Scientifical activity of G. Gat. It is a html file, and
consequently the mathematical formulas are not in the best form.
Some results in papers appeared
1. Gat, G., One of the applications of the
fast Fourier transform,
Bulletins for Applied Mathematics 41 (1986), 102105
ZBL
613.65146.
2. Gat, G., On computing the
generalized fast discrete Fourier
transform, Bulletins for Applied Mathematics 45 (1987), 123134
ZBL 636.65040.
In papers [3,4,5] I introduce a new orthonormal function system,
as a common generalization of some wellknown 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 Vilenkinlike 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 socalled
Sunouchi operator
That is, for Tf_{1}
\leq Cf_{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(12) (1993), 131149
ZBL
805.42019, MR 94d:42035.
In [7] György Gát proves that the Vilenkinlike systems
introduced by him, and the ordinary Vilenkin system are not
equivalent
bases in the space L^{p} (1 < p
< \infty).
7. Gat, G., On the nonequivalence of Vilenkinlike systems, Acta
Math. Acad. Paed. Nyiregyh. 13 (1993), 8795 ZBL 913.42022.
8. Gat, G., Investigation of some operators with respect to
Vilenkinlike systems, Annales Univ. Sci. Budapestiensis 14
(1994), 6170 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), 101107 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
wellknown norm
convergence theorem S_{n}f f \in L^{p}
(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
L^{p}, 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),
311319 ZBL 835.43008, MR 96i:43007.
11. Gat, G., Toledo, R., Lpnorm convergence of series in compact
totally disconnected groups, Analysis Math. 22
(1996), 1324 ZBL
856.42017, MR 97f:42043.
12. Gat, G., Convergence and Summation With Respect to
Vilenkinlike Systems in: Recent Developments in
Abstract Harmonic
Analysiswith Applications in Signal Processing, Nauka, Belgrade
and Elektronski fakultet,
Nis, 1996, pp. 137146.
In [13] György Gát proved for all twodimensional
integrable function that the Fejér means of the twoparameter
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 WalshFejer means,
Annales Univ. Sci. Budapestiensis, Sect. Comp. 16
(1996), 173184
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
FejérLebesgue theorem with respect to the character system of
the group of the 2adic 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 2adic integers, Journal of Approx.
Theory vol 90 (1) (1997), 8896 ZBL 883.42021, MR 98m:42042.
In [15] György Gát proved the socalled fundamental theorem
of the twodimensional dyadic derivate. Namely, if f \in
L^{1}([0,1)^{2}),
then

lim
min(n_{1},n_{2})

d_{(n1,n2)}If
=
f 

almost everywhere, where (n_{1},n_{2})
Î N^{2}
, n_{1}n_{2} Ł
b (b
is some parameter), d_{(n1,n2)},
and I is
the twodimensional dyadic difference
and integral, respectively.
15. Gat, G., On the twodimensional pointwise dyadic calculus,
Journal of Approx. Theory 92 (2) (1998), 191215
ZBL 897.42017, MR
99c:42049.
Define the operator T of G.I. Sunouchi
In [16] one can find:
Theorem. Let f : G_{m} \to
R , f \in L^{1}(G_{m})
, E_{0}f=0. If some conditions
satiesfied
then
f_{H1}
\leq cTf_{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),
311328.
one can read the proof of theorem with respect to the Walsh
system. That is, the theorem is proved if m_{j}=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),
259272 ZBL 888.42017, MR 99f:42057.
17. Gat, G., On the Fejer kernel functions with respect to the
WalshPaley system, Acta Acad. Paed. Agriensis Sectio
Matematicae
24 (1997), 105110 ZBL 888.42015.
18. Gat, G., On a theorem of type HardyLittlewood with respect to
the Vilenkin like systems, Acta Acad. Paed. Agriensis
Sectio
Matematicae 25 (1998), 8389 ZBL 929.42018, MR 1 728 604.
Let k be the socalled
WalshKaczmarz system. This is
nothing else, but a rearrangement of the WalshPaley system we
talked about.
In [19] cikkben one can find the following results of Gát,
concerning the WalshKaczmarz system:
Theorem. s^{k}_{n}f \to f (n®Ą)
almost everywhere for all f \in
L^{1}(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
cf_{H1}.
The latter (H,L) result Gát and Nagy improved [32]. They
proved cf_{H1},
instead of cf_{H1}.
19. Gat, G., On (C; 1) summability of integrable functions with
respect to the WalshKaczmarz system, Studia Math. 130 (2)
(1998),
135148 ZBL 905.42 016, MR 99e:42043.
20. Gat, G., On the CalderonZygmund decomposition lemma on the
WalshPaley group, Acta Math. Acad. Paed. Nyiregyh. 14
(1998),
2530 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: lim_{min(n1,n2)}d_{(n1,n2)}If
=
f a.e., where (n_{1},n_{2}) \in
N^{2} ,
n_{1}n_{2} \leq
b
(b is some given
parameter).
21. Gat, G., Nagy, K., The fundamental theorem of twoparemeter
pointwise derivate on Vilenkin groups, Analysis Math.
25 (1999),
3355 ZBL 0932.42020, MR 1 678 505.
In [22] Gát proves: Let G_{m} be any Vilenkin group
(bounded
or not), 1 < p and f \in
L^{p}(G_{m}). Then, s_{n}f
\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), 136 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), 223233 ZBL 0932.43010, MR 1 704 611.
In [24] I proved by the application of elementary dyadic methods
(without the martingale convergence theorem) a
CalderonZygmund
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), 2734 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 twodimensional
s_{n,m}f WalshPaleyFejér 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, s_{n,m}f
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
WalshFourier series, Proc. Amer. Math. Soc. 128 (2000),
17111720
ZBL 0976.42016, MR 1 657 751.
In [26] Blahota and me proved for twodimensional bounded
ya
systems (this system is a common generalization of
the Vilenkin, UDMD systems), that the socalled 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
VilenkinFejer means, Stud. Sci. Math. Hungar. 36 (2000),
4963
ZBL 0973.42021, MR 2001h:42041.
In [27]:
Let T^{k}
f
is
the Sunouchi operator with respect to the WalshKaczmarz system.
We
proved that T^{k}
gyengén is of weak type (1,1), of
type (H^{1},L^{1}), 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 (12)
(2000), 93101 ZBL 0973.42022, MR 2003e:42040.
In [28] I proved (one dimenzional case) the FejérLebesgue
theorem for Vilenkinlike systems. These systems are
common
generalizations of the Walsh, Vilenkin, the character systems of
madic integers, UDMD, UCP product
systems (and more others...).
28. Gat, G., On (C,1) summability for Vilenkinlike systems,
Studia Math. 144 (2) (2001), 101120 ZBL 0974.42020,
MR
2001k:42033.
29. Gat, G., On the Fejer kernel functions with respect to the
WalshKaczmarz system, Acta Math. Acad. Paed.
Nyiregyh. 17(2)
(2001), 121126 MR 2002m:42030.
30. Gat, G., Divergence of the (C; 1) means of ddimensional
WalshFourier series, Analysis Math. 27 (2001),
157171 ZBL
0996.40002, MR 2002j:42005.
31. Gat, G., Best approximation by Vilenkinlike systems, Acta
Math. Acad. Paed. Nyiregyh. 17(3) (2001), 161169
ZBL 0992.42011,
MR 2002k:42057.
In [32] we proved the FejérLebesgue theorem for the Kaczmarz
rearrangement of the character system of the
pseries 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 pseries field in the Kaczmarz rearrangement,
Analysis Math
28 (2002), 136 MR 2003c:42011.
In [33] Gát gave the largest convergence space concerning the
Namely, in 1989 Schipp and Wade proved that if a
twoparameter
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 twodimensional dyadic
difference of dyadic integrals, Journal of
Approximation Theory
116 (1) (2002), 127 ZBL 0999.42016, MR 1 909 010.
In 1996 Weisz that the 2dimenzional
Sunouchi operator T is of
type (p,p) for all 1 < p < \infty
and of weak type (L^{1}log^{+}
L,L^{1}).
In [34] Gát proved that the theorem can not be improved:
Theorem: let d:[0,+\infty) \to
[0,+\infty) , lim_{t\to\infty}d(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
twodimensional Walsh Paley system, Function, Series,
Operators 
Alexits Memorial Conference, Budapest (Hungary), Colloq Soc. J.
Bolyai 60 Approx. Theory, Budapest,
(Hungary) (2002), 247260.
In [35] I proved for unbounded Vilenkin groups that for any f \in
L^{1}, we
have s_{Mn}f\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), 2543.
In [36] I proved a theorem concerning the term by term dyadic
differentiability of the WalshKaczmarz series:
The WalshPaley 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,
c_{k} o(1/log(k)),
then the original theorem of Schipp
does not hold.
36. Gat, G., On term by term dyadic differentiability of
WalshKaczmarz series, Analysis in Theory and
Applications (form
title: Approximation Theory and Applications) 19 (1) (2003),
5575.