C OMPOSITIO M ATHEMATICA
F U C HENG H SIANG
On the Fourier coefficients of a simple discontinuous function
Compositio Mathematica, tome 18, n
o1-2 (1967), p. 61-64
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61
function
by
Fu
Cheng Hsiang
1
Let
f(x)
be a functionintegrable
in the sense ofLebesgue
over the interval
(-03C0 n)
and defined outside thisby periodicity.
Let its Fourier series be
Then
is the
conjugate
series of the Fourier series off(x).
For a
fixed x,
we write2
Let
03A3~n=1
an be an infinite series with thepartial
sums{sn}.
Let an be the first arithmetical means of sn, thatis,
O’n =
(03A3nv=1 sv)/n.
Let A :(an,m) (m
=1,2,...; n ~ m)
be aregular triangular
matrix. Ifthen we say that the series
l an
or the sequence{sn}
is summableA.
(C, 1)
to the sum 03C4.3
In this note, we prove a theorem for the A method of sum-
mation of the Fourier coefficients of
f(t)
connected with itsjump
at the
point t
= x.62
THEOREM. Let the
jump of
If
Asatisfies
and
as m ~ oo, then the sequence
is summable A.
(C, 1)
tol/03C0.
4
If we denote
by 03C3n
the(C, 1)-transformation
of{nBn(x)},
we
have,
afterMohanty-Nanda [1],
by Riemann-Lebesgue’s
theorem.On account of the
regularity
of the Amethod,
we need establish thatas m ~ oo, where
which is
0(n2t) by expanding
sin nt and cos nt into the power series of n and t. It is known that[3,
p.5]
Thus,
if we writesay,
then,
We denote
where
Dv(t)
is the Dirichlet kernel of the Fourier series off(t).
Itis known that
[3,
p.49] Dv(t)
=O(t-1).
By considering
thatand
we see that the conditions
(i)
and(ii) imply
thatmam,m
=0(1)
as m - co. So
that,
we write64
say,
by
Abel’s transformation.Now,
by
the condition(i). And,
as m ~ 00
by
the condition(ii)
of the theorem.The theorem is thus
completely proved.
REFERENCES
MOHANTY R. and NANDA, M.
[1] On the behaviour of Fourier coefficients, Proc. American Math. Soc., 5 (1954),
79201484.
PETERSEN, G. M.
[2] Summability of a class of Fourier series, Proc. American Math. Soc., 11 (1960),
994 2014 998.
ZYGMUND, A.
[3] Trigonometric series I, Cambridge, 1959.
(Oblatum 24-8-1966) National Taiwan University Taipei, Formosa, China