Some new multipliers of Fourier series
Authors: Buntinas Martin 
Published: June 2010

Let

L
1

{L^1}

be the space of all complex-valued

2
π

2\pi

-periodic integrable functions

f
f

and let

L
1

^

\widehat {{L^1}}

be the space of sequences of Fourier coefficients

f
^

\hat f

. A sequence

λ
\lambda

is an

(

L
1

^

→

L
1

^

)

\left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

multiplier if

λ
⋅

f
^

=

(

λ

(
n
)

f
^

(
n
)

)

\lambda \cdot \hat f = \left ( {\lambda \left ( n \right )\hat f\left ( n \right )} \right )

belongs to

L
1

^

\widehat {{L^1}}

for every

f
f

in

L
1

{L^1}

. The space of even sequences of bounded variation is defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b upsilon equals left-brace lamda vertical-bar lamda Subscript n Baseline equals lamda Subscript negative n Baseline comma sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts StartAbsoluteValue normal upper Delta lamda Subscript k Baseline EndAbsoluteValue plus sup Underscript n Endscripts StartAbsoluteValue lamda Subscript n Baseline EndAbsoluteValue greater-than normal infinity right-brace">

b
υ
=

{

λ

|

λ
n

=

λ

−
n

,

∑

k
=
0

∞

|

Δ

λ
k

|

+

sup

n

|

λ
n

|

>
∞

}

b\upsilon = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 0}^\infty {\left | {\Delta {\lambda _k}} \right |} + {{\sup }_n}\left | {{\lambda _n}} \right | > \infty } \right .} \right \}

, where

Δ

λ
k

=

λ
k

−

λ

k
+
1

\Delta {\lambda _k} = {\lambda _k} - {\lambda _{k + 1}}

and the space of even bounded quasiconvex sequences is defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q equals left-brace lamda vertical-bar lamda Subscript n Baseline equals lamda Subscript negative n Baseline comma sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts k StartAbsoluteValue normal upper Delta squared lamda Subscript k Baseline EndAbsoluteValue plus sup Underscript n Endscripts StartAbsoluteValue lamda Subscript n Baseline EndAbsoluteValue greater-than normal infinity right-brace">

q
=

{

λ

|

λ
n

=

λ

−
n

,

∑

k
=
1

∞

k

|

Δ
2

λ
k

|

+

sup

n

|

λ
n

|

>
∞

}

q = \left \{ {\lambda \left | {{\lambda _n} = {\lambda _{ - n}},\sum \nolimits _{k = 1}^\infty {k\left | {{\Delta ^2}{\lambda _k}} \right | + {{\sup }_n}\left | {{\lambda _n}} \right | > \infty } } \right .} \right \}

, where

Δ
2

λ
k

=
Δ

λ
k

−
Δ

λ

k
+
1

{\Delta ^2}{\lambda _k} = \Delta {\lambda _k} - \Delta {\lambda _{k + 1}}

. It is well known that

q
⊂
b
υ

q \subset b\upsilon

and

q
⊂

(

L
1

^

→

L
1

^

)

q \subset \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

but

b
υ
⊄

(

L
1

^

→

L
1

^

)

b\upsilon \not \subset \left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

. This result is significantly improved by finding an increasing family of sequence spaces

d

υ
p

d{\upsilon _p}

between

q
q

and

b
υ

b\upsilon

which are

(

L
1

^

→

L
1

^

)

\left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

multipliers. Since the

(

L
1

^

→

L
1

^

)

\left ( {\widehat {{L^1}} \to \widehat {{L^1}}} \right )

multipliers are the

2
π

2\pi

-periodic measures, this result gives sufficient conditions for a sequence to be the Fourier coefficients of a measure.