Scientifical activity of G. Gat. It is a html file, and consequently the mathematical formulas are not in the best form.
 György Gát

 Scientifical activity

Some results in papers appeared

1. Gat, G., One of the applications of the fast Fourier transform, Bulletins for Applied Mathematics 41 (1986), 102-105
ZBL 613.65146.
2. Gat, G., On computing the generalized fast discrete Fourier transform, Bulletins for Applied Mathematics 45 (1987), 123-134
ZBL 636.65040.

In papers [3,4,5] I introduce a new orthonormal function system, as a common generalization of some well-known ones, such as
Walsh, Vilenkin systems, the UDMD product systems (introduced by F. Schipp).
In [5] I show that these systems give a new tool in the research of the limit periodic, almost even arithmetical functions.
By this new method I solve some problems concerning arithmetical functions. Basically, the essence of the method is
that the the arithmetical  functions of this kind can be extended to Vilenkin groups., and the system of the extension of
this functions will be a Vilenkin-like systems. So, the procedures known in theory of harmonic analysis on Vilenkin
groups can be adopted for investigation of arithmetical functions.
3. Gat, G., Vilenkin Fourier series and limit periodic arithmetical functions, Colloq Soc. J. Bolyai 58 Approx. Theory, Kecskemet,
(Hungary) (1990), 315332 ZBL 760.42013, MR 94g:42042.
4. Gat, G., Orthonormal systems on Vilenkin groups, Acta Math. Hungar. 58(1- 2) (1991), 193198 ZBL 753.11027, MR 93e:42039.
5. Gat, G., On almost even arithmetical functions via orthonormal systems on Vilenkin groups, Acta Arith. 49(2) (1991), 105123
ZBL 725.11049, MR 92j:11083.

In [6] I proved more problems concerning unbounded Vilenkin systems and Hardy spaces. I mention that I gave a necessary and
sufficient condition for the upper boundedness of the so-called Sunouchi operator That is, for ||Tf||1 \leq C||f||H. For the time being
I know 21 references to this paper.

6. Gat, G., Investigation of certain operators with respect to the Vilenkin system, Acta Math. Hungar. 61(1-2) (1993), 131149
ZBL 805.42019, MR 94d:42035.

In [7] György Gát proves that the Vilenkin-like systems introduced by him, and the ordinary Vilenkin system are not equivalent
bases in the space Lp  (1 < p < \infty).
7. Gat, G., On the non-equivalence of Vilenkin-like systems, Acta Math. Acad. Paed. Nyiregyh. 13 (1993), 87-95 ZBL 913.42022.

8. Gat, G., Investigation of some operators with respect to Vilenkinlike systems, Annales Univ. Sci. Budapestiensis 14
(1994), 61-70 ZBL 923.42018, MR 96a:42034.
9. Gat, G., On a norm convergence theorem with respect to Vilenkin system in the Hardy spaces, Acta Acad. Paed.
Agriensis Sectio Matematicae 22 (1994), 101-107 ZBL 882.42016.

In [11] György Gát and Rodolfo Toledo proved some approximation theorems with respect to the Fejér means of
integrable functions on non Abelian compact totaly disconnected groups. Among others, they proved that the
well-known norm convergence theorem Snf   f \in Lp  (1 < p < \infti) does not. Besides, they proved that if one
take the Fejér means instead the partial sums of the Fourier series, then the norm convergence holds.
In [10] György Gát proved, that the theorem of Carleson does not hold in the non Abelian case, that is there
exists a non commutative Vilenkin group and p > 1, f \in Lp, such as that the partial sums of the Fourier series
do not converge to the function almost everywhere.
10. Gat, G., Pointwise convergence of Fejer means on compact totally disconnected groups, Acta Sci. Math.
(Szeged) 60 (1995), 311-319 ZBL 835.43008, MR 96i:43007.
11. Gat, G., Toledo, R., Lp-norm convergence of series in compact totally disconnected groups, Analysis Math. 22
(1996), 13-24 ZBL 856.42017, MR 97f:42043.

12. Gat, G., Convergence and Summation With Respect to Vilenkin-like Systems in: Recent Developments in
Abstract Harmonic Analysiswith Applications in Signal Processing, Nauka, Belgrade and Elektronski fakultet,
Nis, 1996, pp. 137-146.

In [13] György Gát proved for all two-dimensional integrable function that the Fejér means of the two-parameter
Fourier series converge to the function almost everywhere, provided that the ratio of the indices remain in some cone.
The same result was proved by F. Weisz. His paper appeared in the same year.
13. Gat, G., Pointwise convergence of double Walsh-Fejer means, Annales Univ. Sci. Budapestiensis, Sect. Comp. 16
(1996), 173-184 ZBL 891.42014, MR 99b:42033.

In [14] György Gát proves the more than 25 years old conjecture of M.H. Taibleson Fourier Analysis on local fields
(Princeton University Press).  Namely, he proves that the Fejér-Lebesgue theorem with respect to the character system of
the group of the 2-adic integers. He also proves that the maximal operator of the Fejér means is of type (H,L).
14. Gat, G., On the almost everywhere convergence of Fejer means of functions on the group of 2-adic integers, Journal of Approx.
Theory vol 90 (1) (1997), 88-96 ZBL 883.42021, MR 98m:42042.

In [15] György Gát proved the so-called fundamental theorem of the two-dimensional dyadic derivate. Namely, if f \in L1([0,1)2), then
 lim min(n1,n2) d(n1,n2)If = f
almost everywhere, where (n1,n2) Î N2 ,  |n1-n2| £ b (b is some parameter), d(n1,n2), and I is the two-dimensional dyadic difference
and integral, respectively.
15. Gat, G., On the two-dimensional pointwise dyadic calculus, Journal of Approx. Theory 92 (2) (1998), 191-215
ZBL 897.42017, MR 99c:42049.

Define the operator T of G.I. Sunouchi  In [16] one can find:
Theorem. Let f : Gm \to R ,  f \in  L1(Gm) ,  E0f=0. If some conditions satiesfied  then
 ||f||H1 \leq c||Tf||1 .
The result of Gát is the very first one, with respect to the lower boundednes of this operator. There was nothing known
before this. It is also of interest, that since this paper appeared in the paper

Daly, J., Phillips, K., Walsh Multipliers and a Square Functions for the Hardy Space H1, Acta Math. Hungar. 79 (4) (1998), 311-328.

one can read the proof of theorem with respect to the Walsh system. That is, the theorem is proved if mj=2 for all j and
also in the case of m increasing enough fast. So, there remained a gap. It is quite unusual in the theory of dyadic harmonic
analysis. The usual procedure is as follows. A theorem is proved for the Walsh system, first. Then, for the bounded Vilenkin
systems, and after then a long break... Finally, there is "something" with respect to the unbounded Vilenkin systems. The
methods concerning the Walsh, and the bounded Vilenkin systems are very similar. On the other hand, the discuss the
unbounded Vilenkin systems needs more powerful techniques. The difficulties increase more times....
16. Gat, G., On the lower bound of Sunouchi's operator with respect to the Vilen- kin system, Analysis Math. 23 (1997),
259-272 ZBL 888.42017, MR 99f:42057.

17. Gat, G., On the Fejer kernel functions with respect to the Walsh-Paley system, Acta Acad. Paed. Agriensis Sectio
Matematicae 24 (1997), 105-110 ZBL 888.42015.
18. Gat, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin- like systems, Acta Acad. Paed. Agriensis
Sectio Matematicae 25 (1998), 83-89 ZBL 929.42018, MR 1 728 604.

Let k be the so-called Walsh-Kaczmarz system. This is nothing else, but a rearrangement of the Walsh-Paley system we talked about.
In [19] cikkben one can find the following results of Gát, concerning the Walsh-Kaczmarz system:

Theorem. sknf \to f (n®¥) almost everywhere for all f \in  L1(I).

Theorem. The operator s* is of type (p,p) for each 1 < p \leq \infty, and it is of weak type (1,1). Moreover, ||s*f||1 \leq c|||f|||H1.

The latter (H,L) result Gát and Nagy improved [32]. They proved c||f||H1, instead of c|||f|||H1.
19. Gat, G., On (C; 1) summability of integrable functions with respect to the Walsh-Kaczmarz system, Studia Math. 130 (2)
(1998), 135-148 ZBL 905.42- 016, MR 99e:42043.
20. Gat, G., On the Calderon-Zygmund decomposition lemma on the Walsh-Paley group, Acta Math. Acad. Paed. Nyiregyh. 14
(1998), 25-30 ZBL 908.42011, MR 1 712 506.

In [21] n Nagy and me verified the result in [15] for bounded Vilenkin groups. That is, the fundamental theorem of dyadic
calculus. This reads as: limmin(n1,n2)d(n1,n2)If = f a.e., where (n1,n2) \in N2 ,  |n1-n2| \leq b (b is some given parameter).
21. Gat, G., Nagy, K., The fundamental theorem of two-paremeter pointwise derivate on Vilenkin groups, Analysis Math.
25 (1999), 33-55 ZBL 0932.42020, MR 1 678 505.

In [22] Gát proves: Let Gm be any Vilenkin group (bounded or not), 1 < p and f \in Lp(Gm). Then, snf \to f    a.e.
Earlier, there was nothing known with respect to the almost everywhere convergence of the Fejér means on unbounded
Vilenkin groups. This 36 pages long paper consists the proof of this theorem.
22. Gat, G., Pointwise convergence of the Fejer means of functions on unbounded Vilenkin groups, Journal of Approx.
Theory 101 (1) Nov (1999), 1-36 ZBL 0972.42019, MR 1 724 023.

23. Gat, G., Toledo, R., Fourier coe±cients and absolute convergence on compact totally disconnected groups, Math.
Pannonica 10/2 (1999), 223-233 ZBL 0932.43010, MR 1 704 611.

In [24] I proved by the application of elementary dyadic methods (without the martingale convergence theorem) a
Calderon-Zygmund decomposition lemma for unbounded Vilenkin groups.
24. Gat, G., On the a.e. convergence of Fourier series on unbounded Vilenkin groups, Acta Math. Acad. Paed. Nyiregyh.
15 (1999), 27-34 ZBL 0980.42019, MR 1 706 915.

In [25] Gát proved If there is no restriction for the indices, then the largest convergence space of the two-dimensional
sn,mf Walsh-Paley-Fejér means is L log  L. That is, for an arbitrary measurable function d vanishing at plus infinity, we
have a function in LlogL d(L), such that, sn,mf may converges to f in the sense of Pringsheim only on a set of measure zero.
25. Gat, G., On the divergence of the (C; 1) means of double Walsh-Fourier series, Proc. Amer. Math. Soc. 128 (2000),
1711-1720 ZBL 0976.42016, MR 1 657 751.

In [26] Blahota and me proved for two-dimensional bounded ya systems (this system is a common generalization of
the Vilenkin, UDMD systems), that the so-called conic Fejér means of integrable functions converge to the function
almost everywhere. This is a generalization of the results in [13].
26. Blahota, I., Gat, G., Pointwise convergence of double Vilenkin-Fejer means, Stud. Sci. Math. Hungar. 36 (2000),
49-63 ZBL 0973.42021, MR 2001h:42041.

In [27]:
Let Tk f is the Sunouchi operator with respect to the Walsh-Kaczmarz system. We
proved that Tk gyengén is of weak type (1,1), of type (H1,L1), and (p,p) for all 1 < p \leq 2.
27. Gat, G., Nagy, K., On the Sunouchi operator with respect to the Walsh- Kaczmarz system, Acta Math. Hungar.
vol 89 (1-2) (2000), 93-101 ZBL 0973.42022, MR 2003e:42040.

In [28] I proved (one dimenzional case) the Fejér-Lebesgue theorem for Vilenkin-like systems. These systems are
common generalizations of the Walsh, Vilenkin, the character systems of m-adic integers, UDMD, UCP product
systems (and more others...).
28. Gat, G., On (C,1) summability for Vilenkin-like systems, Studia Math. 144 (2) (2001), 101-120 ZBL 0974.42020,
MR 2001k:42033.

29. Gat, G., On the Fejer kernel functions with respect to the Walsh-Kaczmarz system, Acta Math. Acad. Paed.
Nyiregyh. 17(2) (2001), 121-126 MR 2002m:42030.

30. Gat, G., Divergence of the (C; 1) means of d-dimensional Walsh-Fourier series, Analysis Math. 27 (2001),
157171 ZBL 0996.40002, MR 2002j:42005.

31. Gat, G., Best approximation by Vilenkin-like systems, Acta Math. Acad. Paed. Nyiregyh. 17(3) (2001), 161-169
ZBL 0992.42011, MR 2002k:42057.

In [32] we proved the Fejér-Lebesgue theorem for the Kaczmarz rearrangement of the character system of the
p-series group. We also proved that the maximal operator of the Fejér means is of type (H,L).
32. Gat, G., Nagy, K., Cesaro summability of the character system of the p-series field in the Kaczmarz rearrangement,
Analysis Math 28 (2002), 1-36 MR 2003c:42011.

In [33] Gát gave the largest convergence space concerning the Namely, in 1989 Schipp and Wade proved that if a
two-parameter function belongs to LlogL, then its dyadic integal is differentiable almost everywhere, and the
derivate equals with the function a.e. Gát proved, this theorem is sharp, that is, for an arbitrary measurable at
plus infinity vanishing function d we have a function in LlogLd(L)such that its dyadic integral may be differentiable
on a set of measure zero, at most. In the classical case (ordinary integral and derivate) this result is due to Saks.
He proved this in 1935.
33. Gat, G., On the divergence of the two-dimensional dyadic difference of dyadic integrals, Journal of
Approximation Theory 116 (1) (2002), 1-27 ZBL 0999.42016, MR 1 909 010.

In 1996 Weisz that the 2-dimenzional Sunouchi operator T is of type (p,p) for all 1 < p < \infty and of weak type (L1log+ L,L1).
In [34] Gát proved that the theorem can not be improved:
Theorem: let d:[0,+\infty) \to [0,+\infty) , limt\to\inftyd(t)=0 measurable. Then, there exists an f such that f \in L log+ Ld(L)
and Tf(x) = +\infty a.e.
34. Gat, G., On the Sunouchi operator with respect to the two-dimensional Walsh- Paley system, Function, Series,
Operators - Alexits Memorial Conference, Budapest (Hungary), Colloq Soc. J. Bolyai 60 Approx. Theory, Budapest,
(Hungary) (2002), 247-260.

In [35] I proved for unbounded Vilenkin groups that for any f \in L1, we have sMnf\to f a.e.
35. Gat, G., Cesaro means of integrable functions with respect to unbounded Vilenkin systems, Journal of
Approximation Theory 124 (1) (2003), 25-43.

In [36] I proved a theorem concerning the term by term dyadic differentiability of the Walsh-Kaczmarz series:
The Walsh-Paley version of this result is due to Schipp. The proof is a bit stiffer, since the Kaczmarz
rearrangement is somewhat unfriendly. Exactly the result does not hold, for instance: If, ck  o(1/log(k)),
then the original theorem of Schipp does not hold.
36. Gat, G., On term by term dyadic differentiability of Walsh-Kaczmarz series, Analysis in Theory and
Applications (form title: Approximation Theory and Applications) 19 (1) (2003), 55-75.