A note on the strong summability of the Riesz means of multiple Fourier series
Authors: Kuratsubo Shigehiko 
Published: June 2010

Let

S
t
α

(
x
,
f
)

S_t^\alpha (x,f)

be the Riesz means of order

α
\alpha

of an integrable function

f
(
x
)

f(x)

on

N
N

-dimensional torus

T
N

(
N
⩾
2
)

{T^N}(N \geqslant 2)

, that is,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript t Superscript alpha Baseline left-parenthesis x comma f right-parenthesis equals sigma-summation Underscript StartAbsoluteValue m EndAbsoluteValue squared greater-than t Endscripts left-parenthesis 1 minus StartFraction StartAbsoluteValue m EndAbsoluteValue squared Over t EndFraction right-parenthesis Superscript alpha Baseline ModifyingAbove f With caret left-parenthesis m right-parenthesis e Superscript 2 pi i m x Baseline period">

S
t
α

(
x
,
f
)
=

∑

|
m
|

2

>
t

(

1
−

|
m
|

2

t

)

α

f
^

(
m
)

e

2
π
i
m
x

.

S_t^\alpha (x,f) = \sum \limits _{{{\left | m \right |}^2} > t} {{{\left ( {1 - \frac {{{{\left | m \right |}^2}}}{t}} \right )}^\alpha }\hat f(m){e^{2\pi imx}}.}

\]
E. M. Stein has shown that if

1
>
p
⩽
2

1 > p \leqslant 2

and

α
>

α
p

\alpha > {\alpha _p}

where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript p Baseline equals StartFraction upper N minus 1 Over 2 EndFraction left-parenthesis StartFraction 2 Over p EndFraction minus 1 right-parenthesis minus StartFraction 1 Over p Superscript prime Baseline EndFraction equals StartFraction upper N minus 1 Over 2 EndFraction minus StartFraction upper N Over p Superscript prime Baseline EndFraction comma">

α
p

=

N
−
1

2

(

2
p

−
1

)

−

1

p
′

=

N
−
1

2

−

N

p
′

,

{\alpha _p} = \frac {{N - 1}}{2}\left ( {\frac {2}{p} - 1} \right ) - \frac {1}{{p’}} = \frac {{N - 1}}{2} - \frac {N}{{p’}},

\]
then for any function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis element-of upper L Superscript p Baseline left-parenthesis upper T Superscript upper N Baseline right-parenthesis upper S Subscript t Superscript alpha Baseline left-parenthesis x comma f right-parenthesis">

f
(
x
)
∈

L
p

(

T
N

)

S
t
α

(
x
,
f
)

f(x) \in {L^p}({T^N})S_t^\alpha (x,f)

is strong summable to

f
(
x
)

f(x)

, that is,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript upper T right-arrow normal infinity Endscripts StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline StartAbsoluteValue upper S Subscript t Superscript alpha Baseline left-parenthesis x comma f right-parenthesis minus f left-parenthesis x right-parenthesis EndAbsoluteValue squared d t equals 0">

lim

T
→
∞

1
T

∫
0
T

|

S
t
α

(
x
,
f
)
−
f
(
x
)

|

2

d
t
=
0

\lim \limits _{T \to \infty } \frac {1}{T}\int _0^T {{{\left | {S_t^\alpha (x,f) - f(x)} \right |}^2}dt = 0}

\]
for almost every

x
x

. In this paper we shall show that if

1
⩽
p
⩽
2

1 \leqslant p \leqslant 2

and

−
1
>
α
>

α
p

- 1 > \alpha > {\alpha _p}

, then there exists a function

f
(
x
)
∈

L
p

(

T
N

)

f(x) \in {L^p}({T^N})

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline StartAbsoluteValue upper S Subscript t Superscript alpha Baseline left-parenthesis x comma f right-parenthesis EndAbsoluteValue squared d t minus normal upper Omega left-parenthesis upper T Superscript alpha Super Subscript p Superscript minus alpha Baseline log Superscript minus 2 tau Baseline upper T right-parenthesis as upper T right-arrow normal infinity">

1
T

∫
0
T

|

S
t
α

(
x
,
f
)

|

2

d
t
−
Ω
(

T

α
p

−
α

log

−
2
τ

⁡
T
)

as 

T
→
∞

\frac {1}{T} \int _0^T |S_t^\alpha (x,f)|^2 dt - \Omega (T^{\alpha _p - \alpha } \log ^{-2\tau } T) \quad \text {as $T \to \infty $}

\]
for every

x
x

and every

τ
>
1

/

p

\tau > 1/p

, in particular,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript upper T right-arrow normal infinity Overscript bar Endscripts StartFraction 1 Over upper T EndFraction integral Subscript 0 Superscript upper T Baseline StartAbsoluteValue upper S Subscript t Superscript alpha Baseline left-parenthesis x comma f right-parenthesis EndAbsoluteValue squared d t equals normal infinity">

lim

T
→
∞

¯

1
T

∫
0
T

|

S
t
α

(
x
,
f
)

|

2

d
t
=
∞

\overline {\lim }\limits _{T \to \infty } \frac {1}{T} \int _0^T | S_t^\alpha (x,f) |^2 dt = \infty

\]
for every

x
x

, where we can take for

f
(
x
)
,

f

σ
τ

(
x
)

f(x),{f_{\sigma \tau }}(x)

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove f With caret Subscript sigma tau Baseline left-parenthesis m right-parenthesis equals 1 slash StartAbsoluteValue m EndAbsoluteValue Superscript sigma Baseline log Superscript tau Baseline StartAbsoluteValue m EndAbsoluteValue comma StartAbsoluteValue m EndAbsoluteValue greater-than 1">

f
^

σ
τ

(
m
)
=
1

/

|
m
|

σ

log
τ

|
m
|

,

|
m
|

>
1

{\hat f_{\sigma \tau }}(m) = 1/{\left | m \right |^\sigma }{\log ^\tau }\left | m \right |,\left | m \right | > 1

.