On moduli of smoothness of k-monotone functions and applications

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1 Math. Proc. Camb. Phl. Soc. 2009, 46, 23 c 2008 Cambrdge Phlosophcal Socety do:0.07/s Prnted n the Unted Kngdom Frst publshed onlne 22 August On modul of smoothness of -monotone functons and applcatons BY KIRILL KOPOTUN Department of Mathematcs, Unversty of Mantoba, Wnnpeg, Mantoba, R3T 2N2, Canada. e-mal: opotun@cc.umantoba.ca Receved August 2006; revsed 7 September 2007 Abstract Let M be the set of all -monotone functons on,,.e., those functons f for whch the th dvded dfferences [x 0,...,x ; f ] are nonnegatve for all choces of + dstnct ponts x 0,...,x n,. We obtan estmates whch are exact n a certan sense of th Dtzan Tot L q -modul of smoothness of functons n M L p,, where q < p, and dscuss several applcatons of these estmates.. Introducton Gven an nterval closed, open or half-open I, a functon f : I R s sad to be - monotone on I f ts th dvded dfferences [x 0,...,x ; f ]:= f x /w x, where wx := x x, are nonnegatve for all selectons of + dstnct ponts x 0,...,x n I. We denote the class of all such functons by I, and note that and 2 are convex cones of all monotone and convex functons, respectvely. As usual, L p J, p, denotes the space of all measurable functons f on J such that f Lp J <, where f Lp J := f J x p dx /p f p <, and f L J := ess sup x J f x f f s contnuous on [a, b], then we also use the notaton f C[a,b] := max x [a,b] f x. For smplcty, we wrte L p := L p [, ] and f p := f Lp [,]. It needs to be emphaszed that the classes I essentally depend on whether or not the nterval I s closed. For example, convex functons n the class 2 0, ] do not have to be defned at 0 and hence have to be nether bounded nor ntegrable on 0, ].f x = /x s an example of one such functon. At the same tme, the class 2 [0, ] conssts of all functons f whch are convex and contnuous on 0,, defned at 0 and, and such that lm x 0 + f x f 0 and lm x f x f. Therefore, 2 [0, ] conssts only of bounded functons contnuous everywhere put perhaps the endponts of [0, ] and hence belongng to all L p [0, ] spaces. Clearly, I I where I s the closure of I. All our results n ths paper are gven for the larger functon classes M I := nti, where nts denotes the nteror of a set S. Note that, wth ths notaton, M := M [, ] =M,, etc. Functons from M I enjoy varous smoothness propertes. For example see [, 2], f f :[a, b] R s -monotone for some 2, then, for all j 2, f j x exsts on a, b and s j-monotone. In partcular, f 2 x exsts, s convex, and therefore satsfes a Lpschtz condton on any closed nterval [ξ, ζ] contaned n a, b, s absolutely contnuous on [ξ, ζ], s contnuous on a, b, and has left and rght nondecreasng

2 24 K. KOPOTUN dervatves, f x and f + x on a, b. Moreover, the set E where f x fals to exst s countable, and f s contnuous on a, b \ E. For N, the th classcal modulus of smoothness of a functon f L p J s defned by where h f, x, J := ω f,δ,j p := sup h f,, J L p J, 0<h δ f x h/2 + h, f x ± h/2 J, 0, otherwse, s the th symmetrc dfference. It s rather straghtforward to show that, f q < p and a functon f L p [a, b] has s < changes of monotoncty note that -monotone functons can have at most ponts of monotoncty changes, then ω f,δ,[a, b] q cδ /q /p f Lp [a,b] wth a constant c dependng on s. Snce ω + f,δ,j p 2 max {,/p} ω f,δ,j p, N, the same nequalty holds for all th modul. The dea of the proof s to prove ths nequalty for any monotone functon frst usng consderatons smlar to what we use below to estmate Dtzan-Tot modul of functons n M, represent [a, b] as a unon of ntervals such that f s monotone on each of them, and fnally use Hölder s nequalty f Lq I I /q /p f Lp I, q p, to obtan estmates near ponts of monotoncty changes we omt detals. Also, note that the constant c n cannot be made ndependent of s, and so may no longer be vald for an arbtrary functon f from L p [a, b]. For example, consder f β x := snπβx,0 x, where β N. Clearly, f β has β changes of monotoncty, f β Lp [0,], and assumng that δ /2, we have h f, x q L q [0,] = h/2 h/2 sn πβx + h/2 sn πβx h/2 q dx h/2 = 2 q snπβh/2 q cosπβx q dx snπβh/2 q. Hence, ω f,δ,[0, ] q sup snπβh/2 mn{,βδ}. 0<h δ In partcular, f δ := /β and β, then δ /q+/p ω f β,δ,[0, ] q, and so cannot be bounded by f β Lp [0,]. We also remar that the power of δ n cannot be ncreased. Indeed, for p <, f α := x α,0 < α < /p, s monotone and f α Lp [0,] cα, p. At the same tme, ω f α,δ,[0, ] q cδ /q α, and so f were vald wth δ /q /p+ɛ, ɛ>0, one could choose any α /p ɛ/2 n order to obtan a contradcton. Unform estmates of polynomal approxmaton n terms of classcal modul of smoothness are rather mperfect snce, as s rather well nown, the rate of approxmaton can be mproved near the endponts of an nterval see e.g. [4]. If approxmaton n the unform h/2

3 Modul of smoothness of -monotone functons 25 norm s nvestgated, pontwse estmates yeld constructve characterzaton of the smoothness classes va approxmaton orders acheved by algebrac polynomals. If one desres to obtan exact unform estmates e.g. for approxmaton n the L p norm, p <, then one can no longer use the classcal modul of smoothness, and the new measure of smoothness s needed. Dfferent approaches are possble see e.g. [3, chapter 3] for dscussons and comparsons, but the one that has receved most attenton n recent years s the theory developed by Dtzan and Tot [3] there s also Ivanov s τ-modulus [5] whch s equvalent to the Dtzan Tot modulus for some values of parameters whch we do not dscuss n ths paper. The th Dtzan Tot modulus of f L p s ω ϕ f, t p := sup 0<h t hϕ f, p, where ϕx := x 2 and μ f, x := μ f, x, [, ]. The followng theorem s one of our man results. THEOREM. Let N, q < p, and f M L p. Then ω ϕ f,δ q cϒ δ, q, p f p, 2 where δ 2/q 2/p, f 2, δ ϒ δ, q, p := 2/q 2/p, f = and p < 2q, δ lnδ /q, f = and p = 2q, δ /q, f = and p > 2q. Note that the estmates n Theorem are exact n the sense that the powers of δ n 2 cannot be ncreased. In the cases 2, and = and 2q > p, ths follows, for example, from a smple observaton see also [3, page 35] that, f f ɛ x := + x ɛ /p, ɛ>0, ɛ /p N 0, then f or f s -monotone, f ɛ Lp [0,] cɛ, p, and ω ϕ f ɛ,δ q cδ mn {,2/q 2/p+2ɛ} f 2/q 2/p + 2ɛ. In the case = and 2q p, t s suffcent to notce that ω ϕ sgnx, δ q δ /q. In fact, ω ϕ sgnx, δ q δ /q, for all N, whch shows, n partcular, that Theorem s no longer true n general f the assumpton that f M s removed. Note that a fner analyss and other counterexamples are possble. Perhaps Theorem would not be too nterestng by tself, and our man motvaton n consderng t and wrtng ths note s several applcatons whch we dscuss n Secton 2. In partcular, as we dscuss n Secton 2, Theorem provdes a new method of provng some nown as well as several new results. 2. Applcatons. Recall the followng well-nown result whch holds for all and not only -monotone functons n L q see [3, theorem 7 2 ]: for N and q, there exsts a constant c whch depends only on such that, for any f L q and n, there s a polynomal r n n such that f r n q cω ϕ f, n q. 2

4 26 K. KOPOTUN Now, let q < p, and let f L p M, N, be such that f m L p wth 0 m. Then, usng Theorem and the fact that f m M m, we conclude that for any n, there exsts a polynomal r n of degree n such that f r n q cω ϕ f, n q cn m ω ϕ m f m, n q cn m ϒ /n m, q, p f m p. In other words, f r n q c f m p n m 2/q+2/p, f m 2, or m = and p < 2q, n m /q ln n /2q, f m = and p = 2q, n m /q, m = and p > 2q. 2 2 Remar 2. If = or = 2, then the polynomal r n n 2 2 may be chosen to be from M. Ths mmedately follows see [9, 3, 4] from the fact that, for = or = 2, any f M L q, q, and n, there exsts a polynomal r n n M such that f r n q cω ϕ f, n q, where c s an absolute constant. Now, recallng a consequence of Hölder s nequalty g q g α p g α, α := pq /qp, q < p, and usng Theorem and 2 we have the followng estmates for f M such that f m L p,0 m : f r n q cω ϕ f, n q cn m ω ϕ m f m, n q cn m sup m hϕ f m, α m 0<h n p hϕ f m, α f m, n α f m, n α cn m ω ϕ m cn m ω ϕ m p ωϕ m f m, n α p ϒ /n m,, p α f m α p. 2 3 Recallng that n 2+2/p, f 0 m 2, n 2+2/p, f m = and p < 2, ϒ /n m,, p = n ln n, f m = and p = 2, n, f m = and p > 2, we have, for < q < p and 0 m, f r n q = on m ln, 2 4 where n 2/q+2/p, f 0 m 2, or m = and p < 2, ln := n ln n 2/q, f m = and p = 2, n p q/qp, f m = and p > 2. In partcular, for 0 m, p =, and f M such that f m L,wehave f r n q = o n m mn { m/q,2/q}, < q <. 2 5 Addtonally, f m = 0, then for f M L,wehave f r n q = o n mn {/q,2/q}, < q <. 2 6

5 Modul of smoothness of -monotone functons 27 If = or = 2, then by Remar 2, r n n nequaltes can be chosen to be from M. In the case = 2, 2 6 s the man result n [0]. Fnally, we menton that the above estmates mprove [8, theorem 3]. 2. Recently, Konovalov, Levatan and Maorov [6] nvestgated the orders of best approxmaton by polynomals and rdge functons of certan classes of -monotone radal functons. They obtaned several asymptotcally exact estmates. Our Theorem yelds a dfferent and smpler proof of the upper estmates n [6]. Moreover, we are able to obtan results for monotone and convex polynomal approxmaton as an mmedate consequence of Theorem, some nown estmates, and lower estmates n [6]. In order to dscuss ths further, we need to ntroduce some new notaton. Let M B p denote the ntersecton of M wth the unt ball n L p,.e., M B p s the set of all -monotone functons f on, such that f p, and let n be the space of all algebrac polynomals of degree n. Also, EM B p, n q := sup nf f r n q f M B r n n p denotes the rate of approxmaton of the set M B p by n. It was shown n [6] that, for q p, { n EM mn B p, n q {/q,2/q 2/p}, f = and p 2q, n 2/q+2/p 2 7, f 2, where, for postve sequences a n and b n, a n b n means that c a n b n c 2 a n for some postve constants c and c 2 and all n N. The upper estmates n 2 7 now mmedately follow from 2 2 wth m = 0 tang nto account that, n the case = and p 2q, ϒ δ, q, p = δ mn {/q,2/q 2/p}. Now, let EM B p, n M q := sup nf f r n q f M B p r n n M be the rate of approxmaton of the set M B p by -monotone polynomals n n. Clearly, EM B p, n q EM B p, n M q. Therefore, 2 2, the observaton after t, and lower estmates n 2 7 mmedately mply that { n EM B p, n M mn q {/q,2/q 2/p}, f = and p 2q, n 2/q+2/p, f = 2. Note that, n the case = and p = 2q, we get EM B p, n M p/2 cn 2/p ln n /p, n Auxlary results and proof of Theorem It s convenent to denote D α := {x : x ± αϕx [, ]} = {x : x } α2. + α 2 Then, hϕx f, x = 0fx D h/2. The author s ndebted to the authors of [6] for dscusson of ther new results.

6 28 K. KOPOTUN LEMMA 3. Let 0 <α< and α β α. Then, for any ntegrable functon f, we have D α f x + βϕx dx = + β 2 2αα β/+α 2 +2αα+β/+α 2 βy f y + dy. y2 + β 2 Proof. The functon gx = x + βϕx s strctly ncreasng on D α, snce the only crtcal pont x 0 of g n the case β 0 satsfes x 2 0 = + β 2 + α 2 > α2 + α 2 > α 2 2, + α 2 and hence x 0 D α. Now, solvng the equaton y = x + βϕx we get x = + β 2 y β y 2 + β 2, and t remans to change the varable of ntegraton. The followng auxlary result s nterestng n ts own rght, and s more general than Theorem. Whle we only need the cases = and = 2 n Theorem 3 2 n order to prove Theorem, the proof for arbtrary N s not much longer. However, snce t s somewhat techncal we postpone t untl the last secton. THEOREM 3 2. Let N, f M L and δ /. If s even, then If s odd, then ω ϕ f,δ c f L [, + 2 δ 2 ] + f L [ 2 δ 2,] + δ f. ω ϕ f,δ c f L [, + 2 δ 2 ] + f L [ 2 δ 2,] + sup h f y y 2 /2 L [ + 0<h δ 2 h 2 /2, 2 h 2 /2]. The followng corollary mmedately follows by Hölder s nequalty and the fact n the case for odd that, for p wth /p + /p =, y 2 /2 h +2/p, f p > 2, c Lp lnh /p, f p = 2, [ + 2 h 2 /2, 2 h 2 /2], f p < 2. In partcular, f 3, then y 2 /2 Lp [ + 2 h 2 /2, 2 h 2 /2] ch +2/p, and, f =, then y 2 /2 c h +2/p, f p > 2, Lp [ +h 2 /2, h 2 /2] lnh, f p = 2,, f p < 2. COROLLARY 3 3. Let N, f M L p, p. Then δ 2 2/p, f 2, or = and p < 2, ω ϕ f,δ c f p δ lnδ, f = and p = 2, δ, = and 2 < p. We now generalze the estmates n Corollary 3 3 n the case = and = 2 provdng estmates of ω ϕ f,δ q for all q <. We need two auxlary lemmas.

7 Modul of smoothness of -monotone functons 29 LEMMA 3 4. Let q <, and let f L q be nonnegatve on [, ]. Then, ω ϕ f,δ q ω ϕ f q,δ /q. 3 Proof. The followng nequaltes mmedately follow from the convexty of x q and postvty of + x q x q forx > 0 and q : 2 q a + b q a q + b q a + b q, a 0, b 0 and q. 3 2 Snce by 3 2, a a 2 q a q aq 2, a 0 and a 2 0, q, for any nonnegatve functon f we have μ f, x q μ f q, x, whch mples 3. LEMMA 3 5. Let q <, and let f M 2 L q be nonnegatve on [, ]. Then, ω ϕ 2 f,δ q 2 /q ω ϕ 2 f q,δ /q. 3 3 Proof. If a 0, a 2 0, a 3 0, and a 2a 2 + a 3 0 and q, then usng 3 2 we have a 2a 2 + a 3 q + 2a 2 q a + a 3 q 2 q a q + aq 3, and so a 2a 2 + a 3 q 2 q a q 2aq 2 + aq 3. Ths mples that, for a convex and nonnegatve functon f, 2 μ f, x q 2 q 2 μ f q, x, and 3 3 mmedately follows. Now, tang nto account that, for a nonnegatve f, f q /q p/q = f p, and usng Lemmas 3 4 and 3 5, and Corollary 3 3 wth p/q nstead of p we get the followng result. COROLLARY 3 6. Let = or = 2, q < p, and let f M L p be nonnegatve. Then δ 2/q 2/p, f = 2, or = and p < 2q, ω ϕ f,δ q c f p δ lnδ /q, f = and p = 2q, δ /q, = and p > 2q. We are now ready to prove Theorem. Frst, note that, f 2 holds for functons f and f 2 whch have the same sgn at all ponts n [, ].e., f x f 2 x 0, x, then t s also vald for f + f 2. Indeed, snce we have f p + f 2 p 2 max f, f 2 p 2 f + f 2 p = 2 f + f 2 p, ω ϕ f + f 2,δ q ω ϕ f,δ q + ω ϕ f 2,δ q cϒ δ, q, p f p + f 2 p cϒ δ, q, p f + f 2 p. 3 4 The followng lemma shows that t s suffcent to prove Theorem for functons f M such that f 0 = 0, 0, where f 0 := f 0 any number between f 0 and f + 0 would do, snce ω ϕ f T f, δ q = ω ϕ f,δ q.

8 220 K. KOPOTUN Ths lemma s an mmedate corollary of a stronger [7, theorem ] tang nto account [2] see also [, Theorem 4 6 3]. LEMMA 3 7 [7]. Let N, 0 < p, and f M L p. Denote by T f, x :=! f 0x the McLaurn polynomal of degree, where f 0 := f 0. Then, there exsts a constant c = c, p such that f T f, p c f p. Proof of Theorem. Let f M L p be such that f 0 = 0, 0 2, and f 0 = 0. Then, as s easly shown by nducton see also [7, lemma 7], f M j [0, ] and j f M j [, 0] for all j = 0,...,. Now, let f x := { 0, f x 0, f x, f 0 < x, and f 2 x := { f x, f x 0, 0, f 0 < x, Note that the functons f and f 2 have the same sgn on [, ] f x f 2 x = 0 for all x, and that f = f + f 2. If =, then f x and f 2 x are both nonnegatve functons n M L p, and f 2, then f x and f 2 are both nonnegatve functons n M 2 L p. Therefore, Corollary 3 6 and 3 4 mply that 2 s satsfed for f = f + f 2 tang nto account that ω ϕ f,δ q cω ϕ 2 f,δ q for 2, see [3, theorem 4 3]. It s convenent to denote 4. Proof of Theorem 3 2 Jβ, y := + β 2 + βy. y2 + β 2 Tang nto account that hϕx f, x 0, for every x, and usng Lemma 3 wth α = h/2 and β = /2h, 0, wehave hϕx f, x = f, x dx D h/2 hϕx = f x + /2hϕx dx D h/2 4 h = 2 /4+ 2 h 2 f y J /2h, y dy +4h 2 /4+ 2 h 2 +4 = 2 h 2 /4+ 2 h h 2 /4+ 2 h h 2 /4+ 2 h h 2 /4+ 2 h 2 4 h 2 /4+ 2 h 2 + f y J /2h, y dy =: 4 2 h 2 /4+ 2 h 2 I + I 2 + I 3.

9 Modul of smoothness of -monotone functons 22 Snce J /2h, y 2, 0, wehave I + I h 2 /4+ 2 h 2 4 h 2 /4+ 2 h f y dy +4h 2 /4+ 2 h h 2 /4+ 2 h f L [, +4 2 h 2 /4+ 2 h 2 ] + f L [ 4 2 h 2 /4+ 2 h 2,] c f L [, + 2 h 2 ] + f L [ 2 h,]. 4 2 Now, where A y, h := = j=0 4 2 h 2 /4+ 2 h 2 I 2 = +4 2 h 2 /4+ 2 h 2 J /2h, y + + /2 2 h 2 f ya y, h dy, 4 2 /2hy. 4 3 y2 + /2 2 h 2 Changng the order of summaton.e., lettng j = we get A y, h = j j /2hy. 4 4 j + j /2 2 h 2 y2 + j /2 2 h 2 Therefore, addng 4 3 and 4 4 we have 2A y, h = + /2 2 h 2 In partcular, f s even, then and, f s odd, then A y, h = A y, h = + + /2hy y2 + /2 2 h 2 + /2 2 h /2 2 h 2 /2hy y2 + /2 2 h 2. We now consder the cases for even and odd separately. Case I: even. It s well nown that f g m s contnuous on [a, b] and f x 0, x,...,x m are any m + dstnct ponts n [a, b], then for some ξ a, b, [x 0,...,x m ; f ]=g m ξ/m!. Snce m h f, x = m! hm [x mh/2, x mh/2 + h,...,x + mh/2; f ], we conclude that, f g m s contnuous on [x mh/2, x + mh/2], then for some ξ x mh/2, x + mh/2, m h g, x = hm g m ξ. 4 5.

10 222 K. KOPOTUN We now note that t follows from the defnton of the th symmetrc dfference that, f gt := + t 2, then A y, h = h g, 0 and 4 5 mples that for some ξ h/2, h/2 A y, h =h g ξ. Snce g C [, ], we conclude that g ξ c and so A y, h ch.itnow follows from 4 2 that 4 2 h 2 /4+ 2 h 2 I 2 ch f y dy ch f h 2 /4+ 2 h 2 Hence, recallng 4, hϕx f, x c f L [, + 2 h 2 ] + f L [ 2 h 2,] + h f and so for even we have ω ϕ f,δ c f L [, + 2 δ 2 ] + f L [ 2 δ 2,] + δ f. Case II: odd. Let y [ h 2 /4 + 2 h 2, 4 2 h 2 /4 + 2 h 2 ] be fxed and denote γ := y 2 and t gt := + t 2 γ 2 + t. 2 Then, A y, h = y h g, 0 and so by 4 5 A y, h = y h g ξ, ξ h/2, h/2. To estmate the th dervatve of gt for t [ h/2, h/2] we note that γ h/2 and so t γ. Now, notce that gt = Gt/γ where Gx := + γ 2 x x. 2 + x 2 It s not dffcult to show that G C[,] c, and so g t = γ G t/γ cγ, for all t γ. 4 6 Hence, A y, h ch y 2 /2, and t follows from 4 2 that 4 2 h 2 /4+ 2 h 2 I 2 ch f y y 2 /2 dy +4 2 h 2 /4+ 2 h 2 ch f y y 2 /2. L [ + 2 h 2 /2, 2 h 2 /2]

11 Modul of smoothness of -monotone functons 223 Therefore, recallng 4 we have hϕx f, x c f L [, + 2 h 2 ] + f L [ 2 h 2,] and fnally + h f y y 2 /2 L [ + 2 h 2 /2, 2 h 2 /2] ω ϕ f,δ c f L [, + 2 δ 2 ] + f L [ 2 δ 2,] + sup h f y y 2 /2 L [ + 0<h δ 2 h 2 /2, 2 h 2 /2]. The proof of Theorem 3 2 s now complete. Acnowledgements. The author acnowledges support of NSERC of Canada. The author also thans anonymous referees for several useful suggestons. REFERENCES [] R. A. DEVORE and G. G. LORENTZ. Constructve approxmaton. Grundlehren der Mathematschen Wssenschaften [Fundamental Prncples of Mathematcal Scences] vol. 303 Sprnger-Verlag, 993. [2] N. DYN, G. G. LORENTZ and S. D. RIEMENSCHNEIDER. Contnuty of the brhoff nterpolaton. SIAM J. Numer. Anal , no. 3, [3] Z. DITZIAN and V. TOTIK. Modul of smoothness. Sprnger Seres n Computatonal Mathematcs, vol. 9 Sprnger-Verlag, 987. [4] V. K. DZJADYK. Vvedene v teoryu ravnomernogo prblzhenya funts polnomam [ntroducton to the theory of unform approxmaton of functons by polynomals]. Izdat. Naua, Moscow 977 Russan. [5] K. G. IVANOV. Drect and converse theorems for the best algebrac approxmaton n C[, ] and L p [, ]. C. R. Acad. Bulgare Sc , no. 0, [6] V. N. KONOVALOV, D. LEVIATAN and V. E. MAIOROV. Approxmaton by polynomals and rdge functons of the classes of s-monotone radal functons, preprnt. [7] K. A. KOPOTUN. Whtney theorem of nterpolatory type for -monotone functons. Constr. Approx , no. 2, [8] K. A. KOPOTUN. Approxmaton of -monotone functons. J. Approx. Theory , no. 3, [9] D. LEVIATAN. Monotone and comonotone polynomal approxmaton revsted. J. Approx. Theory , no., 6. [0] M. NIKOLTJEVA HEDBERG and V. OPERSTEIN. A note on convex approxmaton n L p. J. Approx. Theory 8 995, no., [] J. E. PEČARIĆ, F. PROSCHAN and Y. L. TONG. Convex functons, partal orderngs, and statstcal applcatons. Math.Sc.Engrg.vol. 87 Academc Press Inc., 992. [2] A. W. ROBERTS and D. E. VARBERG. Convex Functons Academc Press, 973, Pure and Appl. Math. vol. 57. [3] X. M. YU. Monotone polynomal approxmaton n L p space. Acta Math. Snca N.S , no. 4, [4] X.M.YU. Convex polynomal approxmaton n L p space. Approx. Theory Appl ,

Extending Structures for Lie 2-Algebras

Extending Structures for Lie 2-Algebras mathematcs Artcle Extendng Structures for Le 2-Algebras Yan Tan * and Zhxang Wu Mathematcs Department, Zhejang Unversty, Hangzhou 310027, Chna; wzx@zju.edu.cn * Correspondence: ytanalg@126.com Receved: