The Translation Operator

Introduction

The simplest operator on a function of a group is the one that shifts the argument. On $\mathbb R^n$ the translation by $y$ is $$ (\tau_yf)(x)=f(x-y), $$ so that the family $\{\tau_y\}_{y\in\mathbb R^n}$ is a representation of the additive group: $\tau_y\tau_z=\tau_{y+z}$ and $\tau_0=\mathrm{id}$. Its importance is that it makes the group act on the function space, and that on $L^2(\mathbb R^n)$ it acts by unitaries; the operators that commute with it are exactly the translation-invariant ones, which is why convolution and Fourier multipliers are the same subject. This article develops the family as an operator family: the isometry and unitary properties, the continuity in the group variable, the infinitesimal generator that differentiation supplies, the spectrum and the characters, and the diagonalisation by the Fourier operator. It is the article in which the group of translations and the algebra it generates are placed among the operators of the corpus.

The prerequisites are Measure Theory and Integration for the translation invariance of Lebesgue measure, the $L^p$ spaces and the continuity of translation, Fourier Analysis on Euclidean Spaces for the characters, the transform and its commutation with translation, Convolution Operators, the previous article of this group, for the translation-invariant operators and the multiplier correspondence, and Modes of Convergence for convergence in $L^p$. The unitary and spectral vocabulary — unitary operator, spectrum, Stone's theorem, one-parameter group — is quoted from Banach and Hilbert Spaces, later in this Part, and from Semigroups and Evolution Equations, later in this Part, where the one-parameter groups and their generators are developed; the concrete facts are proved here, and nothing structural is assumed beyond the measure-theoretic continuity of translation. The general locally compact group, its Haar measure and its unitary representations are the subject of Locally Compact Groups and Haar Measure, Representations of Locally Compact Groups and Harmonic Analysis on Groups, in the neighbouring category Analysis on Groups of this Part; the group is $\mathbb R^n$, or a group whose theory is available, and the general case is named as a forward reference only. The difference operator $\tau_h-\mathrm{id}$ and its discrete theory are The Difference Operator, the next article of this group. No geometry is invoked; the group structure is used through translation alone.

The Translation Operator

Definition and the Representation Property

Definition. Let $G$ be a group acting on a set $X$ on the left, $(x,g)\mapsto gx$, let $f$ be a function on $X$, and for $g\in G$ define $$ (\tau_gf)(x)=f(g^{-1}x). $$ The map $\tau_g$ is the translation (or shift) by $g$. On $G$ itself, where $G$ acts by left multiplication, this is $(\tau_gf)(x)=f(g^{-1}x)$; on the additive group $(\mathbb R^n,+)$ it is $(\tau_yf)(x)=f(x-y)$; on $(\mathbb Z,+)$ it is $(\tau_af)_n=f_{n-a}$.

Proposition (representation). The maps $\tau_g$ satisfy $\tau_e=\mathrm{id}$, $\tau_g\tau_h=\tau_{gh}$ and $\tau_g^{-1}=\tau_{g^{-1}}$, so $g\mapsto\tau_g$ is a representation of $G$ by linear operators on the space of functions; the representation is faithful on a space containing a function of two distinct values.

Proof. $(\tau_g\tau_hf)(x)=\tau_hf(g^{-1}x)=f(h^{-1}g^{-1}x)=f((gh)^{-1}x)=(\tau_{gh}f)(x)$ by the inverse law $(gh)^{-1}=h^{-1}g^{-1}$, and the rest is immediate. If $g\neq h$, a function separating $g^{-1}x$ from $h^{-1}x$ at some point $x$ is moved differently by the two. $\blacksquare$

Isometry and Unitarity

Theorem. On $L^p(\mathbb R^n)$, $1\leq p\leq\infty$, the translation is an isometry, $\lVert\tau_yf\rVert_p=\lVert f\rVert_p$, by the translation invariance of Lebesgue measure; on $L^2(\mathbb R^n)$ it is a unitary operator, $\tau_y^\dagger\tau_y=\tau_y\tau_y^\dagger=\mathrm{id}$, with $\tau_y^\dagger=\tau_{-y}=\tau_y^{-1}$ and $\langle\tau_yf,\tau_yg\rangle=\langle f,g\rangle$.

Proof. The change of variable $x\mapsto x+y$ preserves Lebesgue measure, so $\int\lvert f(x-y)\rvert^pdx=\int\lvert f\rvert^p$; the inner product identity is the polarisation, and the adjoint is computed from $\langle\tau_yf,g\rangle=\int f(x-y)\overline{g(x)}dx=\int f(u)\overline{g(u+y)}du=\langle f,\tau_{-y}g\rangle$. $\blacksquare$

On a group with a measure, the same proof needs the measure to be invariant under the left action, which is the defining property of the Haar measure; this is why the general theory belongs to Locally Compact Groups and Haar Measure, and why only groups with an explicit invariant measure — the Euclidean spaces, the integers, the finite groups, the circle — are used in this article and its companions.

Continuity in the Group Variable

Theorem (strong continuity). For $1\leq p<\infty$ and $f\in L^p(\mathbb R^n)$ the map $y\mapsto\tau_yf$ is continuous from $\mathbb R^n$ to $L^p(\mathbb R^n)$; that is, $\lVert\tau_yf-f\rVert_p\to0$ as $y\to0$, and the map is uniformly continuous in the sense that $\lVert\tau_yf-\tau_{y'}f\rVert_p=\lVert\tau_{y-y'}f-f\rVert_p$ depends on $y-y'$ only. For $p=\infty$ with the supremum norm the statement is false.

Proof. The continuity of translation in $L^p$ for $1\leq p<\infty$ is proved by approximating $f$ by a continuous compactly supported function, for which the statement is uniform continuity, and using the density; the translation-invariance of the norm gives the difference form. For $p=\infty$ the indicator of a half-line is moved by the translation by an amount bounded away from $0$ in the supremum norm, so the statement fails. $\blacksquare$

Remark (the strong operator topology). The theorem says that $y\mapsto\tau_y$ is continuous when the space of operators carries the strong operator topology, the topology of pointwise convergence, but not when it carries the operator-norm topology: $\lVert\tau_y-\mathrm{id}\rVert=\sqrt2$ for every $y\neq0$ on $L^2$, since $\lVert\tau_yf-f\rVert^2\leq2\lVert f\rVert^2$ with equality for a function whose support is disjoint from its strict translate. The family is therefore a strongly continuous but not norm-continuous unitary representation, which is exactly the situation covered by Stone's theorem and not by the norm-convergent exponential of a bounded generator.

The Infinitesimal Generator

Differentiation as the Generator

Theorem (generator of the translation group). On $L^2(\mathbb R^n)$ let $A_j$ be the operator defined on the Schwartz class by $A_jf=-\partial_jf$. Then $A_j$ is the infinitesimal generator of the one-parameter unitary group $t\mapsto\tau_{te_j}$: $$ \tau_{te_j}=e^{tA_j},\qquad A_jf=\lim_{t\to0}\frac{\tau_{te_j}f-f}{t}=-\partial_jf , $$ the limit being in $L^2$. The operator $A_j$ is closed, densely defined, and skew-adjoint, $A_j^\dagger=-A_j$; under the Fourier operator it becomes multiplication by the purely imaginary function $2\pi i\xi_j$.

Proof. On the Schwartz class the difference quotient converges to $-\partial_jf$ in $L^2$ by Taylor's formula and the integrability of the second derivative. The operator $-\partial_j$ is skew-adjoint because integration by parts gives $\langle-\partial_jf,g\rangle=\langle f,\partial_jg\rangle=\langle f,-(-\partial_j)g\rangle$ on $\mathcal S$, and $\mathcal S$ is a core. The identification with the Fourier side is $\mathcal F(-\partial_j)\mathcal F^{-1}=M_{-2\pi i\xi_j}$, the differentiation identity of the Fourier article with a sign. The exponential identity and the closedness are the content of Stone's theorem for a strongly continuous one-parameter unitary group, in Semigroups and Evolution Equations, later in this Part. $\blacksquare$

Corollary. The finite differences $\tau_{te_j}-\mathrm{id}$ approximate the generator, and $\tau_{te_j}=\mathrm{id}+tA_j+o(t)$; the forward difference $\Delta_h=\tau_{-h}-\mathrm{id}$ is the discrete analogue of $A_j$, and the two are related by $\Delta_{te_j}=t\partial_j+o(t)=-tA_j+o(t)$ on the domain of $A_j$.

The Spectrum and the Characters

The Characters as Eigenfunctions

Definition. A character of $G$ is a continuous homomorphism $\chi:G\to\mathbb T$ into the circle group; on $\mathbb R^n$ the characters are $e_\xi(x)=e^{2\pi ix\cdot\xi}$ for $\xi\in\mathbb R^n$, and on the circle they are $e_m(x)=e^{2\pi imx}$ for $m\in\mathbb Z$.

Proposition. Every character is an eigenfunction of every translation, with $$ \tau_g e_\xi=\overline{e_\xi(g)}\,e_\xi . $$ Consequently the characters simultaneously diagonalise the whole representation, and the algebra generated by the translations is commutative.

Proof. $\tau_ge_\xi(x)=e_\xi(x-g)=e_\xi(x)e_\xi(-g)=\overline{e_\xi(g)}e_\xi(x)$, using $e_\xi(-g)=\overline{e_\xi(g)}$ for a character. $\blacksquare$

The Spectrum

Theorem. On $L^2(\mathbb R^n)$ the translation $\tau_y$ is unitary; for $y=0$ it is the identity, and for $y\neq0$ its spectrum is the whole unit circle, $\sigma(\tau_y)=\mathbb T$, and it has no eigenvalues. On a finite group $G$ the translation $\tau_g$ has the characters as a basis of eigenvectors, with eigenvalues the $\lvert G\rvert$-th roots of unity; on $\ell^2(\mathbb Z)$ the shift $S=\tau_1$ has spectrum $\mathbb T$ and no eigenvalues.

Proof. The Fourier operator conjugates $\tau_y$ into multiplication by $e_y(\xi)=e^{-2\pi iy\cdot\xi}$ by The Fourier Operator of this group; the spectrum of a multiplication operator $M_m$ on $L^2$ is the essential range of $m$, and for $y\neq0$ the function $\xi\mapsto e^{-2\pi iy\cdot\xi}$ sweeps the circle, so the essential range is $\mathbb T$. An eigenvalue would require $e_y$ to be constant on a set of positive measure, which it is not. On a finite group the characters form an orthonormal basis and the eigenvalues are $\overline{\chi(g)}$, roots of unity; on $\ell^2(\mathbb Z)$ the shift equation $Sf=\lambda f$ gives $f_n=\lambda^{-n}f_0$, which is square-summable only if $f=0$, whatever $\lambda$. $\blacksquare$

Corollary (the diagonalisation). On a finite abelian group the translation operators are diagonal in the character basis; on $\mathbb R^n$ and on $\mathbb Z$ the characters diagonalise the representation in the sense of the spectral theorem or of the Gelfand transform, but the diagonalising functions are not elements of $L^2$, so that the diagonalisation is continuous rather than discrete.

The Commutant and the Convolution Operators

The Operators that Commute with Translation

Theorem. A bounded operator on $L^2(\mathbb R^n)$ commutes with every translation $\tau_y$ if and only if it is a Fourier multiplier operator $T_m$ for a unique $m\in L^\infty(\mathbb R^n)$; a bounded operator on $L^1(\mathbb R^n)$ with the same property is convolution with a finite measure. In particular the convolution operators $T_k$ with $k\in L^1$ are exactly the translation-invariant operators with an integrable profile.

Proof. This is the translation-invariance theorem of Convolution Operators, the previous article of this group: the Fourier operator carries the translations into the multiplications by the characters, and an operator commuting with all of them is a multiplication operator. The analysis case is the measure algebra. $\blacksquare$

Translation as the Atoms of Convolution

Theorem. For $k\in L^1(\mathbb R^n)$ the convolution operator is a superposition of translations, $$ (T_kf)(x)=\int_{\mathbb R^n}k(y)\,(\tau_yf)(x)\,dy , $$ the integral converging in $L^p$ for $f\in L^p$ and $1\leq p<\infty$. The closed algebra generated by the translations in the strong operator topology is the algebra of translation-invariant operators.

Proof. The identity is the definition of the convolution in the two-variable form, and the convergence is Minkowski's integral inequality, $\lVert\int k(y)\tau_yf\,dy\rVert_p\leq\lVert k\rVert_1\lVert f\rVert_p$; the closure statement is the theorem above together with the density of the profiles in the multipliers that are transforms of measures. $\blacksquare$

The General Locally Compact Group

The definitions of this article use only the group operation and an invariant measure. On a locally compact group $G$ with left Haar measure $\mu$, the left translation $(\tau_gf)(x)=f(g^{-1}x)$ is an isometry of $L^p(G,\mu)$ for every $p$ and a unitary of $L^2(G,\mu)$, and $g\mapsto\tau_g$ is strongly continuous, by the left invariance of the Haar measure; the convolution $(f*h)(x)=\int f(y)h(y^{-1}x)\,d\mu(y)$ is the superposition of the translations, and the operators commuting with the representation form its commutant, whose study — the group von Neumann algebra, the unitary representations and their irreducibility — is the subject of Locally Compact Groups and Haar Measure, Representations of Locally Compact Groups and Harmonic Analysis on Groups, in the neighbouring category Analysis on Groups of this Part. The present Article is the abelian and explicit instance of that theory, and the one the operator articles of this corpus use.

Summary

The translation $\tau_g$ on a function space, $(\tau_gf)(x)=f(g^{-1}x)$, is a representation of the group: $\tau_g\tau_h=\tau_{gh}$, $\tau_e=\mathrm{id}$. On $L^p$ of a group with invariant measure it is an isometry, and on $L^2$ a unitary; on $L^p(\mathbb R^n)$, $1\leq p<\infty$, it is strongly continuous in the group variable but not continuous in the operator norm, $\lVert\tau_y-\mathrm{id}\rVert=\sqrt2$ for $y\neq0$. Its infinitesimal generator on the one-parameter subgroup $t\mapsto\tau_{te_j}$ is the skew-adjoint operator $-\partial_j$, which the Fourier operator turns into multiplication by $2\pi i\xi_j$; the characters are eigenfunctions of every translation, $\tau_ge_\xi=\overline{e_\xi(g)}e_\xi$, and they diagonalise the representation, discretely on a finite group and in the spectral sense on $\mathbb R^n$ and $\mathbb Z$. The translation $\tau_y$ for $y\neq0$ is unitary with spectrum the unit circle and no eigenvalues on $L^2(\mathbb R^n)$. The operators commuting with all translations are exactly the multiplier operators, and a convolution operator is an integral superposition of translations; the general locally compact group instance belongs to the group-theoretic category of this Part.

Summary of Notation

Further Reading

• Walter Rudin, Fourier Analysis on Groups (Interscience, 1962), for translation, convolution and the characters on a locally compact abelian group.
• Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis (Van Nostrand, 1953; reprinted Dover, 2011), for the representation theory of the translation group and its commutant.
• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, 1980), for strongly continuous unitary groups, Stone's theorem and the generator.
• Elias M. Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, 1971), for translation, the continuity of translation and the multiplier operators.
• Kôsaku Yosida, Functional Analysis (6th ed., Springer, 1980), for the translation group on $L^p$ and the semigroup generated by differentiation.