Use induction to show the following two statements for the B$-$splines $\{$Bk

i

$($x$)\}($k $>=0)$:

$(1)$ For all i in Z$,$ if x in $($ti

$,$ ti$+$k$+1),$ then Bk

i

$($x$)>0.$

$(2)$

i in Z Bk

i

$($x$)1.$

$[$Hint: The two statements are obviously true for k $=0.$ For $(1),$ we may use the definition for Bk

i

$.$

For $(2),$ we may use the $$recurrence relation of B$-$splines$.]$