▁sh ifts ▁should ▁at ▁least ▁form ▁a ▁frame ▁for ▁the ▁sub space ▁, ▁which ▁impos es ▁certain ▁conditions ▁on ▁the ▁decay ▁at ▁infinity . ▁The ▁generating ▁functions ▁are ▁also ▁known ▁as ▁scaling ▁functions ▁or ▁father ▁wave lets . ▁In ▁most ▁cases ▁one ▁dem ands ▁of ▁those ▁functions ▁to ▁be ▁piece wise ▁continuous ▁with ▁compact ▁support . ▁ ▁Comple ten ess ▁dem ands ▁that ▁those ▁nested ▁sub spaces ▁fill ▁the ▁whole ▁space , ▁i . e ., ▁their ▁union ▁should ▁be ▁dense ▁in ▁, ▁and ▁that ▁they ▁are ▁not ▁too ▁redundant , ▁i . e ., ▁their ▁intersection ▁should ▁only ▁contain ▁the ▁zero ▁element . ▁ ▁Import ant ▁conclus ions ▁ ▁In ▁the ▁case ▁of ▁one ▁continuous ▁( or ▁at ▁least ▁with ▁bounded ▁variation ) ▁compact ly ▁supported ▁scaling ▁function ▁with ▁orthogonal ▁sh ifts , ▁one ▁may ▁make ▁a ▁number ▁of ▁dedu ctions . ▁The ▁proof ▁of ▁existence ▁of ▁this ▁class ▁of ▁functions ▁is ▁due ▁to ▁In grid ▁Da ube ch ies . ▁ ▁Assuming ▁the ▁scaling ▁function ▁has ▁compact ▁support , ▁then ▁ ▁implies ▁that ▁there ▁is ▁a ▁finite ▁sequence ▁of ▁coefficients ▁ ▁for ▁, ▁and ▁ ▁for ▁, ▁such ▁that ▁▁▁ ▁Def ining ▁another ▁function , ▁known ▁as ▁mother ▁wave let ▁or ▁just ▁the ▁wave let ▁ ▁one ▁can ▁show ▁that ▁the ▁space ▁, ▁which ▁is ▁defined ▁as ▁the ▁( closed ) ▁linear ▁h ull ▁of ▁the ▁mother ▁wave let ' s ▁integer ▁sh ifts , ▁is ▁the ▁orthogonal ▁complement ▁to ▁ ▁inside ▁. ▁Or ▁put ▁differently , ▁ ▁is ▁the ▁orthogonal