Abstract
In Bettaibi and Bouzeffour (J. Math. Anal. Appl. 342:1203-1219, 2008), some properties of the third Jackson q-Bessel function of order zero were established. This paper is devoted to studying the q-convolution product by using a q-integral representation of the related q-translation. The central part of this work is first to study the related q-heat semi-group and its hypercontractivity and second to specify the q-analogue of the Wiener algebra.
Highlights
In contrast to the classical theory, the positivity of the translation operator associated to the normalized q-Bessel function of order α is not clear at this stage
It is still an open conjecture to find q ∈ [, ] and α which assure the positivity of the related translation
For α = – /, it was proved that the q-translation is not positive for all q ∈ [, ]
Summary
Proof The result follows from the previous proposition, the properties of the q-generalized translation and the Lebesgue theorem. Theorem (Plancherel formula) Fq is an isomorphism from S∗q(Rq,+) onto itself, Fq– = Fq, and for all f ∈ S∗q(Rq,+), Fq(f ) ,q = f ,q Using this result and the relation ( ), one can state the following proposition. Proposition For ≤ p < ∞, S∗q(Rq,+) is dense in Lpq(Rq,+, x dqx) Proof It suffices to consider functions with compact supports on Rq,+. 4 q-convolution product In [ ], the authors defined the q-convolution product of two suitable functions as f It satisfies the following properties (see [ ]). We shall prove that the notion of q-convolution product can be extended to functions in Lpq(Rq,+, x dqx) space. Tx,q(f )(y) x dqx y dqy g(y) Ty,q(f ) q, y dqy ≤ g ,q f ,q
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