MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS JOANNA MEISSNER 1

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1 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS JOANNA MEISSNER IMUJ Preprnt 9/9 Abstract. Let X C N [, ], ), where N 3 and let V be a lnear subspace of Π N, where Π N denotes the space of algebrac polynomals of degree less than or equal to N. Denote by P S P S X, V ) {P : X V P lnear and bounded P V d V, P S S}, where S denotes a cone of multconvex functons. In [5, 6] the mult-convex projectons were defned and t was shown the explcte formula for projecton wth mnmal norm n P S for V Π N. In ths paper we present a generalzaton of these results n the case of V beng certan, proper subspaces of Π N.. Introducton Let X be a real Banach space and let V X be a fnte-dmensonal subspace. A lnear and contnuous mappng P : X V s called a projecton f P V d V. Denote by PX, V ) the set of all projectons from X onto V. Moreover let S denote a cone, that s a convex set closed under nonnegatve multplcaton. We say that projecton P preserve S-shape f P S S notaton P P S ). Partculary, f S denotes a cone of functons whch certan dervatves are nonnegatve, then projecton preservng S-shape wll be called mult-convex. Faculty of Appled Mathematcs, AGH Unversty of Scence and Technology, Kraków, Poland; emal: messner@wms.mat.agh.edu.pl

2 JOANNA MEISSNER The man problem consder n ths paper s to fnd a mnmal multconvex projecton, that s a projecton n P S of mnmal norm. Note the problem of fndng projecton of mnmal norm so called mnmal projecton) or mnmal shape preservng projecton has been wdely studed by many authors see e.g. [] [38]). Recent papers ex. [6, 5]) gave answer to ths problem n case, where subspace V s entre space of polynomals of degree less than or equal to N see [6]) under assumpton that the norm of mnmal projecton n P s s not greater than. In ths paper we replace the space of polynomals by spaces of ncomplete ones. Also the above mentoned assumpton s not needed. The artcle s dvded nto four sectons. The frst one s an ntroducton. The second one conssts of notaton and mportant results that are used n the paper. In thrd secton we descrbe a basc projecton and ts propertes. The man results concernng recurrence formula are enclosed n fourth secton.. Notaton As t was mentoned before, the man goal of ths paper s to defne a mnmal mult-convex projecton from C N [, ] where N 3) onto certan subspaces of space of polynomals of degree less than or equal to N. To descrbe these subspaces and norms n C N [, ] we use a specal sequence. For n N N 3), let be such that K.) {,... n} : k < k +, K.) k, K.3) k n N and k n N. Now, set a norm on C N [, ] as {k } n {,,..., N} ) f max{ f k ) : {,..., n}}. ) By K.) such expresson defnes a norm and the couple C N [, ], ) s a Banach space, for brevty we wrte X as a C N [, ], )). Also usng our sequence {k } we defne v t) tk, t [, ],,..., n, 3) k! and V span{v,,... n} X. 4) Furthermore, we denote PX, V ) {P : X V : P lnear and bounded P V d V }.

3 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS3 Now defne S {f X : t [; ], {,,..., n} f k ) t) }. 5) Then, P S denotes set of all projectons preservng S, that s P S {P PX, V ) : P S S}. Now we present some result whch wll be use later. Frst two are related to the geometry of consdered spaces. Lemma. Lemma 4.4 [6]). Let g X be gven by g where δ ) t f) f ) t). Set Then W. N δ ) + δ N ), W {F X F g) N + and F }. Lemma. Theorem 4.4 [6]). Let W {F X : F u ),,..., n, F }. Assume µ s a probablstc Borel measure such that uf) f N) t)dµt). Then for any F W and for any Borel measure µ, F u). Next two lemmas concern the form of any projecton n P S. Lemma.3 Lemma 5. [6]). Let Q P S. Then there exsts u X such that n Qf u f)v + uf)v n, 6) where uf) and µ s a probablstc Borel measure. f n) t)dµt) 7) Lemma.4 Corollary 5. [6]). Let Q l P Sl. Then there exst u l, u X, for,..., l such that l Q l f n+l u l f)v [l] + l u l f lk )v [l] + uf)v [l] n+l, where u and v [l] are defned n 8), ). see page 4 and 8)

4 4 JOANNA MEISSNER 3. Mnmal mult-convex projecton basc case We start wth Theorem 3. Explcte formula for P ). Let X C N [, ], ) for N 3) and let {k } n for n N) satsfy K.)-K.3). For f X and t [, ] defne n P ft) u f)v t), where u f) f k ) ),,..., n, u n f) f k n ) ) f k n ) ) ) 8) v t) tk, t [, ],,..., n. k! Then P s mult-convex projecton from X onto V, where V s gven by 4). To prove Theorem 3. we need Observaton 3.. The projecton defned above can be wrtten as P ft) f) + f k ) ) tk k! + f k ) ) tk k! f k n ) ) tkn k n! + t N +f N ) ) N )! + f N ) ) f N ) ) ) t N N! Note that due to propertes of dfferentaton, P s lnear and bounded. Next observaton and two corollares concern the subspace V. Observaton 3.3. For,..., n and j,..., n, k -th dervatve of v j s expressed by formula t k j k, < j v k k j k )! ) j t), j, > j. Corollary 3.4. For,..., n and j,..., n v k ) j ) δ j v k n ) j ) f and only f j n or j n.

5 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS5 Corollary 3.5. For,..., n and j,..., n u v j ) δ j u v n ) u n v j ), u n v n ), where u, v are as n Theorem 3.. Proof of Theorem 3.. We dvde ths proof nto two parts. ) By Corollary 3.5, we obtan n P v j t) δ j v t) v j t). Snce P s lnear and bounded, P P S. ) Now we show that P s mult-convex. It s obvous, that for t [, ], v t). Moreover, v v )t). Let f S. Thus u f), f N ) ) and f N ) ). Hence n P ft) u f)v t) n u f)v t) + f N ) )v n t) + f N ) ) f N ) ) ) v n t) n u f)v t) + f N ) ) v n t) v n t)) +f N ) )v n t) By Observaton 3.3 for j < n, we get: n P f) kj) t) u f)v j t) j n u f)v j t) + f N ) )v n j t) j v n j t)) + f N ) )v n j t).

6 6 JOANNA MEISSNER Also P f) k n ) t) f N ) ) v t) v t)) + f N ) )v t) and P f) kn) t) f N ) ) f N ) ) ) v t) f N ) ) f N ) ) ) f N) s)ds. Ths shows that P f S and consequently P P S. Theorem 3.6. Let X C N [, ], ) for N 3) and {k } n for n N) satsfes K.)-K.). Let P be as n Theorem 3.. Then P has mnmal norm n P S and Proof. Note that t [,] { n P n k!. P f P f n sup u f)v t) + f n ) ) v n t) v n t)) + f n ) )v n t) t [,] { n sup u f)v t) + f n ) ) v n t) v n t)) + f n ) )v n t) } sup t [,] f sup f sup t [,] whch shows that } f v t) + f v n t) v n t) + f v n t) { n t [,] { n } v t) + v n t) v n t) + v n t) } n v t) + v n t) v n t) + v n t) f P n k! Before provng next nequalty we need Lemma 3.7. X C N [, ]. There exsts {f k } k X such that: k!,

7 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS7 ),..., N : lm k f ) k ), ) lm k f N ) k ), 3),..., N : f ) k. Proof. It follows from Lemma 4.4 n [6] see Lemma.) and the Goldstne Theorem. Now applyng sequence {f k } k X we obtan Hence As a result lm P f n kt) k v t) n t k k!. n t k n sup lm P f k t) sup t [,] k t [,] k! k!. n k! P. The last part concerns mnmalty of P n P S. Let Q P s. By Lemma.3 n Qf u f)v + uf)v n, where uf) s gven by 7. By Lemma. for every F W Hence and P s mnmal. Q n n k! + F u) Q P k!, 4. The constructon of projectons for other mult-convex shape Due to a specal constructon, we can obtan by recursve formula projectons preservng others shapes from our projecton P defned n prevous secton. Defne T m : C[, ] C[, ], m k, by s sm s T m fs)... fs )ds ds... ds m. 9)

8 8 JOANNA MEISSNER Observaton 4.. For any f C[, ] and m N T m f) m) f. The operator T m wll be used n our recurrence formula. Now we need some addtonal notaton. In C N+lk [, ] we consder a norm: f l max { f jk, f k +lk ) }. ) {,...,n},j {,...,l } It s clear that, the norm defned n secton s our norm. Let X l C N+lk [, ], l ), so X X. Moreover, put where Let v [l] t), v [l] t) P l P l X l, V l ) V l span{v [l], v [l],..., v [l] n+l }, ) T m v [l ] s)ds,,..., n + l. ) 3) {P : X l V l : P lnear and bounded, P Vl d Vl }.4) Now we wlle defne a cone S l. S l {f X l : t [; ], {,,..., n} f k +lk ) t) }. 5) We denote shape-preservng projecton wth respect to S l as P Sl {P P l : P S l S l }. Theorem 4. Recurrence formula). For fxed N 3 and n N suppose {k } n satsfes K.) K.3). For gven l N, let X l, V l and S l be as above. Suppose P l P Sl. Then an operator defned by P l+ ft) f) + f) belongs to P l+ P Sl+. + T m P l f k ) s)ds Proof. Frst we show that P l+ s a projecton. Note that ) k ) t) v [l+] T m P l f k ) s)ds 6)

9 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS9 and for j,..., n + l +, v [l+] j ) k) t) v [l+] j ) +m) t) ) ) m) v [l+] t) j T m v [l] j s)ds ) ) m) ) m) T m v [l] [l] j t) v j t). Consequently P l+ v [l+] t) v[l+] ) + v [l+] ) + T m P l ) T m P l v [l+] k ) s)ds v [l+] t) T m P l ds v [l+] T m P l ds ) k ) s)ds Also for,..., n + l P l+ v [l+] t) v[l+] ) + v [l+] ) + T m P l ) T m P l v [l+] k ) s)ds v[l+] ) + v [l+] ) + T m P l v [l] ) s)ds v [l+] ) k ) s)ds ) T m P l v [l] s)ds

10 JOANNA MEISSNER v[l+] ) + v [l+] ) + ) T m v [l] s)ds v [l+] v [l+] v [l+] ) ) + v [l+] ) ) ) v [l+] ) ) T m v [l] s)ds + v [l+] t) + v [l+] ) v [l+] t), what ends ths part of the proof. Now assume that we have proved and t) j {,..., l } : P l+ f) jk ) P l f k ) ) j )k ) 7) {,..., n} : P l+ f) k +lk ) P l f k ) ) k +l )k ).8) Let f S l+. Hence f k j+l+)k ) t) for t [, ] and j,,..., n. By 7 8) and defnton of P l P l+ f) k j+lk ) Hence for j,..., n P l f k ) ) k j +l )k ) P l f k ) ) k j +l )k )... P f lk ) ) k j ). ) P f lk ) k j ) n t) u f lk) )v j t) j n u f lk) )v j t) + f ) lk ) N ) )vn j t) j + f lk ) ) N ) ) vn j t) v n j t)) n f k +lk ) )v j t) + f lk+n ) )v n j t) j +f lk +N ) ) v n j t) v n j t)). Also P f lk) ) k n ) t) f lk+n ) ) t) + f lk+n ) )t

11 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS and P f lk ) ) k n) t) f lk +N ) ) f lk +N ) ) ). To end the proof, we need to show 7) and 8) Note that f) + f) t P l+ f) jk) t) + T m P l f k) s)ds whch shows 7). Also ) jk ) T m P l f k) s)ds f) + f) ) jk ) T m P l f k) s)ds ) jk ) + T m P l f k) s)ds T m P l f k) s)ds P l+ f) k +lk ) t) ) k +j )k ) ) +m+j )k ) T m P l f k) s)ds T m P l f k ) t) ) m+j )k ) P l f k ) ) j )k ) t), f) + f) + T m P l f k) s)ds T m P l f k) s)ds T m P l f k ) s)ds ) k +lk ) ) k +lk ) whch shows 8) The proof s complete ) k +k +l )k ) T m P l f k) s)ds T m P l f k ) t) ) m+k +l )k ) P l f k ) ) k +l )k ) t),

12 JOANNA MEISSNER As we know, the recurrence formula gven n Theorem 4. bulds a multconvex projecton. Now we show that f P l s mnmal n P Sl, then P l+ s mnmal n P Sl+, whch s the man result of ths paper. Frst we need two lemmas. Lemma 4.3. Let p Π N, ps) N a s! and let C p be a constant such that sup s [,] { N } a s C p.! Then T m ps)ds T m ps)ds k! C p. Proof. Set ps) n a s!, T mps) N a s +m + m)!, pt) T m ps)ds. Then T m ps)ds T m ps)ds pt) p). Note that pt) N N k! k! a t +m+ + m + )! a t +k + k )! N N a k!! t +k + k )!! a +k ) t+k! k.

13 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS 3 Hence sup pt) t [,] p) N a sup t [,] k! +k ) t+k! k k! sup N a t [,] +k ) t+k! k { N k! sup a t [,] +k ) t +k }! k { N k! sup a t +k } t [,]! N n k! a! k! a! k! C p, as requred. It s worth to notce that above Lemma plays crucal role n ths paper. It permts to generalze [[6], Theorem.4] wthout assumpton that P l. Lemma 4.4. If P l + k!, then P l+ f P l. Proof. Let f l+. By defnton of l and l+, we obtan that f k) l. Hence P l+ f sup f) + f) t + T m P l f k) s)ds T m P l f k) s)ds t [,] sup f + f t + T m P l f k) s)ds T m P l f k) s)ds t [,] f + sup T m P l f k) s)ds T m P l f k) s)ds t [,] + sup T m P l f k) s)ds T m P l f k) s)ds t [,] + k t! + + sup T m P l f k) s)ds T m P l f k) s)ds k! t [,]

14 4 JOANNA MEISSNER By Lemma 4.3, appled to ps) P l f k) s) Π N+lk and C p P l, we obtan: P l+ f + k! + k! + k! P l + P l + k k!! P l k! + + k! P l k! + P l cause k ). Frst, we show that our recurrence formula keeps the norm of projecton P l constant. Theorem 4.5. For fxed P l such that P l P l wth P l + k! and P l+ receved by the recurrence formula 4. we have Proof. Note that P l+ P l. P l+ f l+ max{ P l+ f) jk, P l+ f) k +lk ) : Assume that f l+. Then by Lemma 4.4 ) P l+ f P l. By 7) {,..., n}, j {,..., l }} ) P l+ ) jk ) f P l ) j )k ) f k ) ) P l. By 8) 3) P l+ ) k +lk ) f P l ) k +l )k ) f k ) ) ) P l. Consequently, P l+ P l. By to Lemma.4, any Q l P Sl l Q l f It s obvous that n+l u l f)v [l] + l may be represented as u l f lk )v [l] Q l f l Q l f) lk ). + uf)v [l] n+l.

15 MINIMAL MULTI-CONVEX PROJECTIONS ONTO SUBSPACES OF POLYNOMIALS 5 Lemma 4.6. If Q l P Sl and f l, f X l, then Q l f) lk ) Q, where Q s a correspondng projecton n P S. Proof. By drect calculaton see page 9), we obtan Q l f) lk ) What mples l n+l l n+l u l f)v [l] + u l f lk )v [l] l u l f lk )v [l] n u f lk )v + uf)v n. + uf)v [l] n+l Q l f) lk ) Q. Now we show the man result of ths paper. ) lk ) + uf)v [l] n+l ) lk ) Theorem 4.7 Mnmalty of P l ). Let P be a mnmal multconvex projecton P P S ) and let P l be projecton created by applyng l-tmes recurrence formula 6) see Theorem 4.). Then for any projecton Q l P Sl Q l l P l l. Proof. By Lemma 4.6, for f X, f l, Q l Q l f l Q l f) lk ) Q P P l l, whch proofs the mnmalty of P l n P Sl. Remark. If V Π N then Theorem 4.7 reduces to Theorem.4 n [6]. References [] J. Blatter, E.W. Cheney, Mnmal projectons onto hyperplanes n sequence spaces, Ann. Mat. Pura Appl. 974), 5 7. [] H.F. Bohnenblust, Subspaces of l p,n -spaces, Amer. J. Math., 63), 94), [3] B.L. Chalmers, G. Lewck, Symmetrc subspaces of l wth large projecton constants, Studa Math. 34 ) 999), [4] B.L. Chalmers, G. Lewck, Symmetrc spaces wth maxmal projecton constants, Journal of Functonal Analyss,, 3). [5] B.L. Chalmers, F.T. Metcalf, The determnaton of mnmal projectons and extensons n L, Trans. Amer. Math. Soc.,39 99),

16 6 JOANNA MEISSNER [6] B.L. Chalmers, F.T. Metcalf, A characterzaton and equatons for mnmal projectons and extensons, J. Oper. thy ), [7] B.L. Chalmers, F.T. Metcalf, Determnaton of a mnmal projecton from C[, ] onto the quadratcs, Numer. Funct. Anal. and Optmz., 99),. [8] B.L. Chalmers, M.P. Prophet, Exstence of shape preservng A-acton operators, Rocky Mountan J. Math., 8998), No. 3, [9] B.L. Chalmers, M.P. Prophet, Mnmal shape preservng projectons, Numer. Funct. Anal. and Optmz., 8997), [] B.L. Chalmers, M.P. Prophet, J. Rbando, Smplcal cones and the exstence of shape-preservaton cyclc operators, Lnear Algebra and ts Applcatons, 375 3), [] E.W. Cheney and C. Franchett, Mnmal projectons n L -spaces, Duke Mathematcal Journal 43 3) 976), 5 5. [] E.W. Cheney, C.R. Hobby, P.D. Morrs, F. Schurer, D.E. Wulbert, On the mnmal property of the Fourer projecton, Trans. Amer. Math. Soc ), [3] E.W. Cheney, P.D. Morrs, On the exstence and characterzaton of mnmal projectons, J. Rene Angew. Math ), [4] S.D. Fsher, P.D. Morrs, D.E. Wulbert, Unque mnmalty of Fourer projectons, Trans. Amer. Math. Soc. 65), 98), [5] C. Franchett, Projectons onto Hyperplanes n Banach Spaces, J. Approx. Th. 38, 983), [6] J.R. Isbell, Z. Semaden, Projecton constants and spaces of contnuous functons, Trans. Amer. Math. Soc. 7, 963) [7] J.E. Jamson, A. Kamńska, G. Lewck, One-complemented subspaces of Muselak-Orlcz sequence spaces, J. Approx. Th., 3 4), 37. [8] H. Köng, Spaces wth large projecton constants, Israel Journal Math ), [9] H. Köng, N. Tomczak-Jaegermann, Bounds for projecton constants and - summng norms, Trans. Ams. 3 99), [] H. Köng, N. Tomczak-Jaegermann, Norms of mnmal projectons, Journ. of Funct. Anal. 9, 994), [] G. Lewck, Mnmal projectons onto two dmensonal subspaces of l 4), Journ. Approx. Theory, 88) 997), 9 8. [] G. Lewck, Mnmal extensons n tensor product spaces, Journ. Approx. Theory 97) 999), [3] G. Lewck, On mnmal projectons n l n), Monatsh. Math., 9), ), 9 3. [4] G. Lewck, G. Marno, P. Petramala, Fourer-type mnmal extensons n real L -spaces, Rocky Mountan Journal of Mathematcs, 3,3) ), [5] G. Lewck, M. Prophet Mnmal Shape-Preservng Projectons Onto Π n : Generalzatons and Extensons, Numercal Functonal Analyss and Optmzaton, 7 7 8) 6), [6] G. Lewck, M. Prophet Mnmal mult-convex projectons, Studa Mathematca 78 ) 7), [7] J. Lndenstrauss, On projectons wth norm one-an example, Proc. Amer. Math. Soc. 5 No ),

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