ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION

Transcription

1 ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION STEVEN BÜRGER AND BERND HOFMANN Abstract. We revisit i L 2 -spaces the autocovolutio equatio x x = y with solutios which are real-valued or complex-valued fuctios x(t) defied o a fiite real iterval, say t [, ]. Such operator equatios of quadratic type occur i physics of spectra, i optics ad i stochastics, ofte as part of a more complex task. Because of their weak oliearity deautocovolutio problems are ot see as difficult ad hece little attetio is paid to them wrogly. I this paper, we will idicate o the example of autocovolutio a deficit i low order covergece rates for regularized solutios of oliear ill-posed operator equatios F (x) = y with solutios x i a Hilbert space settig. So for the real-valued versio of the deautocovolutio problem, which is locally ill-posed everywhere, the classical covergece rate theory developed for the Tikhoov regularizatio of oliear ill-posed problems reaches its limits if stadard source coditios usig the rage of F (x ) fail. O the other had, covergece rate results based o Hölder source coditios with small Hölder expoet ad logarithmic source coditios or o the method of approximate source coditios are ot applicable sice qualified oliearity coditios are required which caot be show for the autocovolutio case accordig to curret kowledge. We also discuss the complex-valued versio of autocovolutio with full data o [, 2] ad see that ill-posedess must be expected if ubouded amplitude fuctios are admissible. As a ew detail, we preset situatios of local well-posedess if the domai of the autocovolutio operator is restricted to complex L 2 -fuctios with a fixed ad uiformly bouded modulus fuctio.. Itroductio Regularizatio theory for liear ill-posed operator equatios i Hilbert spaces represetig liear iverse problems seems to be almost complete, icludig results o covergece rates (cf. [8, Chapters 2-9] ad more recetly for example [29, 3, 36]). Moreover, there are ow successful steps toward Baach space theory (cf., e.g., [37] ad refereces therei). However, i the treatmet of oliear iverse problems aimed at solvig operator equatios (.) F (x) = y Date: December 6, Mathematics Subject Classificatio. 65J2, 45G, 47J6, 47A52, 65J5.

2 2 STEVEN BÜRGER AND BERND HOFMANN with oliear forward operators F : D(F ) X Y ad domai D(F ) there is still much to do eve i the Hilbert space settig. With focus o autocovolutio problems we will cosider i this paper oliear equatios (.), where X ad Y are ifiite dimesioal separable Hilbert spaces, ad we deote by the symbols ad, the orms ad the ier products, respectively, i both spaces. It begis with the rarely clarified questio of how ill-posedess is to be defied for oliear problems, whereas for liear problems illposedess is completely well-defied by the fact that the rage of the liear forward operator is ot closed i Y. I [23, Defiitio 2] a cocept of local well-posedess ad ill-posedess was suggested ad we repeat here this idea: Defiitio.. We call a oliear operator equatio (.) locally well-posed at a solutio poit x D(F ) if there is a closed ball B r (x ) := {x X : x x r} aroud x with radius r > such that, for every sequece {x } = B r (x ) D(F ), the limit coditio lim F (x ) F (x ) = implies that lim x x =. Otherwise the equatio is called locally ill-posed at x D(F ), which meas that, for arbitrarily small radii r >, there exist sequeces {x } = B r (x ) D(F ) such that lim F (x ) F (x ) =, but lim x x = fails. Furthermore, i the past 25 years a geeral theory icludig covergece rates results was developed for variatioal (Tikhoov-type) regularizatio methods (cf., e.g., [8, Chapter ] ad [34, Chapter 3]) ad iterative regularizatio methods (cf., e.g., [27]) applied to abstract ill-posed oliear equatios (.). This geeral theory is mostly based o Gâteaux, Frechét or directioal derivatives F (x) of F for elemets x from some eighborhood of a solutio x to (.). There is a collectio of oliearity coditios which are relevat for that theory. I particular, startig from the paper [6], the tagetial coe coditio (.2) F (x) F (x ) F (x )(x x ) C F (x) F (x ) for some costat < C < ad all x B r (x ) D(F ) is playig a promiet role, where the focus of iterative regularizatio methods is o costats < C <. But the verificatio of such qualified oliearity coditios is still missig or caot be prove for large relevat classes of oliear iverse problems. The same ca be said for weaker coditios of the form (.3) F (x) F (x ) F (x )(x x ) C ϕ( F (x) F (x ) ), where ϕ is a cocave idex fuctio ϕ : (, ) (, ). As usual (cf. [2, 3]) we call ϕ a idex fuctio if it is strictly icreasig with the limit coditio lim t + ϕ(t) =.

3 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION 3 However, for wide classes of problems there are good chaces to show at least a local Lipschitz coditio of F at x with the cosequece that (.4) F (x) F (x ) F (x )(x x ) K x x 2 holds for some costat < K < o B r (x ) D(F ). This is of great iterest for the classical form of Tikhoov regularizatio for oliear ill-posed problems i Hilbert spaces, where istead of y = F (x ) oly oisy data y δ Y with y y δ δ with oise level δ > are available. The stable approximate solutios x δ α to x are miimizers of the extremal problem (.5) F (x) y δ 2 + α x x 2 mi, subject to x D(F ), with regularizatio parameter α > ad a prescribed referece elemet x X. Wheever the limit coditios δ 2 (.6) α ad α hold for the regularizatio parameter oe ca show by usig the cocept of x-miimum-orm solutios (see [8, Sect..]) that the regularized solutios x δ α coverge (i the sese of subsequeces, cf. [8, Theorem.3]) for δ with respect to the orm i X to such solutios x which have miimal distace to x uder all solutios to (.) If, moreover, for covex domai D(F ) ad weakly sequetially closed operator F the bechmark source coditio (.7) x = x + 2 F (x ) v with the adjoit operator F (x ) of F (x ) ad with a source elemet v Y is satisfied ad moreover the smalless coditio (.8) K v < is fulfilled, the the results of the semial paper [9] o covergece rates for the Tikhoov regularizatio of oliear ill-posed problems apply ad yield for a a priori choice α(δ) δ of the regularizatio the covergece rate ( ) (.9) x δ α(δ) x = O δ as δ. If the set of x-miimum-orm solutios to (.) is ot uique, the it is a immediate cosequece of the result (.9) that oly oe such solutio x D(F ) to (.) ca satisfy the three coditios (.4), (.7) ad (.8), simultaeously. The papers [9] ad [3] have discussed cosequeces of oliearity coditios of the form (.3) for Baach space regularizatio, but they also apply to the Hilbert space situatio of Tikhoov regularizatio (.5) uder cosideratio here. I this situatio, we obtai for a

4 4 STEVEN BÜRGER AND BERND HOFMANN choice α = α(δ, y δ ) of the regularizatio parameter by the sequetial discrepacy priciple (cf. [, 2]) covergece rates ( ) (.) x δ α(δ,y δ ) x = O ϕ(δ) as δ wheever (.3) is satisfied for some cocave idex fuctio ϕ together with the bechmark source coditio (.7), ad o smalless coditio is required. If the bechmark source coditio fails, but the derivative F (x ) : X Y is a ijective ad bouded liear operator, the uder (.3) the method of approximate source coditios developed i [8] ca be used together with variatioal iequalities combiig solutio smoothess ad oliearity structure i oe tool (cf. [22], [34, Chapt. 3], [, Chapt. 2] ad [5]). This yields covergece rates (.) x δ α(δ,y δ ) x = O (ψ(δ)) as δ, which are lower tha the rates i (.). Takig ito accout [3, Theorem 5.2] ad [2, Theorem 2] it ca be see that the rate fuctio ψ i (.) is a idex fuctio of the form ψ(δ) = d ( Ψ (ϕ(δ)) ) with Ψ(R) := d(r)2 R, essetially based o the decay rate of the cocave decreasig ad strictly positive distace fuctio d(r) := mi{ x x 2 F (x ) w : w Y, w R}, R >, to zero as R which idicates for x the degree of violatio with respect to (.7). The rate (.) ca be arbitrarily slow if x misses the bechmark source coditio sigificatly what goes had i had with a very low decay of d(r) as R. If the bechmark source coditio (.7) fails, but the Fréchet derivative F (x) exists for all x B r (x ) D(F ) ad some r >, by extedig the ideas of [6, 33, 38] two further alteratives for obtaiig covergece rates to (.5) have bee preseted i the paper [26] with focus o low order Hölder source coditios (see also [23, 38]) (.2) x = x + (F (x ) F (x )) ν w, w X, < ν < 2, ad logarithmic source coditios (cf. [7]) (.3) x = x + f(f (x ) F (x )) w, w X, f(t) := ( log t) µ, µ >. As first optio the oliearity coditio (.4) F (x) = R(x, x )F (x ), R(x, x ) I Y Y C R x x κ, < κ, for some costat < C R < ad all x B r (x ) D(F ) is recommeded. The the mea value theorem i itegral form yields (cf. [6,

5 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION 5 p.28]) F (x) F (x ) F (x )(x x ) = ad hece (.5) [F (x +t(x x )) F (x )](x x )dt [R(x + t(x x ), x ) I] F (x )(x x ) dt ( ) C R t κ dt F (x )(x x ) x x κ F (x) F (x ) F (x )(x x ) C R + κ F (x )(x x ) x x κ. Now the iequality (.5) implies o the oe had that (.6) F (x) F (x ) F (x )(x x ) C F (x )(x x ) holds for some costat < C < ad all x B r (x ). O the other had, by usig the triagle iequality, from (.5) we eve derive the tagetial coe coditio (.2) i the case of sufficietly small r >, which is the also a cosequece of (.4). As secod optio the oliearity coditio (.7) F (x) = F (x )R(x, x ), R(x, x ) I X X C R x x κ, < κ, for some costat < C R < ad all x B r (x ) D(F ) has bee suggested, which is very differet from the tagetial coe coditio but ca be verified for iverse problems with boudary measuremets (cf., e.g., [5]). For Hölder ad logarithmic rates uder (.7) we refer to [26, Theorem 2.] ad should metio i this cotext that for the proof of those covergece rates a coditio of form (.7) must be valid with a uiform costat C R for all x ad x lyig i a small ball. O the other had, whe the bechmark source coditio (.7) fails or the source elemet v Y i (.7) violates the smalless coditio (.8) ad if moreover either a coditio (.3) with ay cocave idex fuctio ϕ or the coditio (.7) are satisfied, but oly a oliearity coditio (.4) holds, the to our kowledge the literature provides o covergece rate result. Hece, this situatio of low solutio smoothess i combiatio with a poor structure of oliearity describes a uexplored area with respect to covergece rates for the Tikhoov regularizatio. I Sectio 2 we will show that this situatio may arise for the real-valued autocovolutio problem o the uit iterval. This variety of autocovolutio problems, occurrig for example i the decovolutio of appearace potetial spectra, leads to operator equatios (.) which are locally ill-posed everywhere. As the umerical case studies i [] show, the stregth of ill-posedess is somewhat reduced if for a support of solutios x o [, ] the full data of F (x )

6 6 STEVEN BÜRGER AND BERND HOFMANN are observed o [, 2]. The questio of ill-posedess must be reset i the case of the complex-valued autocovolutio equatios motivated by a applicatio from laser optics (cf. [3]). We will show i Sectio 3 that both locally well-posed ad ill-posed situatios occur for such complex-valued problems with full data i depedece of the domai D(F ) uder cosideratio. 2. Autocovolutio for real fuctios o the uit iterval I this sectio, we cosider the autocovolutio operator F o the space X = Y = L 2 (, ) of quadratically itegrable real fuctios over the uit iterval [, ]. The (.) attais the form (2.) [F (x)](s) := s x(s t)x(t)dt = y(s), s, with F : L 2 (, ) L 2 (, ) ad D(F ) = L 2 (, ). This operator equatio of quadratic type occurs i physics of spectra, i optics ad i stochastics, ofte as part of a more complex task (see, e.g., [2, 6, 35]). A series of studies o deautocovolutio ad regularizatio have bee published for the settig (2.), see for example [7, 24, 25, 32]. Some first basic mathematical aalysis of the autocovolutio equatio ca already be foud i the paper [4]. Moreover, a regularizatio approach for geeral quadratic operator equatios was suggested i the recet paper [2], correspodig umerical case studies have bee preseted i [4]. Example 2.. The simple example of a sequece belogig to B r (x ) L 2 (, ), x (t) = { x (t) if t x (t) + r if < t ( = 2, 3,...), with x x = r, but F (x ) F (x ) 2r / x (t) dt 2r x as, shows that the equatio (2.) is locally ill-posed at every poit x L 2 (, ). This ill-posedess occurs although the correspodig oliear operator F is ot compact (cf. [4, Prop. 4]). However, its liearizatio is compact, sice (2.2) [F (x)h] (s) = 2 s x(s t)h(t)dt, s, h L 2 (, ),

7 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION 7 characterizes the Fréchet derivative F (x) : L 2 (, ) L 2 (, ) of F i all poits x L 2 (, ) ad ay liear covolutio operator mappig x L 2 (, ) a x L 2 (, ) with a L 2 (, ) is compact. Based o Titchmarsh s theorem (cf. [4, Lemma 3]) it ca be show that F (x ) is just a ijective operator if (2.3) sup{ t [, ] : x (t) = a.e. o [, t]} =. If a solutio x to (2.) satisfies the coditio (2.3), the x ad x are the oly solutios of this equatio, i.e. the solutio is twofold. O the other had, it was also show i [4, Theorem 2] that F is weakly cotiuous, hece weakly sequetially closed. Moreover, F (x) is Lipschitz cotiuous ad satisfies the coditio (2.4) F (x) F (x ) F (x )(x x ) = F (x x ) 2 x x 2, for all x, x L 2 (, ), such that the oliearity coditio (.4) is fulfilled with K = ad for arbitrarily large balls B r (x ). A further assertio o oliearity is formulated i the followig propositio. Propositio 2.2. For the autocovolutio operator F mappig i L 2 (, ) ad ay elemet x L 2 (, ) there is o idex fuctio η i combiatio with a radius r > such that (2.5) F (x) F (x ) Ĉ η( F (x )(x x ) ) for some costat < Ĉ < ad all x B r(x ). Proof. To costruct a cotradictio it is eough to fid a sequece {x } = B r (x ) such that F (x )(x x ) as, but lim F (x ) F (x ) >. Alog the lies of Example 4 from [4] we ca cosider the sequece of fuctios x = x + B r (x ) with (t) = 2r si(πt) ad = r >. Takig ito accout the weak covergece x x i L 2 (, ) we have F (x )(x x ) ad for ay idex fuctio η also η( F (x )(x x ) ) as, because F (x ) is a compact operator. However, F is ot compact ad lim F (x ) F (x ) = lim (2x + ) = lim = r 2 6 >. This proves the propositio. Note that we have used i this cotext the limit lim x =, which is agai a cosequece of the compactess of liear covolutio operators. Now the followig corollary of Propositio 2.2 is valid. Corollary 2.3. For the autocovolutio operator from Propositio 2.2 a coditio (.6) ad cosequetly a oliearity coditio (.4) caot hold. Moreover also the tagetial coe coditio (.2) caot hold with a small costat < C <. Proof. From (.6) we would obtai by usig the triagle iequality F (x) F (x ) F (x) F (x ) F (x )(x x ) + F (x )(x x )

8 8 STEVEN BÜRGER AND BERND HOFMANN ( C + ) F (x )(x x ) ad hece (2.5) with η(t) = t, which cotradicts Propositio 2.2. By takig ito accout that (.6) is a cosequece of the oliearity coditio (.4) we see that also (.4) caot hold. Moreover, a tagetial coe coditio (.2) would yield F (x) F (x ) F (x) F (x ) F (x )(x x ) + F (x )(x x ) C F (x) F (x ) + F (x )(x x ), ad i particular with < C < F (x) F (x ) C F (x )(x x ), which is also icompatible with Propositio 2.2. With the followig propositio we will show that also the oliearity coditio (.7) caot hold. Propositio 2.4. For the autocovolutio operator F mappig i L 2 (, ) a oliearity coditio (.7) caot hold. Proof. For x = the assertio is obviously true sice F (x ) is the zero-operator i this case, but there are o-zero operators F (x) for elemets x i ay ball B r (). Hece we ca restrict our proof to the case that x. Now let us assume that coditio (.7) is satisfied. From (.7) we have that, for all x B r (x ), R(x, x ) : X X deotes bouded liear operators with a uiform orm boud ad R(x, x ) I X X = R(x, x ) I X X C R r κ for all those operators ad their adjoits. Let us defie, for all s [, ], the fuctios { { x(s t) for t s x x s (t) :=, x else s(t) := (s t) for t s else The we have for arbitrary v L 2 (, ) s [F (x)v](s) = 2 x(s t)v(t)dt = 2 x s (t)v(t)dt. ad s [F (x )R(x, x )v](s) = 2 x (s t)r(x, x )v(t)dt Hece, = 2 x s (t)v(t)dt = x s(t)r(x, x )v(t)dt = 2 R(x, x ) x s(t)v(t)dt. R(x, x ) x s(t)v(t)dt for all v L 2 (, ),

9 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION 9 which yields for all s the equality (2.6) R(x, x ) x s = x s. To costruct a cotradictio we cosider x := x {} ( =, 2,...) with x {} (t) := x (t) + 2r si(πt). From the last equality we get R(x {}, x ) (x x ) = R(x {}, x ) x R(x {}, x ) x For the orms of x x x x 2 = = ad x {} x {} we obtai (x ( t) x ( t))2 dt + (x (t + ) x (t)) 2 dt + as x (t) 2 dt = x {} x {}. x ( t) 2 dt ad x {} x {} 2 = = + (x {} ( t) x {} ( t))2 dt + x {} ( t) 2 dt (x (t + ) + 2r si(π(t + )) x (t) 2r si(πt)) 2 dt (x (t) + 2r si(πt)) 2 dt ( 2 ) 2dt. 2 2 (x (t + ) x (t)) 2r si(πt) Now the simple iequality 2(a + b) 2 b 2 2a 2 with 2 a := 2 (x (t + ) x (t)) ad b := 2r si(πt) yields ( 2 ) 2dt 2 2 (x (t + ) x (t)) 2r si(πt) ( 4 ( τ si(πt) ) 2 ( x (t + ) x (t) ) ) 2 dt 4r 2 si 2 (πt)dt =2r 2 ( ) 2r 2 as. ( x (t + ) x (t) ) 2 dt ( x (t + ) x (t) ) 2 dt

10 STEVEN BÜRGER AND BERND HOFMANN By (.7) we obtai x {} x {} x x R(x {}, x ) X X x x ( R(x {}, x ) I X X + I X X ). x x (C R r κ + ) Takig the limit this turs to 2r 2, which is a cotradictio. Thus the proof is complete. Ufortuately, by ow we caot prove for ay elemet x that the autocovolutio operator (2.) i L 2 (, ) satisfies the tagetial coe coditio (.2) or its atteuatio (.3) for some cocave idex fuctio ϕ. Takig ito accout the triagle iequality we ca reformulate the correspodig ope problem i the followig form: Ope problem 2.5. For which elemets x do we have a cocave idex fuctio ϕ i combiatio with a radius r > for the autocovolutio operator F mappig i L 2 (, ) such that (2.7) F (x )(x x ) C ϕ( F (x) F (x ) ) holds for some costat < C < ad all x B r (x ). For solutios x, which violate the coditio (2.7) for all cocave idex fuctios ϕ, to our best kowledge covergece rates for the Tikhoov regularizatio (.5) applied to equatio (2.) ca curretly be established if ad oly if (2.8) x (t) = x(t) + t x (s t) v(s)ds, t, v L 2 (, ), v <, holds, which expresses here the bechmark source coditio (.7) together with the smalless coditio (.8). The we have for α(δ) δ the rate (.9) from [8, Theorem.4]. Necessary coditios to accomplish (2.8) cocerig the iterplay of x ad x are formulated i the subsequet propositio. Propositio 2.6. Apart from the trivial case x = x, the coditio (2.8) ca oly hold if x ad if the referece elemet x L 2 (, ) is chose such that (2.9) x x x < ad x x is a cotiuous fuctio o [, ] with x() = x (). Hece, for the appropriate choice of x the value x () must be kow. Furthermore, for the choice x = there is o x which satisfies (2.8).

11 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION Proof. For x = x, (2.8) is always satisfied with v =. By usig the orm-coservig liear trasformatio L : v ṽ i L 2 (, ) defied as [Lv](t) = ṽ(t) := v( t), t, we ca rewrite the equatio i (2.8) as x ( t) x( t) = t ṽ(t s) x (s)ds, t, or short i covolutio form as Lx = Lx + [Lv] x. However, the trasformatio x T x := L(Lx + [Lv] x) is a cotractive, affie liear mappig for fixed v < ad by Baach s fixed poit theorem there is a uiquely determied solutio x L 2 (, ) satisfyig the equatio i (2.8). For x = we have x = as uiquely determied solutio to that equatio for all such source elemets v. Now we ca estimate x x x v < x, for all ozero solutios x, which yields the ecessary coditio (2.9). Moreover, x x is a cotiuous fuctio as the result of the covolutio of the two fuctios ṽ ad x from L 2 (, ), ad thus we have x() = x () as aother ecessary coditio imposed o x to satisfy (2.8). Remark 2.7. If x solves the operator equatio (2.) the there is also a secod differet solutio x. If x, x =, the the referece elemet x L 2 (, ) has the same orm distace to both solutios ad the x-miimum-orm solutio is ot uique. Hece, the bechmark source coditio (2.8) ca apply for at most oe of the solutios x or x. Otherwise by (.9) the regularized solutios x δ α would coverge with α = α(δ) δ simultaeously to both solutios x ad x. However, from (2.9) we have for x ad thus x x 2 = x 2 2 x, x + x 2 < x 2 (2.) x 2 < 2 x, x, x. Now the ecessary coditio (2. for obtaiig (2.8) shows that uder x, x =, x, the bechmark source coditio caot hold at all. 3. Local well-posedess ad ill-posedess occurrig i the complex-valued autocovolutio equatio The complex-valued ad full data aalog to equatio (2.) was motivated by problems of ultrashort laser pulse characterizatio arisig i the cotext of the self-diffractio SPIDER method, ad we refer to [3, 28] for physical details. Takig ito accout L 2 C-spaces of quadratically itegrable complex-valued fuctios over fiite real itervals we

12 2 STEVEN BÜRGER AND BERND HOFMANN set X = L 2 C (, ), Y = L2 C (, ), ad cosider the operator equatio (3.) s x(s t)x(t)dt if s F (x) = y, [F (x)](s) := x(s t)x(t)dt if < s 2 s with F : L 2 C (, ) L2 C (, 2) ad D(F ) = L2 C (, ). Every fuctio x L 2 C (, ) ca be represeted as x(t) = A(t) eiφ(t), t, with the oegative amplitude (modulus) fuctio A = x ad the phase fuctio φ : [, ] R. Example 3.. Sice the Example 2. fails i the full data case, i.e. if y(s) is observed for all s 2, for showig the ill-posedess a ew sequece costructio became ecessary. So it was outlied i [, Prop. 2.3] that the sequece, x (t) = x (t) + Ψ 2 (t), t, Ψ β (t) := r 2β, β >, t β with x x = r ad F (x ) F (x ) as is appropriate for the full data case. This sequece eve applies to x L 2 C (, ) ad shows local ill-posedess everywhere also for the complex-valued case F : L 2 C (, ) L2 C (, 2) (cf. [3, Example 3.]). I Example 3., the perturbatio fuctio Ψ β (t) is real-valued ad has a weak pole at t = +. The, for fixed, the amplitude fuctio x is ot i L (, ) ad hece x eed ot belog to L C (, ). The ext example, however, shows that local ill-posedess everywhere ca also be show by meas of sequeces {x } = L C (, ). Example 3.2. For the complex-valued case the liear operator F (x) : L 2 C (, ) L2 C (, 2) defied as [F (x)h] (s) = 2 [x h] (s), s 2, h L 2 C(, ), is compact ad represets the Fréchet derivative of F at all poits x L 2 C (, ). Hece for weakly coverget sequeces z i L 2 C (, ), i.e. if z, z as holds for all z L 2 C (, ), we have the orm covergece lim z h = for all h L 2 C (, ). Cosequetly, for x = x +z we have F (x ) F (x ) = z (z +2x ) if ad oly if z z as. Whe we set z (t) = r e i 2 t 2, t, for fixed ad arbitrary r > the we have z = r ad z i L 2 C (, ) ad z z as i the orm of L 2 C (, 2). Hece, the problem is locally ill-posed at ay poit x L 2 C (, ). It was formulated i [3] as a ope questio whether the deautocovolutio process remais always istable if oly phase perturbatios occur. This questio is motivated by the laser pulse problem, where a complex-valued measurig tool based kerel fuctio k(s, t) is added to

13 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION3 the itegral equatio (3.), but the amplitude fuctio A = x as part of the solutio x (t) = A(t) e iφ (t), t, ca be measured ad oly the phase fuctio φ is to be determied from observed y L 2 C (, 2). The followig propositio gives a egative aswer to this questio for the simplified case (3.) with trivial kerel k(s, t). Propositio 3.3. For solutios x iφ (t) (t) = A(t) e to the complexvalued autocovolutio equatio (3.) with a fixed amplitude fuctio A L (, ), which is ot almost everywhere o [, ] the zero fuctio, we restrict the domai of the operator F : D(F ) L 2 C (, ) (, 2) to L 2 C D(F ) := {x(t) = A(t) e iφ(t), t, φ : [, ] R} L 2 C(, ). The there exist phase fuctios φ such that (3.) is locally well-posed at x. Proof. We set K := A L (,) >, K := A L (,) ad will show local well-posedess at poits x (t) = A(t) e iφ (t) with φ (t) ω, t, ad a arbitrary real costat ω. The we have for all x B K (x ) D(F ) the local Hölder coditio (3.2) x x 2 3/4 K K F (x) F (x ) with Hölder expoet /2, which yields the local well-posedess at the poit x. Namely, usig the Hölder iequality we have for all x(t) = A(t) e iφ(t) the estimate 2 2 mi(,s) F (x) F (x ) = A(s t)a(t)(e 2iω e iφ(s t) e iφ(t) )dt ds max(,s ) 2 mi(,s) A(s t)a(t)(e 2iω e iφ(s t) e iφ(t) )dt ds 2 max(,s ) ad further by settig ζ(t) := φ(t) ω 2 mi(,s) A(s t)a(t)(e 2iω e iφ(s t) e iφ(t) )dt ds = max(,s ) 2 mi(,s) max(,s ) A(s t)a(t)( e iζ(s t) e iζ(t) )dt ds

14 4 STEVEN BÜRGER AND BERND HOFMANN 2 Re mi(,s) max(,s ) A(s t)a(t)( e i(ζ(s t)+ζ(t)) )dt ds = 2 mi(,s) A(s t)a(t)( cos(ζ(s t) + ζ(t))dtds. max(,s ) By chagig the order of itegratio ad exploitig additio theorems we moreover obtai 2 mi(,s) max(,s ) A(s t)a(t)( cos(ζ(s t) + ζ(t))dtds = t+ t A(s t)a(t)( cos(ζ(s t)) cos(ζ(t))+si(ζ(s t)) si(ζ(t)))dsdt = A(t) +t A(s t)dsdt A(t) cos(ζ(t)) t+ A(s t) cos(ζ(s t))dsdt t t + A(t) si(ζ(t)) t+ A(s t) si(ζ(s t))dsdt t = A(t) A(s)dsdt A(t) cos(ζ(t)) A(s) cos(ζ(s))dsdt = A(s)ds + 2 A(t) si(ζ(t)) A(s) cos(ζ(s))ds A(s)( + cos(ζ(s)))ds A(s) si(ζ(s))dsdt 2 + A(s) si(ζ(s))ds A(s)( cos(ζ(s)))ds. 2

15 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION5 O the other had, we have x x 2 = = = = =2 2K A(t)e iω A(t)e iφ(t) 2 dt A(t) 2 e iζ(t) 2 dt A(t) 2 (( cos(ζ(t))) 2 + si 2 (ζ(t)))dt A(t) 2 ( 2 cos(ζ(t)) + cos 2 (ζ(t)) + si 2 (ζ(t)))dt A(t) 2 ( cos(ζ(t)))dt A(t)( cos(ζ(t)))dt. Owig to cos(ζ(t)) 2 we have also the estimate A(s)( + cos(ζ(s)))ds =2 A(s)ds A(s)( cos(ζ(s)))ds 2K A(s) 2 ( cos(ζ(s))) 2 ds 2K 2 =2K x x. A(s) 2 ( cos(ζ(s)))ds This yields for x B K (x ) A(s)( + cos(ζ(s)))ds K ad hece by combiig the above estimates (3.2), which proves the propositio.

16 6 STEVEN BÜRGER AND BERND HOFMANN Fially, we ote that well-posedess situatios for the real autocovolutio operator F : D(F ) L 2 (, ) L 2 (, 2) ad specific compact domais D(F ) were already outlied i [] by exploitig Fourier trasforms. I these cases, oe ca eve verify the modulus of cotiuity of the iverse operator F. 4. Coclusios ad ope questios It is certaily future work to obtai progress with respect to the Ope Problem 2.5. For the geeral theory it will be of iterest to derive (low order) covergece rates for ill-posed oliear operator equatios (.) whe the solutio x is too osmooth to satisfy the bechmark source coditio (.7) ad if moreover the oliearity structure of the operator F aroud x is too poor to fulfil a coditio (.3) for some idex fuctio ϕ or a coditio of type (.7). As aother ope questio from Sectio 2 we ca ask for smoothess classes M of solutios x of the real-valued autocovolutio equatio i L 2 (, ) such that for all x M a covergece rate (4.) x δ α(δ,y δ ) x = O (θ(δ)) as δ, with some fixed cocave idex fuctio θ is obtaied for the Tikhoov regularizatio (.5). For ill-posed operator equatios Ax = y with bouded liear operators A : X Y possessig a oclosed rage i Y such smoothess classes M yieldig (4.) are usually dese subsets M = {x X : x = Gv, v X} of the Hilbert space X characterized by rages of some bouded liear operators G : X X with ubouded Moore-Perose iverse G where G ad A are coected by some lik coditio (cf. [2]). For oliear problems the smoothess classes have a much more complicated structure. For the real-valued autocovolutio problem (2.) the bechmark source coditio (.7) leadig to (4.) with θ(t) = t is quite illustrative. Here, the smoothess class M collects all elemets x solvig the fixed poit equatio x (t) = x(t)+ t x (s t)v(s)ds, t, where the source elemets v pass through the ope uit ball i L 2 (, ). This fixed poit equatio is uiquely solvable for all such v, because the mappig x x+ 2 F (x) v is cotractive ad Baach s fixed poit theorem applies. Ackowledgemets The authors express their thaks to Qiia Ji (Australia Natioal Uiversity Caberra), Barbara Kaltebacher (Uiv. Klagefurt) ad Jes Flemmig (TU Chemitz) for helpful discussios ad valuable hits. SB ad BH greatly appreciate fiacial support by the Germa Research Foudatio (DFG) uder grats FL 832/- ad HO 454/9-, respectively.

17 ABOUT A DEFICIT IN CONVERGENCE RATES AND AUTOCONVOLUTION7 Refereces [] Azegruber S W, Hofma B, Mathé P: Regularizatio properties of the discrepacy priciple for Tikhoov regularizatio i Baach spaces. Applicable Aalysis (to appear), published electroically 3 Sept. 23, [2] Baumeister J: Decovolutio of appearace potetial spectra. I: Direct ad Iverse Boudary Value Problems (eds Kleima R, Kress R, Martese E), vol. 37 of Methode ud Verfahre der mathematische Physik. Peter Lag, Frakfurt am Mai 99, 3. [3] Boţ R I, Hofma B: A extesio of the variatioal iequality approach for obtaiig covergece rates i regularizatio of oliear ill-posed problems. Joural of Itegral Equatios ad Applicatios 22 (2), [4] Bürger S, Flemmig J: Deautocovolutio: a ew decompositio approach versus TIGRA ad local regularizatio. Paper submitted 23. [5] Burger M, Kaltebacher B: Regularizig Newto-Kaczmarz methods for oliear ill-posed problems. SIAM J. Numer. Aal. 44 (26), [6] Choi K, Laterma A D: A iterative deautocovolutio algorithm for oegative fuctios. Iverse Problems 2 (25), [7] Dai Z, Lamm, P K: Local regularizatio for the oliear iverse autocovolutio problem. SIAM J. Numer. Aal. 46 (28), [8] Egl H W, Hake M, Neubauer A: Regularizatio of Iverse Problems vol. 375 of Mathematics ad its Applicatio. Kluwer Academic Publishers, Dordrecht 996. [9] Egl H W, Kuisch K, Neubauer A: Covergece rates for Tikhoov regularisatio of o-liear ill-posed problems. Iverse Problems 5 (989), [] Fleischer G, Hofma B: O iversio rates for the autocovolutio equatio. Iverse Problems 4 (996), [] Flemmig J: Geeralized Tikhoov Regularizatio ad Moder Covergece Rate Theory i Baach Spaces. Shaker Verlag, Aache 22. [2] Flemmig J: Regularizatio of autocovolutio ad other ill-posed quadratic equatios by decompositio. J. Iverse ad Ill-Posed Problems (23) (to appear). [3] Gerth D, Hofma B, Birkholz S, Koke S, Steimeyer G: Regularizatio of a autocovolutio problem i ultrashort laser pulse characterizatio. Iverse Problems i Sciece ad Egieerig. Published olie 6 March 23: http: //dx.doi.org/.8/ [4] Goreflo R, Hofma B: O autocovolutio ad regularizatio. Iverse Problems 2 (994), [5] Grasmair M: Variatioal iequalities ad higher order covergece rates for Tikhoov regularisatio o Baach spaces. J. Iv. Ill-Posed Problems 2 (23), [6] Hake M, Neubauer A, Scherzer O: A covergece aalysis of the Ladweber iteratio for oliear ill-posed problems. Numer. Math. 72 (995), [7] Hohage T: Regularizatio of expoetially ill-posed problems. Numer. Fuct. Aal. Optimizatio 2 (2), [8] Hofma B: Approximate source coditios i TikhoovPhillips regularizatio ad cosequeces for iverse problems with multiplicatio operators. Math. Methods Appl. Sci. 29 (26), [9] Hei T, Hofma B: Approximate source coditios for oliear ill-posed problems chaces ad limitatios. Iverse Problems 3 (29), 353 (6pp). [2] Hofma B, Mathé P: Aalysis of profile fuctios for geeral liear regularizatio methods. SIAM J. Numer. Aal. 45 (27), 22 4.

18 8 STEVEN BÜRGER AND BERND HOFMANN [2] Hofma B, Mathé P: Parameter choice i Baach space regularizatio uder variatioal iequalities. Iverse Problems (22), 46 (7pp). [22] Hofma B, Kaltebacher B, Pöschl C, Scherzer O: A covergece rates result for Tikhoov regularizatio i Baach spaces with o-smooth operators. Iverse Problems 23 (27), 987. [23] Hofma B, Scherzer O: Factors ifluecig the ill-posedess of oliear problems. Iverse Problems 6 (994), [24] Jao J: Lavret ev regularizatio of ill-posed problems cotaiig oliear ear-to-mootoe operators with applicatio to atocovolutio equatio. Iverse Problems 6 (2), [25] Ji Q: Iexact Newto-Ladweber iteratio i Baach spaces with o-smooth covex pealty terms. Paper submitted 23. [26] Kaltebacher B: A ote o logarithmic covergece rates for oliear Tikhoov regularizatio. J. Iv. Ill-Posed Problems 6 (28), [27] Kaltebacher B, Neubauer A, Scherzer O: Iterative Regularizatio Methods for Noliear Ill-Posed Problems, o. 6 of Rado Series o Computatioal ad Applied Mathematics. Walter de Gruyter, Berli/New York 28. [28] Koke S, Birkholz S, Bethge J, Grebig C, Steimeyer G: Self-diffractio SPIDER. Coferece o Lasers ad Electro-Optics, OSA Techical Digest (CD) (Optical Society of America) 2, Paper CMK3, DOI.364/CLEO.2.CMK3. [29] Lu S, Pereverzev S V: Regularizatio Theory for Ill-Posed Problems, o. 58 of Iverse ad Ill-Posed Problems Series. Walter de Gruyter, Berli/Bosto 23. [3] Mathé P, Pereverzev S V: Geometry of liear ill-posed problems i variable Hilbert scales. Iverse Problems 9 (23), [3] Nair M T: Liear Operator Equatios - Approximatio ad Regularizatio. World Scietific Publ., Sigapore 29. [32] Ramlau R: Morozov s discrepacy priciple for Tikhoov-regularizatio of olieaer operators. Numer. Fuct. Aal. ad Optimz. 23 (22), [33] Scherzer O, Egl H W, Kuisch K: Optimal a posteriori parameter choice for Tikhoov regularizatio for solvig oliear ill-posed problems. SIAM J. Numer. Aal. 3 (993), [34] Scherzer O, Grasmair M, Grossauer H, Haltmeier M, Leze F: Variatioal Methods i Imagig. Volume 67 of Applied Mathematical Scieces. Spriger, New York 29. [35] Schleicher K-Th, Schuz S W, Gmeier, R, Chu, H-U: A computatioal method for the evaluatio of highly resolved DOS fuctios from APS measuremets. Joural of Electro Spectroscopy ad Related Pheomea 3 (983), [36] Schuster T: The Method of Approximate Iverse: Theory ad Applicatios. Spriger, Berli/Heidelberg 27. [37] Schuster T, Kaltebacher B, Hofma B, Kazimierski K S: Regularizatio Methods i Baach Spaces, o. of Rado Series o Computatioal ad Applied Mathematics. Walter de Gruyter, Berli/Bosto 22. [38] Tautehah U, Ji Q: Tikhoov regularizatio ad a posteriori rules for solvig oliear ill-posed problems. Iverse Problems 9 (23), 2. Faculty of Mathematics, TU Chemitz, 97 Chemitz, Germay address: steve.buerger@mathematik.tu-chemitz.de address: berd.hofma@mathematik.tu-chemitz.de