A A computable computable function

Martin Ziegler

xx∈∈ computablecomputable ⇔⇔ ||xx--aa_nn/2/2ⁿⁿ⁺¹⁺¹||≤≤22^--nn forfor recursiverecursive ((aa_nn))⊆⊆

f

Uniformly Computable Real Functions

A A computable computable function

function must must be be continuous continuous

x' x

Martin Ziegler

Computable Weierstrass Theorem

Theorem:

Theorem: For For ff:[0,1]:[0,1]→→ thethe followingfollowing areare equivalentequivalent:: a)a) ThereThere isis an an algorithmalgorithm convertingconverting anyany seqseq. . qq_nn∈∈ _n+1n+1

withwith ||xx--qq_nn|| ≤ ≤ 22^{--n n} intointo pp_mm∈∈ _m+1m+1 withwith ||ff((xx))--pp_mm|| ≤ ≤ 22^-m-m b) b) ThereThere isis an an algorithmalgorithm printingprinting a a sequencesequence (of (of degreesdegrees

and and coefficientcoefficient listslists of) (of) (PP_nn))⊆⊆ [X[X] ] withwith ||||ff--PP_nn|||| ≤ ≤ 22^-n-n c) c) TheThe real real sequencesequence ff((qq), ), qq∈∈ ∩∩[0,1], [0,1], isis computablecomputable

&

& ff admitsadmits a a computablecomputable modulusmodulus of uniform of uniform continuity.continuity

Call Call ( (

) )

computable iff iff an an algorithm algorithm can can print print , , on on input input

, , some some

with with | |

- -

| |

2 2

. .

:= :=

nn nn

, ,

:= := { {

/2 /2

: :

} }

||xx--yy||≤≤22^--µ(µ(m)m) ⇒⇒ ||ff((xx))--ff((yy)|)|≤≤22^--mm

Proof: Proof: a) a) ⇒ ⇒ c) c) ⇒ ⇒ b) b)

Martin Ziegler

uncomputable

uncomputable in in generalgeneral Exercises: Computable Real Functions

a) a) ff computablecomputable ⇒⇒ samesame forfor anyany restrictionrestriction

b) b) expexp, , sinsin, , coscos, ln(1+, ln(1+xx) ) areare computablecomputable functionsfunctions c) c) For For a a computablecomputable sequencesequence aa=(=(aa_nn),),

thethe powerpower seriesseries xx→→∑∑

nn aa_nn··xxⁿⁿ isis computablecomputable on on ((--r,rr,r)) forfor r r << RR((aa) := 1/) := 1/limsuplimsup_nn ||aa_nn||^1/n1/n

d) d) LetLet ff∈∈C[0,1]C[0,1] bebe computablecomputable. . ThenThen so so areare

∫∫ff: : xx→→∫∫

00 xx

ff((tt) ) dtdt and and max(max(ff):):xx→→max{max{ff((tt):):tt≤≤xx}.}.

e) e) IfIf ((x,mx,m))→f→f_mm((xx)) computablecomputable withwith ||ff_nn--ff_mm||_∞∞≤≤22^-n-n+2+2^--mm

then

then limlim_nn ff_nn is computable.is computable.

f) For

f) For computablecomputable aa∈∈ ^{, ,} ff:[0,:[0,aa]]→→ ^{, and , and}

gTo computegTo :[:[aacompute,1],1]→→ ^withwithff:: →→ff((aa: : )=)=convertconvertgg((aa)), , theirtheiranyany sequencesequencejoinjoin isis computablecomputableqq_nn∈∈ _{nn+1 +1} withwith ||xx--qq_nn|| ≤ ≤ 22^{--n n} intointo pp_mm∈∈ _m+1m+1 withwith ||ff((xx))--pp_mm|| ≤ ≤ 22^--mm