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Convolution theorems of Titchmarsh type on discrete Rn

Published online by Cambridge University Press:  20 January 2009

Yngve Domar
Yngve Domar
Affiliation:
Department of MathematicsUppsala UniversityThunbergsvagen 3, S-752 38Uppsala, Sweden
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Summary

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This paper contains results related to Titchmarsh's convolution theorem and valid for , the additive group of Rn with the discrete topology. The method of proof consists in transferring the problem to Rn with the usual topology by a procedure which has been used earlier, for instance in Helson [3].

In Section 1, the classical support theorems are generalized to . In [1], Titchmarsh's convolution theorem [6] on R was generalized to convolutions of functions belonging to certain weighted Lp-spaces on R. Section 2 contains a corresponding generalization to weighted l2(Rd).

It should be observed that convolutions of elements f and g in l1() can be interpreted as convolutions of bounded discrete measures on Rn. Hence, in that case the support theorem (Theorem 4.33 of Hörmander [5]) is directly applicable to give the results of our Theorems 1 and 3. So the novelty in our theorems lies in the fact that they apply for instance to the case when it is only assumed f, g ∈l2(), together with support conditions. It is not known whether it suffices to assume f∈l1(), g∈lp(), when p > 2.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 3 , October 1989 , pp. 449 - 457
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

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