3. Approximation Theorem for Convolution Operators Supported by SubgroupsWe refer to [7] for and the canonical map of into (Section p. 101). We denote by the Banach space of all bounded operators of.

We define a family of linear maps of into where is an arbitrary closed

subgroup of. We precise that is the involution of and that for, and we have.

Definition 1. Let be a locally compact group, an arbitrary closed subgroup, and. For we set for

.

Then and where. If then and is contained in [8] .

Lemma 2. Let be a locally compact group, a closed amenable subgroup, , , and an open neighborhood of in. Then there is with, , and such that

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for every and every.

Proof. Let with for every. There is a compact

symmetric neighborhood of in with and such that for every. There is open neighborhood of in such that and

are both smaller than for every and for every. We can choose with, for every and such that.

Let be a symmetric compact neighborhood of in contained in with

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for every and such that

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for every and for every (for and we denote by the function defined on by).

We put, and where is the canonical map of onto.

By the preceding Lemma there is with and such that

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is smaller than

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and also smaller than

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for every. We finally put , and.

1) For every and every we have

.

We show at first that

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From

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we obtain indeed

.

For every we have

.

We have

.

But for every

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and therefore

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consequently

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For every we have

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As above

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taking in account that we obtain

.

Proof of Using and one obtains an estimate for. We finish then the proof of 1) using.

2) For every and every we have

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By the Corollary 6 of section 7.2 p.112 of [7]

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Consequently

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But by definition of for every we have

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3) End of the proof of Lemma 2. We are now able to define the functions and of the Lemma and. Using and we get

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Clearly and. It remains to show that. We have

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But for

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hence and similarly, we finally get.

Theorem 3 Let be a locally compact group, a closed amenable subgroup, , a sequence of, a sequence of, and an open neighborhood of in. Suppose that

the series converges. Then there is with, ,

and such that

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for every

Proof. We choose with

1) There is with , and such that

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for every.

There are and sequences of with

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and

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for every. From the convergence of follows the existence of such that

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By Lemma 2 there is with, ,

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and such that

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for every and every. Consequently

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2) End of the proof of Theorem 3. It suffices to put and to obtain and

4. The Main ResultDefinition 2 Let be a locally compact group, an arbitrary closed subgroup, and For we put

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where and are sequences of such that converges and such that

.

Then is a linear map of into, for and one has

, and [8] .

Corollary 4 Let be a locally compact group, a closed amenable subgroup, , , and a neighborhood of in. Then there are with

, and.

Proof. There are sequences of such that converges and such that

. Let be an open neighborhood of in such that. By Theorem 3 there

is with, , and such that

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for every.

Consider an arbitrary with. From

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and

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we get and therefore

The following theorem is the main result of the paper.

Theorem 5 Let be a locally compact group, a closed amenable subgroup, , , and a neighborhood of in. Then there is with and

Proof. This proof is identical with the one of Proposition ii) p. 115 of [1] . Let be an open neighborhood of in such that the closure of in is compact and contained in. Using the Corollary 4 we show by induction the existence of a sequence of and of a sequence of such that, , , ,

and The function satisfies all the requirements.