View source for Generalized displacement operators

''generalized shift operators, hypergroup''

A concept arising from the axiomatization of certain properties of displacement operators on spaces of functions on a group. Such important mathematical concepts as the [[Convolution of functions|convolution of functions]], a [[Group algebra|group algebra]], a [[Positive-definite function|positive-definite function]], an [[Almost-periodic function|almost-periodic function]], etc., can be formulated in terms of group-displacement operators. In the framework of the theory of generalized displacement operators far-reaching generalizations can be obtained of the fundamental principles and results concerning the concepts listed above. In particular, the theory of generalized displacement operators has substantial applications in abstract harmonic analysis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]).

The terms  "generalized displacement operator"  and  "hypergroup"  are due to J. Delsarte (see [[#References|[1]]]–[[#References|[3]]]). Important ideas and a number of original results in this field are also due to him. The systematic construction of the theory of generalized displacement operators was mainly given in the work of B.M. Levitan (see, for example, [[#References|[4]]]–[[#References|[7]]]).

==Basic concepts.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438001.png" /> be an arbitrary set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438002.png" /> be some vector space of complex-valued functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438003.png" />. Suppose that to each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438004.png" /> there corresponds a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438005.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438006.png" /> such that for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438007.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438008.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g0438009.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380010.png" />. The linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380012.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380013.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380014.png" />). The linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380015.png" /> are called generalized displacement operators if the following conditions (axioms) are satisfied: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380016.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380017.png" /> (associativity axiom); and 2) there exists a neutral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380021.png" /> is the identity operator. In this case the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380022.png" /> is called a hypergroup, so that the concepts of  "a set of generalized displacement operators"  and a  "hypergroup"  are equivalent. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380023.png" /> are often called right displacement operators, while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380024.png" /> are called left displacement operators.

Generalized displacement operators arise in an obvious way on any vector space of functions on an arbitrary semi-group with an identity that is invariant with respect to displacements, and in particular on a group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380026.png" /> is the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380028.png" /> in the semi-group, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380029.png" />; then the associativity axiom reduces to the associativity of multiplication in the semi-group, and the neutral element is the identity in the semi-group, so that the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380030.png" /> form a family of generalized displacement operators. Non-trivial examples are given below.

In the general case, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380031.png" /> do not form generalized displacement operators, since the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380032.png" /> need not be the identity. However, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380033.png" /> is a projector, and its range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380034.png" /> is called the fundamental subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380035.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380036.png" /> form a family of generalized displacement operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380037.png" />, and the symmetry between left and right displacement operators is restored. The second of the axioms for generalized displacement operators is often strengthened by requiring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380038.png" />, that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380039.png" />. Conditions 1) and 2) are the most general axioms for generalized displacement operators. More restricted classes of generalized displacement operators can be selected by imposing additional conditions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380041.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380042.png" /> are called commutative, in this case the hypergroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380043.png" /> is also called commutative. If further assumptions are made about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380044.png" />, then new conditions arise in a natural way for the generalized displacement operators. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380045.png" /> is a locally compact space with a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380046.png" />, then it is usually required that the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380048.png" /> act compatibly on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380049.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380050.png" />, and on the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380052.png" />, so that additional conditions of continuity type are imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380054.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380055.png" /> is a smooth manifold, then conditions of differentiability type are imposed, etc. Different versions of the axioms are given in [[#References|[1]]], [[#References|[3]]]–, [[#References|[8]]], [[#References|[15]]]–[[#References|[20]]].

==Examples of generalized displacement operators related to groups.==
Delsarte generalized displacement operators. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380056.png" /> be a topological group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380057.png" /> be a compact group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380058.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380059.png" /> be the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380060.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380061.png" />. In the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380062.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380063.png" />, the generalized displacement operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380064.png" /> are defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380065.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380067.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380068.png" /> under the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380070.png" /> is the product of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380072.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380073.png" />. The identity element of the group is the neutral element. Both axioms for generalized displacement operators are satisfied; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380074.png" /> is commutative, then Delsarte generalized displacement operators are also commutative. The fundamental subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380075.png" /> consists of all functions constant on the orbits relative to the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380076.png" />, and the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380078.png" /> coincide on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380079.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380081.png" /> lie in the same orbit. Hence the orbit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380082.png" /> can also be given the structure of a hypergroup, by identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380083.png" /> with the space of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380084.png" /> and putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380086.png" /> is any element in the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380087.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380088.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380089.png" />, and the group of automorphisms consists of two elements (reflection in the origin and the identity mapping), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380090.png" />. In this case the fundamental subspace consists of the even functions, and the orbit space is identified with the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380091.png" />. Another special case of Delsarte generalized displacement operators is obtained when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380093.png" />; here the fundamental subspace consists of the central functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380094.png" />, and the hypergroup generated by the orbits, that is, by the conjugacy classes, is commutative.

===Double cosets with respect to a compact subgroup.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380095.png" /> be a locally compact group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380096.png" /> be a compact subgroup, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380097.png" /> be the space of double cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380098.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g04380099.png" />, such a coset contains all elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800100.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800101.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800102.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800103.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800104.png" /> coincides with the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800105.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800106.png" /> be the space of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800107.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800108.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800110.png" />. A generalized displacement operator is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800111.png" /></td> </tr></table>

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800112.png" /> can be identified with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800113.png" /> of continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800114.png" />, and, as for Delsarte generalized displacement operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800115.png" /> can be given the structure of a hypergroup. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800116.png" /> is a linear semi-simple Lie group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800117.png" /> is its maximal compact subgroup, then the hypergroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800118.png" /> is commutative and is closely connected with the spherical functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800119.png" /> (in particular, all spherical functions lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800120.png" />).

In the examples described above, other function spaces may be considered instead of that of the continuous functions (see [[#References|[8]]], [[#References|[13]]], [[#References|[15]]]–[[#References|[19]]]).

===Hypercomplex systems.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800121.png" /> be a finite hypercomplex system, that is, a finite-dimensional associative algebra with a fixed basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800122.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800123.png" /> is identified with the space of functions on the finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800124.png" />, by associating the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800125.png" /> to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800126.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800127.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800128.png" /> is the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800130.png" /> in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800131.png" />. The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800132.png" /> form a family of generalized displacement operators if and only if one of the elements of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800133.png" /> is a right identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800134.png" />. In this way a correspondence is established between arbitrary generalized displacement operators in the space of functions on a finite set and finite hypercomplex systems. Thus, the concept of a generalized displacement operator can be regarded as a far-reaching generalization of the classical concept of a hypercomplex system. Important examples of generalized displacement operators that can naturally be treated as hypercomplex systems with a countable basis or with a basis with the power of the continuum are considered in, for example, [[#References|[4]]], [[#References|[5]]] and [[#References|[8]]].

==Generators and Lie's theorems for generalized displacement operators.==
Let the hypergroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800135.png" /> be a differentiable (or complex-analytic) manifold and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800136.png" /> be a differentiable (respectively, holomorphic) function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800137.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800138.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800139.png" /> be the local coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800140.png" />, where the coordinate system is chosen so that the neutral element has coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800141.png" />. The generators (infinitesimal operators) of the right displacement of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800142.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800143.png" /> are the linear operators of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800144.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800146.png" />. The generators of the left displacement are defined similarly by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800147.png" /></td> </tr></table>

From the associativity axiom it can be deduced that any left displacement generator commutes with all right displacement generators (as well as with the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800148.png" />). Differentiation of the associativity condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800149.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800150.png" /> an appropriate number of times and putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800151.png" /> gives the system of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800152.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800153.png" />. This system should be regarded as the generalization of Lie's first direct theorem to the case of generalized displacement operators (see [[#References|[3]]] for Delsarte generalized displacement operators and [[#References|[5]]] for the general case). Not all the equations (*) need necessarily be involved in determining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800154.png" />. For example, for displacements on a Lie group, the generators of the first order already uniquely determine the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800155.png" /> (that is, the group multiplication). In the general case certain generators of lower orders may degenerate, for example, into multiplication by a constant, so that the corresponding equations in (*) contain no useful information. This gives rise to the following problem: Select a minimal number of equations from (*) that uniquely determine the generalized displacement operators. Here, degenerate generators increase the number of initial conditions. If a finite system of the form (*) under certain initial conditions, including the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800156.png" />, uniquely determines the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800157.png" />, and if the operators on the left-hand side of the system commute with all the operators on the right-hand side, then the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800158.png" /> are generalized displacement operators. This assertion is the analogue of Lie's first converse theorem [[#References|[5]]]. Analogues of Lie's second and third (direct and converse) theorems have been proved for a certain class of generalized displacement operators (see [[#References|[5]]]). In particular, generalized displacement operators have been constructed on the space of infinitely-differentiable functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800159.png" /> variables, for which the right (left) displacement generators generate any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800160.png" />-dimensional Lie algebra. An explicit description of these generators has been obtained in the form of integro-differential operators of the second order [[#References|[10]]]. By means of similar techniques one has constructed generators of any order [[#References|[11]]] acting on the space of entire analytic functions (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800161.png" /> variables) and generating any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800162.png" />-dimensional real Lie algebra; the generalized displacement operators can be re-established in terms of these generators. Generalized displacement operators can be constructed not only starting from Lie algebras, but also from commutation relations of a more general kind (see [[#References|[7]]], [[#References|[12]]]). Thus, generalized displacement operators on the line were already constructed in [[#References|[1]]], starting from an explicitly-given second-order operator by means of a series, analogous to Taylor series, which gives an expansion of the ordinary displacement in powers of the differentiation operator. Commutative generalized displacement operators on the line with a second-order generator of Sturm–Liouville type have been described in detail (see [[#References|[4]]] and [[#References|[5]]]) and applications have been indicated to Sturm–Liouville operators and equations. A complete classification has been given [[#References|[14]]] of generalized displacement operators with a generator of Sturm–Liouville type (including non-commutative ones) on the space of differentiable functions on the line.

==Representations of generalized displacement operators and hypergroup algebras.==
Representation theory is not so well developed for generalized displacement operators as it is for groups, but it is constructed in a similar way. This analogy extends quite far; for example, with the aid of the concept of a generator, representations of generalized displacement operators can be studied by an infinitesimal method, just as this is done for Lie groups (see [[#References|[3]]], [[#References|[5]]], [[#References|[11]]]). It is convenient to treat representations of generalized displacement operators as representations of associative hypergroup algebras, analogous to group algebras (see [[Infinite-dimensional representation|Infinite-dimensional representation]]). If the hypergroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800163.png" /> is locally compact, then the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800164.png" /> of complex Radon measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800165.png" /> with compact support acquires the structure of a hypergroup algebra with respect to the generalized convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800166.png" /> of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800167.png" />, defined by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800168.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800169.png" /> is an arbitrary continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800170.png" />. The structure of a hypergroup can be defined similarly on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800171.png" /> of generalized functions of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800172.png" /> (or on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800173.png" /> of analytic functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800174.png" />), if the hypergroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800175.png" /> is a differentiable (complex-analytic) manifold. In the natural topologies, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800176.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800177.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800178.png" /> are topological algebras, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800179.png" />-function concentrated at the neutral element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800180.png" /> is the right identity in each of these algebras. The converse is also valid: If in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800181.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800182.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800183.png" />), endowed with the natural topology, the structure of a topological associative algebra with a right identity that is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800184.png" />-function has been given, then there exists a (unique) hypergroup structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800185.png" /> such that the generalized convolution is the same as multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800186.png" /> (respectively, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800187.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800188.png" />). Continuous representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800189.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800190.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800191.png" />) can be interpreted as continuous (infinitely-differentiable, holomorphic) representations of the corresponding generalized displacement operators [[#References|[20]]].

The theory of symmetric representations of Banach hypergroup algebras with an involution is an analogue of the theory of unitary representations of groups. The most complete results (see [[#References|[4]]]–) have been obtained for representations of commutative and compact generalized displacement operators. Under certain conditions the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800192.png" /> of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800193.png" /> that are summable with respect to a positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800194.png" /> can be endowed with the structure of a Banach hypergroup algebra with an involution. One of these conditions is that the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800195.png" /> be invariant under generalized displacements (for different versions of the precise definitions see [[#References|[4]]]–, [[#References|[15]]]–[[#References|[19]]]). Under natural assumptions, the uniqueness (up to a scalar multiple) has been proved of a measure that is invariant under right or left generalized displacements; there are also sufficient conditions for the existence of such a measure (conditions like compactness, commutativity or discreteness of the hypergroup, see [[#References|[8]]], [[#References|[16]]]–[[#References|[18]]]). However, the problem of the existence of an invariant measure for generalized displacement operators of general form remains open (1982). Along with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800196.png" />, an important role is played by a Banach hypergroup algebra of measures of bounded variation and a hypergroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800197.png" />-algebra.

Banach hypergroup algebras and their symmetric representations have been studied in [[#References|[4]]],

[[#References|[8]]], [[#References|[15]]]–[[#References|[19]]]. Algebras of analytic functionals related to certain generalized displacement operators on the line were studied in . For generalized displacement operators of general type, topological hypergroup algebras and their representations were considered in [[#References|[20]]], where problems of spectral analysis and synthesis were treated as problems in the study of ideals of hypergroup algebras. The technique of hypergroup algebras was applied in [[#References|[12]]] to solve problems in mathematical physics in the framework of the operator methods of V.P. Maslov.

==Harmonic analysis.==
The following model reveals the structure of commutative generalized displacement operators (see [[#References|[4]]], [[#References|[5]]]). Let two positive measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800198.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800199.png" /> be given on spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800200.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800201.png" />, respectively, and, with the aid of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800202.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800203.png" />, let a generalized Fourier transformation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800204.png" /></td> </tr></table>

be given, which defines an isomorphism of the Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800206.png" />. Assume that the inversion formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800207.png" /></td> </tr></table>

is valid. If the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800208.png" /> is discrete, then this formula gives an expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800209.png" /> in a generalized Fourier series. It turns out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800210.png" /> has the structure of a hypergroup if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800211.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800212.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800213.png" />. In this case generalized displacement operators are defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800214.png" /></td> </tr></table>

Hence, generalized displacement operators arise naturally in problems concerned with expansion in orthogonal systems of functions, in the spectral theory of operators, etc., which ensure a wide range of applications of the theory of generalized displacement operators (see, for example, [[#References|[4]]], [[#References|[5]]], [[#References|[8]]], [[#References|[15]]]–[[#References|[19]]]). In the case of commutative generalized displacement operators, Bochner's theorem on representations of positive-definite functions and the law of Pontryagin duality have been generalized, a generalized Fourier transformation has been defined, and an analogue of the [[Plancherel theorem|Plancherel theorem]] has been proved

(for various versions of these results see [[#References|[15]]]–[[#References|[17]]]). With the aid of representation theory, harmonic analysis may also be constructed for non-commutative generalized displacement operators. E.g., non-commutative analogues of the Plancherel theorem and the inversion formula have been obtained for generalized displacement operators, which include as a special case the corresponding results for locally compact groups; an analogue of the Peter–Weyl theorem is valid for compact generalized displacement operators. Versions of harmonic analysis in the spirit of the theory of almost-periodic and mean-periodic functions have been considered for generalized displacement operators (see [[#References|[1]]], [[#References|[2]]], [[#References|[4]]], [[#References|[8]]], , [[#References|[14]]], [[#References|[20]]]). An analogue of Wiener's Tauberian theorem has been obtained for commutative generalized displacement operators (cf. [[Wiener Tauberian theorem|Wiener Tauberian theorem]]), and questions of spectral synthesis have been considered (see [[#References|[21]]], [[#References|[22]]]). For applications of generalized displacement operators to harmonic analysis on groups see [[#References|[8]]], [[#References|[13]]], [[#References|[16]]], [[#References|[19]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Delsarte,   "Sur une extension de la formule de Taylor"  ''J. Math. Pures Appl.'' , '''17'''  (1938)  pp. 213–231</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Delsarte,   "Une extension nouvelle de la théorie des fonctions presque-périodiques de Bohr"  ''Acta Math.'' , '''69'''  (1938)  pp. 259–317</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Delsarte,  , ''Colloque Internat. CNRS'' :  71  (1956)  pp. 29–45</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B.M. Levitan,   "The application of generalized displacement operators to linear differential equations of the second order"  ''Transl. Amer. Math. Soc. Ser. 1'' , '''10'''  (1962)  pp. 408–541  ''Uspekhi Mat. Nauk'' , '''4''' :  1  (1949)  pp. 3–112</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  B.M. Levitan,   "The theory of generalized displacement operators" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top">  B.M. Levitan,   "Normed rings generated by the generalized shift operator"  ''Dokl. Akad. Nauk SSSR'' , '''47'''  (1945)  pp. 3–6  (In Russian)</TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top">  B.M. Levitan,   "A theorem on the representation of positive-definite functions for the generalized shift operator"  ''Dokl. Akad. Nauk SSSR'' , '''47'''  (1945)  pp. 163–165  (In Russian)</TD></TR><TR><TD valign="top">[6c]</TD> <TD valign="top">  B.M. Levitan,   "Plancherel's theorem for the generalized shift operator"  ''Dokl. Akad. Nauk SSSR'' , '''47'''  (1945)  pp. 323–326  (In Russian)</TD></TR><TR><TD valign="top">[6d]</TD> <TD valign="top">  B.M. Levitan,   "A duality law for the generalized shift operator"  ''Dokl. Akad. 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====Comments====
See also [[#References|[a2]]]. If the pair of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800215.png" /> and a compact subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043800/g043800216.png" /> forms a Gel'fand pair, then the generalized displacement operators associated with it commute. With expansions and dual expansions in [[Jacobi polynomials|Jacobi polynomials]] a commutative hypergroup structure can be associated. See [[#References|[a1]]] for a class of Sturm–Liouville operators yielding generalized displacement operators.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Chébli,   "Opérateurs de translation généralisée et semi-groupes de convolution"  J. Faraut (ed.) , ''Théorie du potentiel et analyse harmonique'' , ''Lect. notes in math.'' , '''404''' , Springer  (1974)  pp. 36–59</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.I. Vainerman,   "Duality of algebras with an involution and generalized shift operators"  ''J. Soviet Math.'' , '''42'''  (1988)  pp. 2113–2138  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''24'''  (1986)  pp. 165–206</TD></TR></table>