φ of functions from a set X to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

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</mrow>

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

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</math><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">⁠ is called a family of generalized shift operators if the following properties hold:

<ol><li><a href="Associative_property" title="Associative property">Associativity</a>: let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R^{y}f)(x)=(L^{x}f)(y).}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (R^{y}f)(x)=(L^{x}f)(y).}</annotation>

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</math><img src="./5318671acbe5902cd5153508158e5cdf1873e431.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.593ex; height:2.843ex;" alt="{\displaystyle (R^{y}f)(x)=(L^{x}f)(y).}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{x}R^{y}=R^{y}L^{x}.}">

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

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</math><img src="./5c861db1de0dc92ca4d7d846263d845cd665fd4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.883ex; height:2.343ex;" alt="{\displaystyle L^{x}R^{y}=R^{y}L^{x}.}" loading="lazy"></li>

<li>There exists e in X such that Le is the <a href="Identity_function" title="Identity function">identity operator</a>.</li></ol>

In this case, the set X is called a <a href="Hypergroup" class="mw-redirect" title="Hypergroup">hypergroup</a>.

<ul><li><a href="Arithmetic_shift" title="Arithmetic shift">Arithmetic shift</a></li>

<li><a href="Logical_shift" title="Logical shift">Logical shift</a></li>

<li><a href="Clock_and_shift_matrices" class="mw-redirect" title="Clock and shift matrices">Clock and shift matrices</a></li>

<li><a href="Finite_difference#Calculus_of_finite_differences" title="Finite difference">Finite difference</a></li>

<li><a href="Translation_operator_(quantum_mechanics)" title="Translation operator (quantum mechanics)">Translation operator (quantum mechanics)</a></li></ul>

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

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ShiftOperator.html">"Shift Operator"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-mar-2">^ <a href="#cite_ref-mar_2-0">a</a> <a href="#cite_ref-mar_2-1">b</a> <cite id="CITEREFMarchenko2006" class="citation book cs1"><a href="Vladimir_Marchenko" class="mw-redirect" title="Vladimir Marchenko">Marchenko, V. A.</a> (2006). "The generalized shift, transformation operators, and inverse problems". Mathematical events of the twentieth century. Berlin: Springer. pp. 145–162. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-29462-7_8">10.1007/3-540-29462-7_8</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-23235-3</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2182783">2182783</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Jordan, Charles, (1939/1965). Calculus of Finite Differences, (AMS Chelsea Publishing), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0828400336</bdi> .

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> M Hamermesh (1989), Group Theory and Its Application to Physical Problems

(Dover Books on Physics), Hamermesh ISBM 978-0486661810, Ch 8-6, pp 294-5,

<a rel="nofollow" class="external text" href="https://physics.stackexchange.com/questions/331635/undefined-phase-flow/331841#331841">online</a>.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> p 75 of Georg Scheffers (1891): Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen, Teubner, Leipzig, 1891. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3743343078</bdi> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7-86AQAAIAAJ&q=+75&pg=PR6">online</a>

</li>

<li id="cite_note-acz-6">^ <a href="#cite_ref-acz_6-0">a</a> <a href="#cite_ref-acz_6-1">b</a> Aczel, J (2006), Lectures on Functional Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0486445236</bdi> .

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> "A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ibid.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFLevitanLitvinov2001" class="citation cs2"><a href="Boris_Levitan" title="Boris Levitan">Levitan, B.M.</a>; Litvinov, G.L. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Generalized_displacement_operators">"Generalized displacement operators"</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></cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBredikhina2001" class="citation cs2">Bredikhina, E.A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Almost-periodic_function">"Almost-periodic function"</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></cite>

</li>

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

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

<ul><li><cite id="CITEREFPartington2004" class="citation book cs1">Partington, Jonathan R. (March 15, 2004). Linear Operators and Linear Systems. Cambridge University Press. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fcbo9780511616693">10.1017/cbo9780511616693</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-83734-7</bdi>.</cite></li>

<li>Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.</li></ul></div><div><div class="zim-footer">

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