METHODS AND APPLICATIONS OF ANALYSIS. © 2000 International Press

LEMMA 3.4. Let f e 1"a and g eH®. Then f *7 g € H^. If g can be continued analytically on a starlike domain ft centered at 0, then f *7 g is well-defined and analytic on Q.

LEMMA 3.5. Let g G T-^X^ for some a > 7. T/ien /or every r G (7, a) there is a constant B > 0 such that fy-oo ry-oo /700 /•7-oo |a:fc(ae5)(a:)| V < ( / I^WI d,a: + r*B) -7-00 ^J — y-oo -7-00 v^ — 7-00 ' 7 v-*- Q) for every e and k in Z>o. Hence deg G H^X™. If g G H^Xy 6^ no^ necessarily in X™, the above conclusion still holds for k = 0 and eG Z>o. Thend^en^.

PROPOSITION 3.7. Let f e 1% and g e T^Zy. Then f *7> g is well-defined and absolutely q-integrable on L(j), and f *7/ g G Ti^Xy.

COROLLARY 3.8. Let f e X% and let g G Ti^Xy. Then dk(f *7# ^) and f *7, 9*^ are also absolutely q-integrable on L(j) for every k G Z>o.

Lemma 3.4 and Proposition 3.7, i.e., / and g need to be of left type, and h has to belong to U®. In order to understand when (/*7#)*7/i is well-defined, and to prove associativity, we need to investigate the behaviour of /ie,7'(/ *7 g). We will use the following lemmas.

LEMMA 4.1. Let f e X™ such that djf G X™ for every j G Z>o. Then r {d«f){x) x* dqx = { (-1)^-^-^ ^ /-TOO m ^ ^ ifb>a, J-'y-oo 10 otherwise. -7-00 In particular, ik f\ _ / -|\fcLe + ^Jg- , :'7vJ/)"(_1) ~H7" (4.1) Me+ft)7(5fc/) = (-lfL ^ "'He.-rV) for every k € Z>o : and ni,1(dkf) =0 ifl<k.

LEMMA 4.2. If f £ HD2^a then Xk fflf € HDl%i0l for every keZ>o.

LEMMA 4.3. Let f G VJ^X^ and let g be defined, together with its q-derivatives, on a domain O. Let 0 ^ x e ft be such that, for some Rx > 0, \deg(x)\ = 0(RI) as e -¥ oo. Then, for every k G Z>o, dk(f *7 g), (dkf *7 g) and (/ *7 dkg) are well defined at x and dk(f *7 g)(x) = (dkf *7 g)(x) = (/ *T dkg)(x). In particular, the result holds if g G /HDIy and x G 1/(7); or i/ g is analytic on a starlike domain fi centered at 0 and x G fi.

PROPOSITION 4.4. Let f e HDiy, andg e ?#i~. Then Xk dl{f ^g) G H^T. for all kj G Z>o.

LEMMA 4.5. Let f G 'HPl%, g G HDiy, k G Z>o. Then k (4.2) Mfe,- ' (/ *7 5) = 53 e=:0 Lej« Pc-yifiVk-erfig)- Thus, 1/7 = 7' iften JJ(f*7g)Xk = Jy(g^f)Xk for every k £ Z>o andf^f^g) = (/7/)(/^)- Proo/. By Proposition 4.4 (/ *7 g) Xk is absolutely ^-integrable on Hj'). Then w.y (/*,.)=^ /, (/ *7 5) ^ft=/, (E (" 1) [Xr(/) xfc ^ Again by Lemma 3.5 and by dominated convergence we may interchange integration

PROPOSITION 4.6. Le* / G nBT^a and g G WD^)/?. Tften / *7 g G «D2^fl?

THEOREM 4.8. Let f e nDl^ and g G HDiy. Let h be defined, together with its q-derivatives, on a domain ft and let x G ft be such that, for some Rx > 0, \(d€h)(x)\ = 0(Re x) as e -» 00. Then ((/ *7 g) *y h)(x) = (/ *7 (g *y h))(a?); /n particular, the equality holds for every x e SI if h is analytic on a starlike domain fi centered atO and it holds for every x G 1/(7") tf h e WPl™,. Proo/. By the previous results all series involved converge absolutely for x as in the hypothesis. We will show tha

COROLLARY 4.9. The class Ti0!^ is an algebra (not necessary unital) with respect to *7. Its subclass U3!" is also an algebra (not necessary unital) and it is a left ideal ofUDI"

PROPOSITION 5.1. Let f e l™a. Ifa>l then \f(±q-jj)\ = 0 for every j e Z; ifa = l then \f(±q~jj)\ = 0 for every j € Z sufficiently large; ifO<a<l then, for some c> 0, \f(±q~j'y)\ = 0(q^j c?) as j ->• oo with (3 = j^. Conversely, let f be a function on £(7) which is bounded on {±^'7 | j > 1} and let (3 > 1. //, for some c > 0, |/(±g-^7)| = O(^V) as j -+ 00 then f e I™ with

LEMMA 5.7. Let f G HDI7%. Tften, ttere are (7,0,^ > 0 5^cft that Ve,7(dkf)<Cqie2ceRk for all k, e G Z>o. /n particular, dkf G T-PX

PROPOSITION 5.8. Let f e flD2£a and g e nDIJ^0. Then f *7 g e nDI^0.

COROLLARY 5.9. The class nDl^ is a subalgebra ofHDl%. Its subclass Ti3!™ is a left ideal ofH3!^, nBX^ and 'HDI™.

COROLLARY 5.11. The classes nsX^>c (force [0,1); andH8!^ (force (0,1]; are left ideals ofHsX^ and ofnsX^. Similar properties hold for 'HPl™. U

COROLLARY 5.12. Let f,ge nDX^, h e nD. Then (/ *7 g *7 h)(x) = (g*7f *7 h)(x) for every x where the product is defined. In particular, for every pair of ideals ICJ with I,Je {HSX^, nsX^ nDX^ nDX}, I is a left module over J/[J, J]*, where [J, J]* denotes the commutator ideal.

LEMMA 6.1. Let f e U^X^>1/2 be such that J f Xk = 0 for every k e Z>0. Then f(x)=0 for every x in some neighbourhood of zero. In particular, if f e T~L X^ui>112, then f(x) = 0 in each point x of the strip around R where f is analytic.

THEOREM 6.4. 7{SI*">C is a commutative algebra for every c E [1/2,1).

THEOREM 6.5. Let g(x) := e^-ar 2) (so g E ^iX^^ by Example 3.2 (a),). Let f E T^Z^i/z be an entire function, not identically zero. Then (Q~nf) *7 g ^ 9 *7 (Q~nf) for n sufficiently large.

PROPOSITION 7.1. (a) // / G T" then T-^f is well-defined and it is an entire analytic function. (b) // moreover f G X*" then J*7/ is also well-defined and T^j — J: 1f'. (c) Let f eXy. Then f^f = 0 iff ^M) = 0 for al1 keZ>o. (d) Let f G nsX^>l/2. Then T^ = 0 iff f = 0. (e) Let f G UDXy, g G HDXy. Then f *7 g G «DI^ and ^(f *7 g) = {Tyf)(TYg).

PROPOSITION 8.1. Letg be a function defined on the half q-lattice Le{^) = {eqk'j} for some 7 > 0 and e G {±1}. Suppose that there exist constants C > 0 and r > 7 such that CJr 1 («) i(^)(^)is(r.g,T)rt(1_,)t for every k > 0 and for every t £ Z for which qtry < r. Then the limit lp := \iTak-^oo(dpg)(eqkj) exists and is finite for every p € Z>o, and there exists a unique analytic function g on {x G C | |x| < r} such that g = g on {eq^^j \ qk/y < r}. If g is defined on the whole q-lattice £(7) and i