SHANNON SAMPLING SERIES WITH AVERAGED KERNELS1 1 ...

6 A.KIVUNUKK, G.TAMBERG
Proof: Let f ∈ T V (R), hence f is bounded on R. Since sm ∈ B 1 π, the derivative s ′ m ∈ B 1 π also, and by
Nikolskii inequality
k∈Z |s′ m(u − k)| < ∞. Therefore, there exists the bounded derivative
(SW,mf) ′ (t) = W
∞
k=−∞
f( k
W )s′ m(W t − k),
which implies SW,mf ∈ AC(R). Now we have for the total variation
Since by (15) the derivative
we get
(SW,mf) ′ (t) = W
m
∞
k=−∞
k=−∞
VR[SW,mf] =
s ′ m(t) = 1
m
f( k
W )
∞
−∞
s(t + m
2
s(W t − k + m
2
|(SW,mf) ′ (t)|dt. (20)
m
) − s(t −
2 )
,
m
) − s(W t − k −
2 )
= W
∞
f(
m
k=−∞
k
∞ m
)s(W t − k + ) −
W 2
k=−∞
= W
∞
f(
m
k − m
) − f(k
W W )
s(W t − k + m
2 )
= W
m−1
m
∞
j=0 k=−∞
f(
k − j
W
Now by (20) and Theorem 4 we obtain for the total variation
VR[SW,mf] 1
m−1
m
j=0 k=−∞
∞
− j − j − 1
f(k ) − f(k
W W
)
f( k
m
)s(W t − k −
W 2 )
− j − 1
) − f(k
W
)
s(W t − k + m
2 ).
∞
−∞
s(W t − k + m
2 )
d(W t)
1
m−1
VR[f]s1 = s1VR[f] SW VR[f]. (21)
m
j=0
Under the assumptions of Theorem 5 we have s1 = SW,1, therefore
VR[SW,1f] SW,1VR[f]
4. Examples of operators with variation detracting property
4.1 Hann’s sampling operators
We can provide more examples of operators with variation detracting property (9).
2 πu
Take the Hann window λH(u) := cos . Then by (3)
sH(t) = 1
sinc(t − 1) + 2 sinc t + sinc(t + 1)
4
and its corresponding averaged kernel by (15) is
sH,1(t) = 1
4
2
Sci(t + 1
2
) − Sci(t − 1
2
= 1
r0(t − 1/2) + r0(t + 1/2) =
2
sinc t
3
3
) + Sci(t + ) − Sci(t −
2 2 )
,
2(1 − t 2 )
(22)