Series Analysis and Schwartz Algebras of Spherical Convolutions on Semisimple Lie Groups

TL;DR

This paper explores the harmonic analysis of spherical convolutions on semisimple Lie groups, focusing on the Harish-Chandra transform and Schwartz algebras, and clarifies the role of the Trombi-Varadarajan Theorem in this context.

Contribution

It provides exact descriptions of the Harish-Chandra transform of Schwartz functions and demonstrates how the Trombi-Varadarajan Theorem applies to spherical convolution transforms.

Findings

Explicit characterization of the Harish-Chandra transform for Schwartz functions

Connection established between spherical convolutions and $L^{p}$-Schwartz algebras

Proof of the role of Trombi-Varadarajan Theorem in harmonic analysis

Abstract

We give the exact contributions of Harish-Chandra transform, $(\mathcal{H}f)(\lambda),$ of Schwartz functions $f$ to the harmonic analysis of spherical convolutions and the corresponding $L^{p}-$ Schwartz algebras on a connected semisimple Lie group $G$ (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of $L^{p}-$ Schwartz functions.

Equations

$${\mathfrak{g}}_{\lambda }=\{X\in \mathfrak{g}:[H,X]=\lambda (H)X,\forall H\in \mathfrak{a}\},$$

$$(s\lambda )(H)=\lambda (s^{-1}H),$$

$$G=\underset{s\in \mathfrak{w}}{⨆}B{m}_{s}B$$

$${\int }_K\phi (xky)dk=\phi (x)\phi (y),x,y\in G.$$

$${\phi }_{\lambda }(x)={\int }_K{e}^{(i\lambda -p)H(xk)}dk,\lambda \in {\mathfrak{a}}_{\mathbb{C}}^{\ast },$$

$$\Omega {\phi }_{\lambda }=-(⟨\lambda ,\lambda ⟩+⟨\rho ,\rho ⟩){\phi }_{\lambda },$$

$$\sum _{i,j=1}^m{\alpha }_i{\bar{\alpha }}_{j}f(x_i^{-1}{x}_j)\ge 0.$$

$${\mathfrak{F}}^1={\mathfrak{a}}^{\ast }+i{C}_{\rho }$$

$${\phi }_0(x):={\int }_K\exp (-\rho (H(xk)))dk$$

$$\sigma (x)=\Vert X\Vert$$

$$1\le \Xi (a)e^{\rho (\log a)}\le c(1+\sigma (a){)}^d.$$

$${\mu }_{a,b;r}(f):=\underset{G}{\sup }[|f(a;x;b)|\Xi (x{)}^{-2/p}(1+\sigma (x){)}^r]<\infty$$

$$C_c^{\infty }(G)\subset {\mathcal{C}}^p(G)\subset L^p(G)$$

$$\mid f(y)\mid \le c_r\Xi (y^{-1}x)(1+\sigma (y^{-1}x){)}^{-r}\forall x,y\in G.$$

$$\mid f(y)\mid \le c_r\Xi (y^{-1})(1+\sigma (y^{-1}){)}^{-r}=c_r\Xi (y)(1+\sigma (y){)}^{-r},\forall y\in G,$$

$$\hat{f}(\lambda )={\int }_{G}f(x){\phi }_{-\lambda }(x)dx,$$

$$x\mapsto (f\ast {\phi }_{\lambda })(x)=:s_{\lambda ,f}(x)$$

$$(1.)\;\;\;\;\;\;\;\;\;\;x\mapsto k_{1}(\lambda):=s_{\lambda,f}(x),\;\mbox{for any $f\in{\cal C}^{p}(G)$}$$

$$(2.)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\mapsto k_{2}(f):=s_{\lambda,f}(x),\;\mbox{for any $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$},$$

$$(3.)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f\mapsto l_{1}(\lambda):=s_{\lambda,f}(x),\;\mbox{for any $x\in G$}$$

$$(4.)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;f\mapsto l_{2}(x):=s_{\lambda,f}(x),\;\mbox{for any $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$},$$

$$(5.)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\lambda\mapsto m_{1}(f):=s_{\lambda,f}(x),\;\mbox{for any $x\in G$}$$

$$(6.)\;\;\;\;\;\;\;\;\;\;\lambda\mapsto m_{2}(x):=s_{\lambda,f}(x),\;\mbox{for any $f\in{\cal C}^{p}(G)$},$$

$$C_{\rho }=\left\{\sum _{i=1}^n{\lambda }_i(s_i\rho ):{\lambda }_i\ge 0,\sum _{i=1}^n{\lambda }_i=1,s_i\in \mathfrak{w}\right\}$$

$$int({\mathfrak{F}}^{ϵ})=\bigcup _{0<{ϵ}^{\prime }<ϵ}{\mathfrak{F}}^{{ϵ}^{{}^{\prime }}}$$

$$f\mapsto l_{1}(\lambda):=s_{\lambda,f}(x),\;\mbox{for any $x\in G$},$$

$$\Vert \Psi {\Vert }_{{\mathcal{Z}}_G({\mathfrak{F}}^{ϵ})}:=\underset{int({\mathfrak{F}}^{ϵ})\times G}{\sup }|D_1\Psi D_2|.$$

$$s_{\lambda ,f}(x):=(f\ast {\phi }_{\lambda })(x),x\in G.$$

$$s_{\lambda ,f}(x\exp tX)=\sum _{n=0}^{\infty }\frac{t^n}{n!}[{\tilde{X}}^n{s}_{\lambda ,f}](x),$$

$$s_{\lambda ,f}(\exp X)=\sum _{n=0}^{\infty }\frac{1}{n!}[{\tilde{X}}^n{s}_{\lambda ,f}](e)=s_{\lambda ,f}(e)+\sum _{n=1}^{\infty }\frac{1}{n!}[{\tilde{X}}^n{s}_{\lambda ,f}](e)$$

Full text

**Series Analysis and Schwartz Algebras of Spherical Convolutions on Semisimple Lie Groups.

**

**Olufemi O. Oyadare

**

Abstract. We give the exact contributions of Harish-Chandra transform, $(\mathcal{H}f)(\lambda),$ of Schwartz functions $f$ to the harmonic analysis of spherical convolutions and the corresponding $L^{p}-$ Schwartz algebras on a connected semisimple Lie group $G$ (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of $L^{p}-$ Schwartz functions.

Subject Classification: $43A85,\;\;22E30,\;\;22E46$

Keywords: Harish-Chandra Transforms; Semisimple Lie groups; Harish-Chandra’s Schwartz algebras

 

$\bf{1}\;\;\;\;$** Introduction

**

Let $G$ be a connected semisimple Lie group with finite center, and denote the Harish-Chandra-type Schwartz spaces of functions on $G$ by ${\cal C}^{p}(G)$, $0<p\leq 2.$ We know that ${\cal C}^{p}(G)\subset L^{p}(G)$ for every such $p$, and if $K$ is a maximal compact subgroup of $G$ such that ${\cal C}^{p}(G//K)$ represents the subspace of ${\cal C}^{p}(G)$ consisting of the $K-$bi-invariant functions, Trombi and Varadarajan ([$9.$]) have shown that the spherical Fourier transform $f\mapsto\widehat{f}$ is a linear topological isomorphism of ${\cal C}^{p}(G//K)$ onto the spaces $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})$, $\epsilon=\left(2/p\right)-1,$  

 

$\overline{\textmd{Department\;of\;Mathematics,}}$ Obafemi Awolowo University, Ile-Ife, $220005,$ Nigeria.

E-mail: femi_oya@yahoo.com  

 

consisting of rapidly decreasing functions on certain sets ${\mathfrak{F}}^{\epsilon}$ of elementary spherical functions.

We show the existence of a hyper-function on both $G$ and ${\mathfrak{F}}^{1}$ (here named a spherical convolution) whose restriction to the group identity element, $e,$ coincides with the spherical Fourier transforms, $f\mapsto\widehat{f},$ of Schwartz functions $f$ on $G$ and which affords us the opportunity of embarking on a more inclusive harmonic analysis on $G.$ Indeed $[8a.]$ contains a more general Plancherel formula for the collection of these functions. As a function on $G$ its series expansion is in the present paper studied. We show that, aside from the fact that the spherical Fourier transforms, $\widehat{f}(\lambda),$ is the constant term of this series expansion, there is a region in $G$ where the spherical convolution is essentially $\widehat{f}(\lambda).$ Various algebras of these functions are thus studied and ultimately embedded in $L^{2}(G).$ It is however clear that the results in $[8.]$ and in the present paper may be extended to include what may be termed as the Harish-Chandra-type Schwartz spaces of Eisenstein Integrals on $G.$ The author has recently used the idea of a spherical convolution to give an explicit computation of the image of $\mathcal{C}^{2}(G)$ under the Harish-Chandra transform, $[8b.]$ thus giving a concrete realization of the abstract results of Arthur, $[2.],$ and showing the direct contribution of the Plancherel formula to Harish-Chandra transform on $G.$

The following is the breakdown of each of the remaining sections of the paper. $\S 2.$ contains the preliminaries to the research containing the structure theory, spherical functions and Schwartz algebras on $G,$ while the series analysis of spherical convolutions on $G$ is the subject of $\S 3.$ The relationship existing among the Schwartz algebras of functions and those of spherical convolutions is considered in $\S 4.$

 

$\bf{2}\;\;\;\;$** Preliminaries

**

For the connected semisimple Lie group $G$ with finite center, we denote its Lie algebra by $\mathfrak{g}$ whose Cartan decomposition is given as $\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{p}.$ Denote by $\theta$ the Cartan involution on $\mathfrak{g}$ whose collection of fixed points is $\mathfrak{t}.$ We also denote by $K$ the analytic subgroup of $G$ with Lie algebra $\mathfrak{t}.$ $K$ is then a maximal compact subgroup of $G.$ Choose a maximal abelian subspace $\mathfrak{a}$ of $\mathfrak{p}$ with algebraic dual $\mathfrak{a}^{*}$ and set $A=\exp\mathfrak{a}.$ For every $\lambda\in\mathfrak{a}^{*}$ put

[TABLE]

and call $\lambda$ a restricted root of $(\mathfrak{g},\mathfrak{a})$ whenever $\mathfrak{g}_{\lambda}\neq\{0\}.$

Denote by $\mathfrak{a}^{\prime}$ the open subset of $\mathfrak{a}$ where all restricted roots are $\neq 0,$ and call its connected components the Weyl chambers. Let $\mathfrak{a}^{+}$ be one of the Weyl chambers, define the restricted root $\lambda$ positive whenever it is positive on $\mathfrak{a}^{+}$ and denote by $\triangle^{+}$ the set of all restricted positive roots. Members of $\triangle^{+}$ which form a basis for $\triangle$ and can not be written as a linear combination of other members of $\triangle^{+}$ are called simple. We then have the Iwasawa decomposition $G=KAN$, where $N$ is the analytic subgroup of $G$ corresponding to $\mathfrak{n}=\sum_{\lambda\in\triangle^{+}}\mathfrak{g}_{\lambda}$, and the polar decomposition $G=K\cdot cl(A^{+})\cdot K,$ with $A^{+}=\exp\mathfrak{a}^{+},$ and $cl(A^{+})$ denoting the closure of $A^{+}.$

If we set $M=\{k\in K:Ad(k)H=H$, $H\in\mathfrak{a}\}$ and $M^{\prime}=\{k\in K:Ad(k)\mathfrak{a}\subset\mathfrak{a}\}$ and call them the centralizer and normalizer of $\mathfrak{a}$ in $K,$ respectively, then (see $[5.]$, p. $284$); (i) $M$ and $M^{\prime}$ are compact and have the same Lie algebra and (ii) the factor $\mathfrak{w}=M^{\prime}/M$ is a finite group called the Weyl group. $\mathfrak{w}$ acts on $\mathfrak{a}^{*}_{\mathbb{C}}$ as a group of linear transformations by the requirement

[TABLE]

$H\in\mathfrak{a}$, $s\in\mathfrak{w}$, $\lambda\in\mathfrak{a}^{*}_{\mathbb{\mathbb{C}}}$, the complexification of $\mathfrak{a}^{*}$. We then have the Bruhat decomposition

[TABLE]

where $B=MAN$ is a closed subgroup of $G$ and $m_{s}\in M^{\prime}$ is the representative of $s$ (i.e., $s=m_{s}M$). The Weyl group invariant members of a space shall be denoted by the superscript w while $\mid\mathfrak{w}\mid$ represents the cardinality of $\mathfrak{w}.$

Some of the most important functions on $G$ are the spherical functions which we now discuss as follows. A non-zero continuous function $\varphi$ on $G$ shall be called a (zonal) spherical function whenever $\varphi(e)=1,$ $\varphi\in C(G//K):=\{g\in C(G)$: $g(k_{1}xk_{2})=g(x)$, $k_{1},k_{2}\in K$, $x\in G\}$ and $f*\varphi=(f*\varphi)(e)\cdot\varphi$ for every $f\in C_{c}(G//K),$ where $(f\ast g)(x):=\int_{G}f(y)g(y^{-1}x)dy.$ This leads to the existence of a homomorphism $\lambda:C_{c}(G//K)\rightarrow\mathbb{C}$ given as $\lambda(f)=(f*\varphi)(e)$. This definition is equivalent to the satisfaction of the functional relation

[TABLE]

It has been shown by Harish-Chandra [$6.$] that spherical functions on $G$ can be parametrized by members of $\mathfrak{a}^{*}_{\mathbb{C}}.$ Indeed every spherical function on $G$ is of the form

[TABLE]

$\rho=\frac{1}{2}\sum_{\lambda\in\triangle^{+}}m_{\lambda}\cdot\lambda,$ where $m_{\lambda}=dim(\mathfrak{g}_{\lambda}),$ and that $\varphi_{\lambda}=\varphi_{\mu}$ iff $\lambda=s\mu$ for some $s\in\mathfrak{w}.$ Some of the well-known properties of spherical functions are $\varphi_{-\lambda}(x^{-1})=\varphi_{\lambda}(x),$ $\varphi_{-\lambda}(x)=\bar{\varphi}_{\bar{\lambda}}(x),$ $\mid\varphi_{\lambda}(x)\mid\leq\varphi_{\Re\lambda}(x),$ $\mid\varphi_{\lambda}(x)\mid\leq\varphi_{i\Im\lambda}(x),$ $\varphi_{-i\rho}(x)=1,$ $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},$ while $\mid\varphi_{\lambda}(x)\mid\leq\varphi_{0}(x),\;\lambda\in i\mathfrak{a}^{*},\;x\in G.$ Also if $\Omega$ is the Casimir operator on $G$ then

[TABLE]

where $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}$ and $\langle\lambda,\mu\rangle:=tr(adH_{\lambda}\ adH_{\mu})$ for elements $H_{\lambda}$, $H_{\mu}\in{\mathfrak{a}}.$ This differential equation may be written simply as $\Omega\varphi_{\lambda}=\gamma(\Omega)(\lambda)\varphi_{\lambda},$ where $\lambda\mapsto\gamma(\Omega)(\lambda)$ is the well-known Harish-Chandra homomorphism. The elements $H_{\lambda}$, $H_{\mu}\in{\mathfrak{a}}$ are uniquely defined by the requirement that $\lambda(H)=tr(adH\ adH_{\lambda})$ and $\mu(H)=tr(adH\ adH_{\mu})$ for every $H\in{\mathfrak{a}}$ ([$5.$], Theorem $4.2$). Clearly $\Omega\varphi_{0}=0.$

Due to a hint dropped by Dixmier $[4.]$ $(cf.\;[9.])$ in his discussion of some functional calculus, it is necessary to recall the notion of a ‘positive-definite’ function and then discuss the situation for positive-definite spherical functions. We call a continuous function $f:G\rightarrow\mathbb{C}$ (algebraically) positive-definite whenever, for all $x_{1},\dots,x_{m}$ in $G$ and all $\alpha_{1},\dots,\alpha_{m}$ in $\mathbb{C},$ we have

[TABLE]

It can be shown $(cf.\;[5.])$ that $f(e)\geq 0$ and $|f(x)|\leq f(e)$ for every $x\in G$ implying that the space ${\cal P}$ of all positive-definite spherical functions on $G$ is a subset of the space ${\mathfrak{F}}^{1}$ of all bounded spherical functions on $G.$

We know, by the Helgason-Johnson theorem ($[7.]$), that

[TABLE]

where $C_{\rho}$ is the convex hull of $\{s\rho:s\in\mathfrak{w}\}$ in $\mathfrak{a}^{*}.$ Defining the involution $f^{*}$ of $f$ as $f^{*}(x)=\overline{f(x^{-1})}$, it follows that $f=f^{*}$ for every $f\in{\cal P}$, and if $\varphi_{\lambda}\in{\cal P}$, then $\lambda$ and $\bar{\lambda}$ are Weyl group conjugate, leading to a realization of $\mathcal{P}$ as a subset of $\mathfrak{w}\setminus\mathfrak{a}^{*}_{\mathbb{C}}.$ ${\cal P}$ becomes a locally compact Hausdorff space when endowed with the weak ${}^{*}-$topology as a subset of $L^{\infty}(G)$.

Let

[TABLE]

be denoted as $\Xi(x)$ and define $\sigma:G\rightarrow\mathbb{C}$ as

[TABLE]

for every $x=k\exp X\in G,\;\;k\in K,\;X\in\mathfrak{a},$ where $\|\cdot\|$ is a norm on the finite-dimensional space $\mathfrak{a}.$ These two functions are spherical functions on $G$ and there exist numbers $c,d$ such that

[TABLE]

Also there exists $r>0$ such that $c=:\int_{G}\Xi(x)^{2}(1+\sigma(x))^{r}dx<\infty$ ($[11.],$ p. $231$). For each $0\leq p\leq 2$ define ${\cal C}^{p}(G)$ to be the set consisting of functions $f$ in $C^{\infty}(G)$ for which

[TABLE]

where $a,b\in\mathfrak{U}(\mathfrak{g}_{\mathbb{C}}),$ the universal enveloping algebra of $\mathfrak{g}_{\mathbb{C}},$ $r\in\mathbb{Z}^{+},x\in G,$ $f(x;b):=\left.\frac{d}{dt}\right|_{t=0}f(x\cdot(\exp tb))$ and $f(a;x):=\left.\frac{d}{dt}\right|_{t=0}f((\exp ta)\cdot x).$ We call ${\cal C}^{p}(G)$ the Schwartz space on $G$ for each $0<p\leq 2$ and note that ${\cal C}^{2}(G)$ is the well-known (see $[1.]$) Harish-Chandra space of rapidly decreasing functions on $G.$ The inclusions

[TABLE]

hold and with dense images. It also follows that ${\cal C}^{p}(G)\subseteq{\cal C}^{q}(G)$ whenever $0\leq p\leq q\leq 2.$ Each ${\cal C}^{p}(G)$ is closed under involution and the convolution, $*.$ Indeed ${\cal C}^{p}(G)$ is a Fr$\acute{e}$chet algebra ($[10.],$ p. $69$). We endow ${\cal C}^{p}(G//K)$ with the relative topology as a subset of ${\cal C}^{p}(G).$

We shall say a function $f$ on $G$ satisfies a general strong inequality if for any $r\geq 0$ there is a constant $c=c_{r}>0$ such that

[TABLE]

We observe that if $x=e$ then, using the fact that $\Xi(y^{-1})=\Xi(y)$ and $\sigma(y^{-1})=\sigma(y),\;\forall\;y\in G,$ such a function satisfies

[TABLE]

showing that a function on $G$ which satisfies a general strong inequality satisfies in particular a strong inequality (in the classical sense of Harish-Chandra, $[11.]$). Members of $\mathcal{C}^{2}(G)=:\mathcal{C}(G)$ are those functions $f$ on $G$ for which $f(g_{1};\cdot;g_{2})$ satisfies the strong inequality, for all $g_{1},g_{2}\in\mathfrak{U}(\mathfrak{g}_{\mathbb{C}}).$ We may then define $\mathcal{C}^{(x)}(G)$ to be those functions $f$ on $G$ for which $f(g_{1};\cdot;g_{2})$ satisfies the general strong inequality, for all $g_{1},g_{2}\in\mathfrak{U}(\mathfrak{g}_{\mathbb{C}})$ and a fixed $x\in G.$ It is clear that $\mathcal{C}^{(e)}(G)=\mathcal{C}(G)$ and that $\bigcup_{x\in G}\mathcal{C}^{(x)}(G),$ which contains $\mathcal{C}(G),$ may be given an inductive limit topology. The seminorms defining this topology will be explicitly given in $\S 4.$

For any measurable function $f$ on $G$ we define the spherical Fourier transform $\widehat{f}$ as

[TABLE]

$\lambda\in\mathfrak{a}^{*}_{\mathbb{C}}.$ It is known (see $[3.]$) that for $f,g\in L^{1}(G)$ we have:

1. $(i.)$

$(f*g)^{\wedge}=\widehat{f}\cdot\widehat{g}$ on ${\mathfrak{F}}^{1}$ whenever $f$ (or $g$) is right - (or left-) $K$-invariant; 2. $(ii.)$

$(f^{*})^{\wedge}(\varphi)=\overline{\widehat{f}(\varphi^{*})},\varphi\in{\mathfrak{F}}^{1}$; hence $(f^{*})^{\wedge}=\overline{\widehat{f}}$ on ${\cal P}:$ and, if we define $f^{\#}(g):=\int_{K\times K}f(k_{1}xk_{2})dk_{1}dk_{2},x\in G,$ then 3. $(iii.)$

$(f^{\#})^{\wedge}=\widehat{f}$ on ${\mathfrak{F}}^{1}.$

We shall denote the spherical Fourier transform $\widehat{f}(\lambda)$ of $f\in\mathcal{C}(G)$ by $(\mathcal{H}f)(\lambda)$ and refer to it as the Harish-Chandra transforms of $f.$ Its major properties are well-known and may be found in $[9.].$ It should be noted that $(\mathcal{H}f)(\lambda)=\widehat{f}(\lambda)=\int_{G}f(y)\varphi_{-\lambda}(y)dy=\int_{G}f(y)\varphi_{\lambda}(y^{-1})dy=\int_{G}f(y)\varphi_{\lambda}(y^{-1}e)dy$

$=(f\ast\varphi_{\lambda})(e).$ That is, the Harish-Chandra transforms of $f$ is the restriction of the function

[TABLE]

on $G$ to the identity element. It is therefore worthwhile to explore $s_{\lambda,f}(x)$ in some details for all $x\in G$ in order to put its behaviour at $x=e$ (as the Harish-Chandra transforms of $f$) in a proper and larger perspective.

The beauty of studying the entirety of the function $s_{\lambda,f}(x),$ for $\lambda\in\mathfrak{a}^{*}_{\mathbb{C}},\;f\in\mathcal{C}^{p}(G),\;x\in G,$ which we shall explore in this paper, is that it could be viewed as a transformation in six ($6$) different ways; As

[TABLE]

and

[TABLE]

(from where the Plancherel formula for the space of functions $x\mapsto k_{2}(f)$ has recently been computed in $[8a.]$) both of which are maps on $G;$ or as

[TABLE]

(which, at $x=e,$ led Harish-Chandra to the consideration of $f\mapsto(\mathcal{H}f)(\lambda):\;cf.\;[9.]$) and

[TABLE]

both of which are maps on ${\cal C}^{p}(G);$ or as

[TABLE]

and

[TABLE]

both of which are maps on $\mathfrak{a}^{*}_{\mathbb{C}}.$ Hence the function $x\mapsto s_{\lambda,f}(x)$ may rightly be called an hyper-function on $G$ whose major contribution to harmonic analysis would be to absorb other known functions of the subject and put their results in proper perspectives, as we shall establish here for the *Harish-Chandra transform.

In order to know the image of the spherical Fourier transform when restricted to ${\cal C}^{p}(G//K)$ we need the following spaces that are central to the statement of the well-known result of Trombi and Varadarajan [$9.$]. Let $C_{\rho}$ be the closed convex hull of the (finite) set $\{s\rho:s\in\mathfrak{w}\}$ in $\mathfrak{a}^{*}$, i.e.,

[TABLE]

where we recall that, for every $H\in\mathfrak{a},\;\;(s\rho)(H)=\frac{1}{2}\sum_{\lambda\in\triangle^{+}}m_{\lambda}\cdot\lambda(s^{-1}H).$

Now for each $\epsilon>0$ set ${\mathfrak{F}}^{\epsilon}=\mathfrak{a}^{*}+i\epsilon C_{\rho}.$ Each ${\mathfrak{F}}^{\epsilon}$ is convex in $\mathfrak{a}^{*}_{\mathbb{C}}$ and

[TABLE]

([$9.$], Lemma $(3.2.2)$). Let us define $\mathcal{Z}({\mathfrak{F}}^{0})=\mathcal{S}(\mathfrak{a}^{*})$ and, for each $\epsilon>0,$ let $\mathcal{Z}({\mathfrak{F}}^{\epsilon})$ be the space of all $\mathbb{C}$-valued functions $\Phi$ such that $(i.)$ $\Phi$ is defined and holomorphic on $int({\mathfrak{F}}^{\epsilon}),$ and $(ii.)$ for each holomorphic differential operator $D$ with polynomial coefficients we have $\sup_{int({\mathfrak{F}}^{\epsilon})}|D\Phi|<\infty.$

The space $\mathcal{Z}({\mathfrak{F}}^{\epsilon})$ is converted to a Fr$\acute{e}$chet algebra by equipping it with the topology generated by the collection, $\|\cdot\|_{\mathcal{Z}({\mathfrak{F}}^{\epsilon})},$ of seminorms given by $\|\Phi\|_{\mathcal{Z}({\mathfrak{F}}^{\epsilon})}:=\sup_{int({\mathfrak{F}}^{\epsilon})}|D\Phi|.$ It is known that $D\Phi$ above extends to a continuous function on all of ${\mathfrak{F}}^{\epsilon}$ ([$9.$], pp. $278-279$). An appropriate subalgebra of $\mathcal{Z}({\mathfrak{F}}^{\epsilon})$ for our purpose is the closed subalgebra $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})$ consisting of $\mathfrak{w}$-invariant elements of $\mathcal{Z}({\mathfrak{F}}^{\epsilon})$, $\epsilon\geq 0.$ The following (known as the Trombi-Varadarajan Theorem) is the major result of $[9.]:$ Let $0<p\leq 2$ and set $\epsilon=\left(2/p\right)-1.$ Then the spherical Fourier transform $f\mapsto\widehat{f}$ is a linear topological algebra isomorphism of ${\cal C}^{p}(G//K)$ onto $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$ That is, the topological algebra $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})$ is an isomorphic copy or a realization of ${\cal C}^{p}(G//K).$

In order to find other isomorphic copies or realizations of ${\cal C}^{p}(G//K)$ under the more inclusive general transformation map

[TABLE]

we shall now introduce a more general algebra, $\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon}),$ of $\mathbb{C}-$valued functions on $int({\mathfrak{F}}^{\epsilon})\times G$ which, when restricted to $int({\mathfrak{F}}^{\epsilon})\times\exp(N_{0}),$ coincides with $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$ The form of this new algebra is suggested by Theorem $3.5.$ Set $\mathcal{Z}_{G}({\mathfrak{F}}^{0})=\mathcal{S}(\mathfrak{a}^{*})\times G$ and let $\mathcal{Z}_{G}({\mathfrak{F}}^{\epsilon}),\;\epsilon>0,$ be the collection of all $\mathbb{C}-$valued functions $\Psi$ ($(\lambda,x)\mapsto\Psi(\lambda,x),\;\forall\;(\lambda,x)\in int({\mathfrak{F}}^{\epsilon})\times G$) such that

$(i.)$ $\Psi$ is holomorphic in the variable $\lambda,$ analytic in $x$ and spherical on $G;$

$(ii.)$ $\sup_{int({\mathfrak{F}}^{\epsilon})}|D_{1}\Psi|<\infty$ and $\sup_{G}|\Psi D_{2}|<\infty,$ for every holomorphic differential operator $D_{1}$ with polynomial coefficients and every left-invariant differential operator $D_{2}$ on $G$ and

$(iii.)$ the restriction of $\Psi$ to $int({\mathfrak{F}}^{\epsilon})\times\{e\}$ (or to $int({\mathfrak{F}}^{\epsilon})\times\exp(N_{0}(A^{+})),$ for some zero neighbourhood $N_{0}(A^{+})$ in $\mathfrak{g},$ as will later be seen in Theorem $3.5$) is (a non-zero constant multiple of) the Harish-Chandra transform, $(\mathcal{H}f)(\lambda)=\hat{f}.$

It may be shown, in exact manner as for $\mathcal{Z}({\mathfrak{F}}^{\epsilon})$ above, that the space $\mathcal{Z}_{G}({\mathfrak{F}}^{\epsilon})$ is converted to a Fr$\acute{e}$chet algebra by equipping it with the topology generated by the collection, $\|\cdot\|_{\mathcal{Z}_{G}({\mathfrak{F}}^{\epsilon})},$ of seminorms given by

[TABLE]

An appropriate subalgebra of $\mathcal{Z}_{G}({\mathfrak{F}}^{\epsilon})$ for our purpose is the closed subalgebra $\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ consisting of $\mathfrak{w}$-invariant elements of $\mathcal{Z}_{G}({\mathfrak{F}}^{\epsilon})$, $\epsilon\geq 0.$ By the time Theorem $3.5$ is established it will be clear that $\bar{\mathcal{Z}}_{\{x\}}({\mathfrak{F}}^{\epsilon})\simeq\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}),$ for every $x$ in some zero neighbourhood $N_{0}(A^{+})$ in $\mathfrak{g}.$ In particular, $\bar{\mathcal{Z}}_{\{e\}}({\mathfrak{F}}^{\epsilon})\simeq\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$  

 

$\bf{3}\;\;\;\;$** Series Analysis of Spherical Convolutions

**

Let $f\in\mathcal{C}(G)$ and $\lambda\in\mathfrak{a}^{\ast}_{\mathbb{C}},$ we recall from $[8a.]$ the definition of spherical convolutions, $s_{\lambda,f},$ on $G$ corresponding to the pair $(\lambda,f)$ as

[TABLE]

We already know that $s_{\lambda,f}(e)=(\mathcal{H}f)(\lambda),$ where $e$ is the identity element of $G$ and $\lambda\in i\mathfrak{a}^{\ast}.$ This relation between a function on $G$ at the identity element and another function on $i\mathfrak{a}^{\ast}$ suggests we study the full contribution of the Harish-Chandra transforms, $(\mathcal{H}f)(\lambda),$ of $f$ to the properties of $x\mapsto s_{\lambda,f}(x)$ and to seek other functions on $i\mathfrak{a}^{\ast}$ which have not been known in the harmonic analysis of $G,$ but still contribute to a deeper understanding of the structure of $G.$

In order to explore the nature of this idea we consider opening up the spherical convolutions $x\mapsto s_{\lambda,f}(x)$ via its *Taylor’s series expansion.

Lemma 3.1. Let $N_{0}$ be a neighbourhood of origin in $\mathfrak{g}$ and $t$ be sufficiently small in $\mathbb{R}$ (say $0\leq t\leq 1$). Then

[TABLE]

where for every $\;X\in N_{0}$ we set $[\tilde{X}^{n}s_{\lambda,f}](x)=\frac{d^{n}}{du^{n}}s_{\lambda,f}(x\exp uX)_{|_{u=0}}$

Proof. The proof follows from a direct application of Taylor’s series expansion, $[5.],\;p.\;105.\;\;\Box$

At $x=e$ and $t=1$ the formula in the Lemma becomes

[TABLE]

[TABLE]

This observation leads quickly to the following result which gives the exact contribution of the Harish-Chandra transforms to the study of spherical convolutions.

Lemma 3.2. *The Harish-Chandra transforms, $\lambda\mapsto(\mathcal{H}f)(\lambda),\;f\in\mathcal{C}(G),$ is the constant term in the (Taylor’s) series expansion of spherical convolutions, $x\mapsto s_{\lambda,f}(x)$ around $x=e,$ for every $\lambda\in\mathfrak{a}^{*}.\;\;\Box$

It may be deduced, from the expansion leading to the proof Lemma $3.2,$ that the only time the remaining terms in $s_{\lambda,f}(\exp X),$ after the (non-zero) constant term $(\mathcal{H}f)(\lambda),$ could vanish is when the differential operator $\tilde{X}=0.$ That is, when $X=0.$ It therefore follows that the well-known (Harish-Chandra) harmonic analysis on $G$ $([1.],[2.],[9.]\;\mbox{and}\;[11.])$ has always been that of the consideration of the map $X\mapsto s_{\lambda,f}(\exp X)$ at only $X=0,$ which is the origin of $\mathfrak{g}$ or which corresponds to the identity point of $\exp(\mathfrak{g}).$ Hence, since the constant term, $(\mathcal{H}f)(\lambda),$ of $s_{\lambda,f}(\exp X)$ corresponds indeed to the consideration of the constant term in the asymptotic expansion of (zonal) spherical functions, $\varphi_{\lambda},$ it also follows that other terms in the expansion of $\varphi_{\lambda}$ may be needed to completely understand $f\mapsto s_{\lambda,f}(x).$

The expression for $s_{\lambda,f}(\exp X)$ therefore suggests that a full harmonic analysis of $G$ may be attained from a close study of the remaining contributions of the transform of $f$ given as

[TABLE]

for all $X\in N_{0},\;n\in\mathbb{N}\cup\{0\},\;x\in G,\;f\in\mathcal{C}(G)$ and sufficiently small values of $t,$ in the same manner that its constant term,

[TABLE]

had been considered.

However before considering the transformational properties of spherical convolutions we note the following lemmas which lead to a more inclusive view of the Trombi-Varadarajan Theorem and prepares the ground for its generalization.

Lemma 3.3. Let $N_{0}$ be a neighbourhood of origin in $\mathfrak{g},$ $\lambda\in\mathfrak{a}^{\ast}_{\mathbb{C}}$ and $t$ be sufficiently small in $\mathbb{R}$ (say $0\leq t\leq 1$). Then

[TABLE]

for every $\;X\in N_{0},\;x\in G,\;f\in\mathcal{C}(G).$

Proof. We note here that

[TABLE]

[TABLE]

Hence

[TABLE]

The particular case of setting $x=e$ and $t=1$ in Lemma $3.3$ introduces the Harish-Chandra transforms, $(\mathcal{H}f)(\lambda),$ into the analysis of this series, proving the following.

Lemma 3.4. Let $N_{0}$ be a neighbourhood of origin in $\mathfrak{g},$ $f\in\mathcal{C}(G)$ and $\lambda\in\mathfrak{a}^{\ast}.$ Then the spherical convolution function, $x\mapsto s_{\lambda,f}(x)$ is a non-zero constant multiple of the Harish-Chandra transforms, $(\mathcal{H}f)(\lambda),$ on $\exp(N_{0}).$

Proof. Set $x=e$ and $t=1$ into Lemma $3.3$ to have

[TABLE]

with $\sum^{\infty}_{n=0}\frac{1}{n!}\gamma(\frac{d^{n}}{du^{n}})(\lambda)_{|_{u=0}}=1+[\sum^{\infty}_{n=1}\frac{1}{n!}\gamma(\frac{d^{n}}{du^{n}})(\lambda)_{|_{u=0}}]\neq 0.\;\;\Box$

Let us denote the non-zero constant in Lemma $3.4$ above by $\kappa.$ The following theorem is a consequence of normalizing the spherical convolutions in Lemma $3.4.$

Theorem 3.5. (Trombi-Varadarajan Theorem for Spherical Convolutions) *Let $0<p\leq 2,$ set $\epsilon=\left(2/p\right)-1$ and $x\in\exp(N_{0}).$ Set $\widehat{f}_{x}(\lambda)=\frac{1}{\kappa}s_{\lambda,f}(x)$ for $f\in{\cal C}^{p}(G//K).$ Then the spherical convolution transforms $f\mapsto\widehat{f}_{x}$ is a linear topological algebra isomorphism of ${\cal C}^{p}(G//K)$ onto $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).\;\;\Box$

We recover the Trombi-Varadarajan Theorem for Harish-Chandra transforms by setting $x=e$ in Theorem $3.5.$ Indeed, Theorem $3.5$ above says that every $x\in\exp(N_{0})$ (and not just $x=e$) gives a topological algebra isomorphism between ${\cal C}^{p}(G//K)$ and $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$ However if $x\in G\setminus\exp(N_{0}),$ for any neighborhood $N_{0}$ of zero in $\mathfrak{g},$ Trombi-Varadarajan Theorem may not be appropriate and it may be necessary to seek a more general realization of ${\cal C}^{p}(G//K)$ under the map f\mapsto l_{1}(\lambda):=s_{\lambda,f}(x),\;\mbox{for any x\in G}. Before considering another major result of this paper, giving the fine structure of spherical convolution functions, we state a result on the finiteness of a central integral usually used in the estimation of many other integrals of harmonic analysis on semisimple Lie groups.

To this end we define, for every $x\in G,$ the function $x\mapsto d(x)$ as

[TABLE]

We observe here that

[TABLE]

which is a constant whose proof of finiteness may be found in $[11.],\;p.\;231.$ This constant is crucial to all harmonic analysis of $\mathcal{C}(G)$ and, in particular, to the embedding of $\mathcal{C}(G)$ in $L^{2}(G).$ It is therefore important to understand the nature of $d(x)$ for all $x\in G$ in order to employ it in a more inclusive harmonic analysis on $G.$ We consider the nature of this integral in the following.

Lemma 3.6. Let $x\in G.$ Then there exist $r\geq 0$ such that

[TABLE]

Proof. We already know that $\Xi(y^{-1}x)\leq 1.$ Also

[TABLE]

It follows therefore that

[TABLE]

The last integral in the above inequality is finite if we embark on its computation via the polar decomposition, $G=K\cdot cl(A^{+})\cdot K,$ of $G.\;\;\Box$

Theorem 3.7. Let $N_{0}$ be a neighbourhood of origin in $\mathfrak{g}$ where $f$ is a measurable function on $G$ which satisfies the general strong inequality. The integral defining the spherical convolution function, $x\mapsto s_{\lambda,f}(x),$ is absolutely and uniformly convergent for all $x\in\exp(N_{0}),\;\lambda\in i\mathfrak{a}^{\ast}.$ Moreover the transforms $\lambda\mapsto s_{\lambda,f}(x)$ of $f,$ with $\;x\in\exp(N_{0}),$ is a continuous function on $i\mathfrak{a}^{\ast}.$ If $r\geq 0$ is such that $d(x)=\int_{G}\Xi^{2}(y^{-1}x)(1+\sigma(y^{-1}x))^{-r}dy<\infty,\;x\in G,$ then

[TABLE]

Proof. We recall that $\mid\varphi_{\lambda}(x)\mid\leq\varphi_{0}(x)=\Xi(x),\;x\in G,\;\lambda\in i\mathfrak{a}^{\ast}.$ Hence

[TABLE]

$=d(x)\cdot\mu_{1,1,r}(f).$ Continuity follows from the use of the Lebesgue’s dominated convergence theorem$.\;\;\Box$

The following well-known result on the foundational properties of the Harish-Chandra transforms, $\lambda\mapsto(\mathcal{H}f)(\lambda),\;\lambda\in i\mathfrak{a}^{\ast},$ now follows from the general outlook given by Theorem $3.7.$

Corollary 3.8. $([9.])$ Let $f$ be a measurable function on $G$ which satisfies the strong inequality. The integral defining the Harish-Chandra transforms,

[TABLE]

is absolutely and uniformly convergent for all $\lambda\in i\mathfrak{a}^{\ast}$ and is continuous on $i\mathfrak{a}^{\ast}.$ If $r\geq 0$ is such that $d=\int_{G}\Xi^{2}(y)(1+\sigma(y))^{-r}dy<\infty,$ then

[TABLE]

Proof. Set $X=0$ in Theorem $3.7$ to have the first results. The inequality follows if we set $x=e$ and observe that $d(e)=\int_{G}\Xi^{2}(y^{-1})(1+\sigma(y^{-1}))^{-r}dy=d.\;\;\Box$

We now consider the image of ${\cal C}^{p}(G//K)$ under the full spherical convolution map, f\mapsto l_{1}(\lambda):=s_{\lambda,f}(x),\;\mbox{for any x\in G}. In order to discuss this we have two options. One of the options is to introduce wave-packet that will still have its domain as $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon})$ while using an appropriate Plancherel measure on $\mathfrak{F^{\epsilon}}.$ This option has been explored in $[8a.],\;p.\;34,$ where the $L^{2}$ Plancherel measure, $d\zeta_{x,\lambda}$ on $\mathfrak{F^{1}}$ for the spherical convolution function (when viewed as a function on $G$) was defined to absorb the group variable, $x.$ The results therein suggest that the image of ${\cal C}^{p}(G//K)$ under the full spherical convolution map is indeed possible.

The second option is to retain the spherical Bochner measure, $d\lambda,$ on (a subset of) ${\mathfrak{F}}^{\epsilon}$ and define the wave-packet as a map on the Fr$\acute{e}$chet algebra $\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon}).$ This will reflect the nature of the full spherical convolution map as a transform of members of $\mathcal{C}^{p}(G//K)$ whose arguments are (generally) taken from $int({\mathfrak{F}}^{\epsilon})\times G$ (and not just from $int({\mathfrak{F}}^{\epsilon})$ as in the first option).

To this end recall the Fr$\acute{e}$chet algebra $\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon}),\;\forall\;\epsilon>0,$ let $\Psi\in\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ and set

[TABLE]

where $N_{0}$ is a zero neighbourhood in $\mathfrak{g}.$ It is clear that $N_{0}(A^{+})$ is also a zero neighbourhood in $\mathfrak{g}$ and that $\Psi=\Psi(\lambda,x),$ for all $(\lambda,x)\in int({\mathfrak{F}}^{\epsilon})\times G.$ It follows, from Theorem $3.5,$ that $\bar{\mathcal{Z}}_{\{x\}}({\mathfrak{F}}^{\epsilon})\simeq\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}),$ for every $x\in\exp(N_{0}(A^{+})).$ We then have the following.

Lemma 3.9. *For every $x\in\exp(N_{0}(A^{+}))$ and $\Psi\in\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon}),$ we have that $\Psi(\lambda,x)=\Phi(\lambda),$ for some $\Phi\in\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$

We now employ these remarks to define a map from $\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ to ${\cal C}^{p}(G//K)$ as follows. Let $a\in\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ and $\lambda\mapsto c(\lambda)$ be the Harish-Chandra $c-$function defined on $\mathfrak{F}_{I}:=i\mathfrak{a}^{*}.$ We associate to every $a\in\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ the function $\varphi_{a}$ on $G$ defined as

[TABLE]

It should be noted here that

[TABLE]

[TABLE]

[TABLE]

which is due to the invarianve of $d\lambda,$ and that

[TABLE]

$\forall\;x\in G,\;k_{1},k_{2}\in K,$ being a property inherited from $a$ and $\varphi_{\lambda}.$

The (extra) requirement of being spherical on $G$ placed on members of $\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ may at first be seen as a restriction, when compared to the requirements on members of $\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$ It however turns out that this extra requirement is what is needed to assure us of the generalization of the classical wave-packets (of Trombi-Varadarajan) on $G$ to all of $x\mapsto\varphi_{a}(x).$ This is established as follows.

Lemma 3.10. *Let $a\in\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon})$ and $N_{0}(A^{+})$ be as defined above. Then, for every $x\in\exp(N_{0}(A^{+})),$ the map $x\mapsto\varphi_{a}(x)$ is the classical wave-packet of $G.$

Proof. We observe that, with $\exp tH\in\exp(N_{0}(A^{+})),$

[TABLE]

for some $\Phi\in\bar{\mathcal{Z}}({\mathfrak{F}}^{\epsilon}).$ Here we have employed the spherical property of $a$ on $G$ in the second equality and Lemma $3.9$ in the third equality$.\;\Box$

The above Lemma shows that the definition and properties of the map $x\mapsto\varphi_{a}(x),$ $x\in G,$ is consistent with the relationship (in Lemma $3.4$) existing between spherical convolutions, $s_{\lambda,f}(x)$ and the Harish-Chandra transfroms, $(\mathcal{H}f)(\lambda).$ Hence in order to extend Trombi-Varadarajan Theorem (which gives the image of the algebra $\mathcal{C}^{p}(G//K)$ under $f\mapsto(\mathcal{H}f)(\lambda)$) to all $x\in G$ (under the spherical convolution tranform), it will be necessary to show that $x\mapsto\varphi_{a}(x)$ is the wave-packet of $f\mapsto s_{\lambda,f}(x)$ for all $x\in G.$ According to Lemma $3.10,$ this needs only be done for those $x=k_{1}\exp tHk_{2}$ in $G$ with $\exp tH\notin\exp(N_{0}(A^{+})),$ for any neighbourhood, $N_{0},$ of zero in $\mathfrak{g}.$ We however give a self-contained discussion of these results, the first of which is given below.

Theorem 3.11. $\varphi_{a}\in\mathcal{C}^{p}(G//K)$* for every $a\in\bar{\mathcal{Z}}_{G}({\mathfrak{F}}^{\epsilon}).$

In order to finish the establishment of this Theorem we need some lemmas which give appropriate background for it. Indeed we derive an appropriate bound for $\mid\varphi_{a}(h;u)\mid,$ where $u\in\mathfrak{U}(\mathfrak{g}_{\mathbb{C}})$ and $h$ is well-chosen, and the appropriate collection of seminorms are also in place. These will be considered in a forthcoming paper on Trombi-Varadarajan Theorem via the eigenfunction expansion of spherical convolution, which includes the extension of Theorem $3.5$ to all $x\in G.$  

 

 

$\bf{4}\;\;\;\;$** Algebras of Spherical Convolutions

**

We now consider the various algebras of spherical convolutions that have emanated in the course of this research and their relationship with the Harish-Chandra Schwartz algebra, $\mathcal{C}(G),$ on $G$ as well as its distinguished commutative subalgebra, $\mathcal{C}(G//K),$ of (elementary) spherical functions.

Define $\mathcal{C}_{\lambda}(G)=\{s_{\lambda,f}:f\in\mathcal{C}(G)\}$ and set $\mathcal{C}_{\lambda,0}(G)=\{s_{\lambda,\varphi_{\lambda}}\},$ for all $\lambda\in\mathfrak{a}^{\ast}_{\mathbb{C}}.$ It is clear that $\bigcup_{\lambda\in\mathfrak{a}^{\ast}_{\mathbb{C}}}\mathcal{C}_{\lambda}(G)$ is contained in $\mathcal{C}(G).$ We may therefore topologize $\bigcup_{\lambda\in\mathfrak{a}^{\ast}_{\mathbb{C}}}\mathcal{C}_{\lambda}(G)$ by giving it the relative topology from the topology defined on $\mathcal{C}(G)$ by the seminorms, $\mu_{a,b,r}.$

Lemma 4.1. The inclusions

[TABLE]

*are all proper$.\;\;\Box$

Theorem 4.2. $\bigcup_{\lambda\in\mathfrak{F}^{1}}\mathcal{C}_{\lambda}(G)$* is a closed subalgebra of $\mathcal{C}(G).$*

Proof. We recall that $\mu_{a,b;r}(f\ast\varphi_{\lambda})\leq c\mu_{1,b;r+r_{0}}(f)\cdot\mu_{a,1;r}(\varphi_{\lambda}),$ where $c:=\int_{G}\Xi^{2}(x)(1+\sigma(x))^{-r_{0}}dx<\infty$ for some $r_{0}\geq 0.$ However

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence $\mu_{a,b;r}(f\ast\varphi_{\lambda})<\infty,\;\forall\;\lambda\in\mathfrak{F}^{1}.\;\;\Box$

It may be recalled that members of $\mathcal{C}(G)$ are exactly those functions on $G$ whose left and right derivatives satisfy the strong inequality. In the light of this observation we define $\mathcal{C}^{(x)}(G)$ as exactly those functions on $G$ whose left and right derivatives satisfy the general strong inequality, for each $x\in G.$ Explicitly we set $\mathcal{C}^{(x)}(G)$ as

[TABLE]

$x\in G.$ A collection of seminorms on each of $\mathcal{C}^{(x)}(G)$ may be given by

[TABLE]

It is however clear that $\mathcal{C}^{(e)}(G)=\mathcal{C}(G),$ so that $\mathcal{C}(G)\subset\bigcup_{x\in G}\mathcal{C}^{(x)}(G).$

Theorem 4.3. The natural inclusion $\bigcup_{x\in G}\mathcal{C}^{(x)}(G)\subset L^{2}(G)$ has a dense image.

Proof. It is known that the natural inclusion of $\mathcal{C}(G)$ in $L^{2}(G)$ has a dense image, $[1.].$ The result therefore follows if we recall that, as sets of functions,

[TABLE]

where the second inclusion holds from the fact that $d(x)<\infty,\;x\in G.\;\;\Box$

References.

[1.]

Arthur, J.G., Harmonic analysis of tempered distributions on semisimple Lie groups of real rank one, Ph.D. Dissertation, Yale University, $1970.$

[2.]

Arthur, J.G., Harmonic analysis of the Schwartz space of a reductive Lie group, I. II. (preprint, $1973$).

[3.]

Barker, W.H., The spherical Bochner theorem on semisimple Lie groups, J. Funct. Anal., vol. 20 ($1975$), pp. $179-207.$

[4.]

Dixmier, J., Op$\acute{\mbox{e}}$rateurs de rang fini dans les repr$\acute{\mbox{e}}$sentations unitaires,* Publ. math. de l’ Inst. Hautes $\acute{\mbox{E}}$tudes Scient.,* tome 6 ($1960$), pp. $13-25.$

[5.]

Helgason, S., “Differential Geometry, Lie Groups and Symmetric Spaces,” Academic Press, New York, $1978.$

[6.]

Helgason, S., “Groups and Geometric Analysis; Integral Geometry, Invariant Differential Operators, and Spherical Functions,” Academic Press, New York, $1984.$

[7.]

Helgason, S. and Johnson, K., The bounded spherical functions on symmetric spaces, Advances in Math, 3 ($1969$), pp. 586-593.

[8.]

Oyadare, O. O., ($a.$) On harmonic analysis of spherical convolutions on semisimple Lie groups, Theoretical Mathematics and Applications, 5(3) ($2015$), pp. $19-36.$ ($b.$) Fourier transform of Schwartz algebras on groups of Harish-Chandra class. Under review.

[9.]

Trombi, P.C. and Varadarajan, V.S., Spherical transforms on semisimple Lie groups, Ann. of Math., 94 ($1971$), pp. $246-303.$

[10.]

Varadarajan, V.S., The theory of characters and the discrete series for semisimple Lie groups, in Harmonic Analysis on Homogeneous Spaces, (C.C. Moore (ed.)) Proc. of Symposia in Pure Maths., vol. 26 ($1973$), pp. $45-99.$

[11.]

Varadarajan, V.S., “An introduction to harmonic analysis on semisimple Lie groups,” Cambridge University Press, Cambridge, $1989.$