. ▁B urt , ▁Edward ▁H . ▁Adel son ▁and ▁James ▁L . ▁Crow ley . ▁ ▁Definition ▁A ▁mult ires ol ution ▁analysis ▁of ▁the ▁Leb es gue ▁space ▁ ▁consists ▁of ▁a ▁sequence ▁of ▁nested ▁sub spaces ▁ ▁that ▁satisfies ▁certain ▁self - similar ity ▁relations ▁in ▁time - space ▁and ▁scale - fre quency , ▁as ▁well ▁as ▁comple ten ess ▁and ▁regular ity ▁relations . ▁▁ ▁Self - similar ity ▁in ▁time ▁dem ands ▁that ▁each ▁sub space ▁V k ▁is ▁invariant ▁under ▁sh ifts ▁by ▁integer ▁multi ples ▁of ▁ 2 k . ▁That ▁is , ▁for ▁each ▁ ▁the ▁function ▁g ▁defined ▁as ▁ ▁also ▁contained ▁in ▁. ▁ ▁Self - similar ity ▁in ▁scale ▁dem ands ▁that ▁all ▁sub spaces ▁ ▁are ▁time - scale d ▁versions ▁of ▁each ▁other , ▁with ▁scaling ▁respectively ▁d ilation ▁factor ▁ 2 k - l . ▁I . e ., ▁for ▁each ▁ ▁there ▁is ▁a ▁ ▁with ▁. ▁ ▁In ▁the ▁sequence ▁of ▁sub spaces , ▁for ▁k > l ▁the ▁space ▁resolution ▁ 2 l ▁of ▁the ▁l - th ▁sub space ▁is ▁higher ▁than ▁the ▁resolution ▁ 2 k ▁of ▁the ▁k - th ▁sub space . ▁▁ ▁Reg ular ity ▁dem ands ▁that ▁the ▁model ▁sub space ▁V 0 ▁be ▁generated ▁as ▁the ▁linear ▁h ull ▁( algebra ically ▁or ▁even ▁top olog ically ▁closed ) ▁of ▁the ▁integer ▁sh ifts ▁of ▁one ▁or ▁a ▁finite ▁number ▁of ▁generating ▁functions ▁ ▁or ▁. ▁Those ▁integer