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), 102-105
2. Gat, G., On computing the
generalized fast discrete Fourier
transform, Bulletins for Applied Mathematics 45 (1987), 123-134
In papers [3,4,5] I introduce a new orthonormal function system,
as a common generalization of some well-known ones, such as
Vilenkin systems, the UDMD product systems (introduced by F.
In  I show that these systems give
a new tool in the research
of the limit periodic, almost even arithmetical functions.
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
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,
(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
725.11049, MR 92j:11083.
In  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
That is, for ||Tf||1
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
805.42019, MR 94d:42035.
In  György Gát proves that the Vilenkin-like systems
introduced by him, and the ordinary Vilenkin system are not
bases in the space Lp (1 < p
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.
Matematicae 22 (1994), 101-107 ZBL 882.42016.
In  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
convergence theorem Snf f \in Lp
(1 < p < \infti)
does not. Besides, they proved
that if one
Fejér means instead the partial sums of the Fourier series, then
the norm convergence holds.
In  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
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
Analysiswith Applications in Signal Processing, Nauka, Belgrade
and Elektronski fakultet,
Nis, 1996, pp. 137-146.
In  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
13. Gat, G., Pointwise convergence of double Walsh-Fejer means,
Annales Univ. Sci. Budapestiensis, Sect. Comp. 16
ZBL 891.42014, MR 99b:42033.
In  György Gát proves the more than 25 years old
conjecture of M.H. Taibleson Fourier Analysis on local
(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  György Gát proved the so-called fundamental theorem
of the two-dimensional dyadic derivate. Namely, if f \in
almost everywhere, where (n1,n2)
, |n1-n2| Ł
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
Define the operator T of G.I. Sunouchi
In  one can find:
Theorem. Let f : Gm \to
R , f \in L1(Gm)
, E0f=0. If some conditions
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),
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
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
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
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
In  cikkben one can find the following results of Gát,
concerning the Walsh-Kaczmarz system:
Theorem. sknf \to f (n®Ą)
almost everywhere for all f \in
Theorem. The operator s*
for each 1 < p \leq \infty,
and it is of weak type (1,1).
Moreover, ||s*f||1 \leq
The latter (H,L) result Gát and Nagy improved . They
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)
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
25-30 ZBL 908.42011, MR 1 712 506.
In  n Nagy and me verified the result in  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
(b is some given
21. Gat, G., Nagy, K., The fundamental theorem of two-paremeter
pointwise derivate on Vilenkin groups, Analysis Math.
33-55 ZBL 0932.42020, MR 1 678 505.
In  Gát proves: Let Gm be any Vilenkin group
or not), 1 < p and f \in
Lp(Gm). Then, snf
Earlier, there was nothing known with respect
to the almost everywhere convergence of the Fejér means on
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  I proved by the application of elementary dyadic methods
(without the martingale convergence theorem) a
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.
(1999), 27-34 ZBL 0980.42019, MR 1 706 915.
In  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
plus infinity, we
have a function in LlogL d(L),
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),
ZBL 0976.42016, MR 1 657 751.
In  Blahota and me proved for two-dimensional bounded
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
everywhere. This is a generalization of the results in .
26. Blahota, I., Gat, G., Pointwise convergence of double
Vilenkin-Fejer means, Stud. Sci. Math. Hungar. 36 (2000),
ZBL 0973.42021, MR 2001h:42041.
the Sunouchi operator with respect to the Walsh-Kaczmarz system.
proved that Tk
gyengén is of weak type (1,1), of
type (H1,L1), and (p,p)
for all 1 < p \leq
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  I proved (one dimenzional case) the Fejér-Lebesgue
theorem for Vilenkin-like systems. These systems are
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,
29. Gat, G., On the Fejer kernel functions with respect to the
Walsh-Kaczmarz system, Acta Math. Acad. Paed.
(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),
0996.40002, MR 2002j:42005.
31. Gat, G., Best approximation by Vilenkin-like systems, Acta
Math. Acad. Paed. Nyiregyh. 17(3) (2001), 161-169
In  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
32. Gat, G., Nagy, K., Cesaro summability of the character system
of the p-series field in the Kaczmarz rearrangement,
28 (2002), 1-36 MR 2003c:42011.
In  Gát gave the largest convergence space concerning the
Namely, in 1989 Schipp and Wade proved that if a
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
a set of measure zero, at most. In the classical case (ordinary
integral and derivate) this result is due to Saks.
He proved this
33. Gat, G., On the divergence of the two-dimensional dyadic
difference of dyadic integrals, Journal of
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+
In  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
f such that f \in L log+
and Tf(x) = +\infty
34. Gat, G., On the Sunouchi operator with respect to the
two-dimensional Walsh- Paley system, Function, Series,
Alexits Memorial Conference, Budapest (Hungary), Colloq Soc. J.
Bolyai 60 Approx. Theory, Budapest,
(Hungary) (2002), 247-260.
In  I proved for unbounded Vilenkin groups that for any f \in
35. Gat, G., Cesaro means of integrable functions with respect to
unbounded Vilenkin systems, Journal of
Approximation Theory 124
(1) (2003), 25-43.
In  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,
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
title: Approximation Theory and Applications) 19 (1) (2003),