Approximation by max-product type nonlinear operators

Transcription

1 Stud. Uiv. Babeş-Bolyai Math. 5620, No. 2, Approximatio by max-product type oliear operators Sori G. Gal Abstract. The purpose of this survey is to preset some approximatio ad shape preservig properties of the so-called oliear more exactly subliear ad positive, max-product operators, costructed by startig from ay discrete liear approximatio operators, obtaied i a series of recet papers joitly writte with B. Bede ad L. Coroiau. We will preset the mai results for the max-product operators of: Bersteitype, Favard-Szász-Miraja-type, trucated Favard-Szász-Mirajatype, Basaov-type, trucated Basaov-type, Meyer-Köig ad Zellertype, Bleima-Butzer-Hah-type, Hermite-Fejér iterpolatio-type o Chebyshev odes of first id, Lagrage iterpolatio-type o Chebyshev ots of secod id, Lagrage iterpolatio-type o arbitrary ots, geeralized samplig-type, samplig sic-type, Cardaliaguet- Euvrard eural etwor-type. Mathematics Subject Classificatio 200: 4A30, 4A25, 4A29, 4A20, 4A35, 4A05, 94A20, 94A2, 92B20. Keywords: Degree of approximatio, shape preservig properties, oliear max-product operators of: Berstei-type, Hermite-Fejér ad Lagrage iterpolatio-type o Chebyshev, Jacobi ad equidistat odes, Whittaer sic-type, samplig-type, eural etwor Cardaliaguet-Euvrard-type.. Itroductio The idea of costructio of these operators goes bac to a paper of Bede, B., Nobuhara, H., Fodor, J. ad Hirota K. [], it is applied to the ratioal approximatio operators of Shepard. How could be applied to ay liear ad discrete Berstei-type operator I have show i my boo Gal [8], pp , Ope Problem 5.5.4, also a geeral form for the estimate i terms of the modulus of cotiuity is obtaied. The costructio is based o a simple idea, exemplified for the case of Berstei polyomials, as follows.

2 342 Sori G. Gal Let B fx = p,xf/ be with p, x = x x ad f : [0, ] R. If i the obvious formula B fx = p,xf/ p, x [0, ],,x we replace the operator with the max operator deoted by, the we obtai the so-called max-product Berstei oliear subliear, piecewise ratioal operator by Gal [8], p. 325 recall B M fx = p,xf/ p, x [0, ],,x p, xf/ := max 0 {p,xf/}. The same idea of costructio ca be applied to ay discrete liear Berstei-type operator or to ay discrete liear iterpolatio operator, obtaiig thus the correspodig oliear max-product operators well-defied because the deomiators of these ew operators always are strictly positive. Surprisigly, the max-product operators do ot lose the approximatio properties of the correspodig liear operators to which they are attached. Moreover, for large classes of fuctios, they improve the order of approximatio to the Jacso-type order. The most importat improvemet is i the case of iterpolatio o ay arbitrary system of odes, whe for the whole class of cotiuous fuctios the Jacso order ω f; / is achieved. Also, the max-product Berstei-type operators preserve the mootoicity ad the quasi-covexity of the fuctios. I this survey we will preset the mai results for the max-product operators of: Berstei-type, Favard-Szász-Miraja-type, trucated Favard- Szász-Miraja-type, Basaov-type, trucated Basaov-type, Meyer-Köig ad Zeller-type, Bleima-Butzer-Hah-type, Hermite-Fejér iterpolatiotype o Chebyshev odes of first id, Lagrage iterpolatio-type o Chebyshev ots of secod id, Lagrage iterpolatio-type o arbitrary ots, geeralized samplig-type, samplig sic-type, Cardaliaguet-Euvrard eural etwor-type. 2. Approximatio by max-product operators of Berstei-type Deote C + [0, ] = {f : [0, ] R + ; f is cotiuous o [0, ]}. This sectio cotais the approximatio ad shape preservig properties for a series of importat max-product Berstei-type operators.

3 Approximatio by max-product type oliear operators 343 Theorem 2.. For f C + [0, ], defie the max-product Berstei operator by Gal [8], p. 325 B M fx = p,xf/ p, x [0, ].,x i Bede-Coroiau-Gal [4] For ay j {0,,..., } ad x [ we have B M fx = f,,j x. f,,j x = j x x j +, j+ + ] j f. This form suggested the deomiatio of max-product operator for B M that is the maximum of the product of the values of f o odes with some ratioal fuctios. ii Bede-Coroiau-Gal [4] B M fx is a cotiuous, piecewise covex ad piecewise ratioal fuctio o [0, ]. iii Bede-Coroiau-Gal [4] For all x [0, ], N we have B M fx fx 2ω f;, + ω f; δ = sup{ fx fy ; x, y [0, ], x y δ}. iv Coroiau-Gal [5] There exists f C + [0, ] such that the order i iii is exactly / +, that is o the whole class C + [0, ], the order i iii caot be improved. v Coroiau-Gal [5] If f C + [0, ] is strictly positive o [0, ] the B M f f C f { [ω f; vi Coroiau-Gal [5] If f Lip the by v ] 2 + ω f; }. B M f f C f, N. vii Coroiau-Gal [5] If f Lip α, the v gives the approximatio order / 2α, which for α 2/3, ] is essetially better tha the geeral approximatio order O[ω f; / ] = O[/ α/2 ] give by iii. viii Bede-Coroiau-Gal [4] If f : [0, ] R + is a cocave fuctio the we have the Jacso-type estimate B M fx fx 2ω f;, N. ix Coroiau-Gal [5] If f C + [0, ] is strictly positive the the poitwise estimate holds x x B M fx fx 24ω f,,

4 344 Sori G. Gal for all x [0, / + ] [/ +, ], ad B M fx fx ω f, m f + 4 ω f,, for all x [/ +, / + ]. x Bede-Coroiau-Gal [4] f : [0, ] R is called quasi-covex quasicocave o [0, ] if it satisfies the iequality for all x, y, λ [0, ] fλx + λy max{fx, fy}. B M f, N, preserve the quasi-covexity, quasi-cocavity ad mootoicity of f. Remars. Comparig with the approximatio by the Berstei polyomials, clearly for large classes of fuctios, B M gives essetially better estimates. 2 The problem of fidig the saturatio class for B M is still ope. Clearly it is differet from the saturatio class of the Berstei polyomials. For f C + [0, we defie the Bleima-Butzer-Hah max-product operators by Gal [8], p. 326 H M fx = x f x + Theorem 2.2. Bede-Coroiau-Gal [8] i If f : [0, R + is cotiuous, the for ay + max{ + 2x, 6x + x} we have H M fx fx 5ω f, + x 3 2 x, x [0,, +. ω f, δ = sup{ fx fy ; x, y [0,, x y δ}. ii If f : [0, R + is a odecreasig cocave fuctio, the for x [0,, 2x, H M + x2 fx fx 2ω f;. iii H M f, N, preserve the mootoicity ad the quasi-covexity of f. For f C + [0, we defie the Meyer-Köig ad Zeller max-product operators by Gal [8], p Z M fx = x f/ +, x [0,, N. x +

5 Approximatio by max-product type oliear operators 345 Theorem 2.3. Bede-Coroiau-Gal [5] i If f : [0, ] R + is cotiuous o [0, ], the for 4 we have Z M fx fx 8ω f, x x, x [0, ], ω f, δ = sup{ fx fy ; x, y [0, ], x y δ}. ii If f : [0, ] R + is a cotiuous, odecreasig cocave fuctio, the Z M fx fx ω f;, x [0, ], N. iii Z M f, N, preserve the mootoicity ad the quasi-covexity of f. For f C + [0, ad f C + [0, ], we defie the Favard-Szász-Miraja max-product Gal [8], p. 326 ad the trucated Favard-Szász-Miraja max-product operators Bede-Coroiau-Gal [7] by ad F M fx = T M fx = x! f x! x! f x! respectively. Theorem 2.4. Bede-Coroiau-Gal [0], [7] i F M fx fx 8ω f,, x [0,, N, x [0, ], N, x, N, x [0,, ad ω f, δ = sup{ fx fy ; x, y [0,, x y δ}, T M fx fx 6ω f,, N, x [0, ]. the the ii If f : [0, R + is a odecreasig cocave fuctio o [0,, F M fx fx ω f;, x [0,, N. iii If f : [0, ] R + is a odecreasig cocave fuctio o [0, ], T M fx fx 6ω f,, N, x [0, ].

6 346 Sori G. Gal iv F M f ad T M f, N, preserve the mootoicity ad the quasi-covexity of f o the correspodig itervals. For f C + [0, ad f C + [0, ], we defie Basaov max-product Gal [8], p. 326 ad the trucated Basaov max-product operators Bede- Coroiau-Gal [9] by, respectively ad U M fx = V M fx = b, xf, b, x b, xf, x [0, ], N,, b, x b, x = + x / + x +. Theorem 2.5. Bede-Coroiau-Gal [6], [9] i For 3 ad x [0, we have xx + V M fx fx 2ω f,, ω f, δ = sup{ fx fy ; x, y [0,, x y δ}. Also, for N, 2, x [0, ] we have U M fx fx 24ω f,, + ω f, δ = sup{ fx fy ; x, y [0, ], x y δ}. ii If f : [0, [0, is a odecreasig cocave fuctio o [0,, the for 3, x [0,, V M fx fx 2ω f; x +. iii If f : [0, ] [0, is a odecreasig cocave fuctio o [0, ], the U M fx fx 2ω f;, x [0, ], N. iv V M f ad U M f, N, preserve the mootoicity ad the quasi-covexity of f o the correspodig itervals. Remar. The estimates i Theorems 2., iii, ad Theorems , i, were obtaied by usig the followig geeral result: Theorem 2.6. Gal [8], p. 326, Bede-Gal [3] Let I R be a bouded or ubouded iterval, CB + I = {f : I R + ; f cotiuous ad bouded o I},

7 Approximatio by max-product type oliear operators 347 ad L : CB + I CB + I, N be a sequece of positive homogeous operators, satisfyig i additio the followig properties: i Mootoicity if f, g CB + I satisfy f g the L f L g for all N; ii Subliearity L f + g L f + L g for all f, g CB + I. The for all f CB + I, N ad x I we have fx L fx [ ] δ L ϕ x x + L e 0 x ω f; δ I + fx L e 0 x, δ > 0, e 0 t = for all t I, ϕ x t = t x. Remars. The above Theorem 2.6 is a geeralizatio of the classical oe for Positive Liear Operators, because the Positivity + Liearity imply the Positivity + Subliearity + Positive homogeeity +Mootoicity, but the coverse implicatio does ot hold, taig ito accout that the max product operators are couterexamples. 2 The Jacso-type estimates for subclasses of fuctios i Theorems , were obtaied by direct reasoigs. 3 The saturatio results for the above max-product Berstei-type operators are iterestig ope questios. 3. Approximatio by iterpolatio max-product operators I this sectio we preset the approximatio properties of a series of maxproduct iterpolatio operators. Cosider the Hermite-Fejér iterpolatio polyomial of degree 2 + attached to f : [, ] R ad to the Chebyshev ots of first id, x, = cos π, with H 2+ fx = h, xfx,, 2 T + x h, x = xx,, + x x, T + x = cos[ + arccosx]-chebyshev polyomials. Because H 2+ fx = h,xfx, h,,x

8 348 Sori G. Gal by the max-product method the correspodig max-product Hermite-Fejér iterpolatio operator is h, xf x, H M 2+ fx = h, x Remar. We have H M 2+ fx,j = fx,j, for all j {0,..., }. Theorem 3.. Coroiau-Gal [4] If f : [, ] R + is cotiuous o [, ] the for all x [, ] ad N H M 2+ f f 4ω f,.. + Remar. For f Lip [, ], we have H M c 2+ f f +, while it is well-ow that H 2+ f f l+ +. Let ow x, [, ], {,..., }, be arbitrary ad cosider the Lagrage iterpolatio polyomial of degree attached to f ad to the odes x,, L fx = l, xfx,, with = l, x = x x,...x x, x x,+...x x, x, x,...x, x, x, x,+...x, x,. Because = l,x =, for all x R, we ca write = L fx = l,xfx, = l, for all x I.,x Therefore, its correspodig max-product iterpolatio operator will be give by l, xf x, L M fx = = l, x =, x I. Remar. We have L M fx, = fx,, =,...,. Theorem 3.2. Coroiau-Gal [2] If x, = cos, π =,..., ad f : [, ] R + the L M f f 28ω f,, 3.

9 Approximatio by max-product type oliear operators 349 Remars. For the liear Lagrage polyomials we have the worst estimate L f f Cω f; l, N. 2 The case of other id of odes e.g. equidistat, or roots of orthogoal polyomials, etc ca be foud i the joit paper [7] with L. Coroiau published i this proceedigs. Now, cosider the trucated Whittaer sic series defied by W fx = six π x π f π, x [0, π], ad the trucated max-product Whittaer operator give by six π W M x π f π fx =, x [0, π] six π x π Remar. Clearly, W M fjπ/ = fjπ/, for all j {0,..., }. Theorem 3.3. Coroiau-Gal [6] If f : [0, π] R + is cotiuous the W M fx fx 4ω f;, N, x [0, π]. [0,π] Remar. If lim ω f; / l = 0 the W fx fx uiformly iside of 0, π ad poitwise i [0, π], while it is ow that W, for all 2. 3π 4. Approximatio by samplig ad eural etwors max-prod operators This sectio cotais approximatio results for some max-product samplig operators ad for some max-product eural etwors operators. Defiitio 4.. Bardaro-Butzer-Stes-Viti [2] A fuctio ϕ CR is called a time-limited erel for a samplig operator, if: i There exist T 0, T R, T 0 < T, such that ϕt = 0 for all t [T 0, T ]; ii ϕu =, for all u R. = If ϕ is a time-limited erel ad W > 0, the S W,ϕ ft = f W = ϕw t, t R, will be called a geeralized samplig operator. Taig ito accout Defiitio 4., ii, we ca write = S W,ϕ ft = f W ϕw t = ϕw t, t R. Remar. If e.g. ϕt = sict = siπt πt, the S W,ϕ ft becomes the Whittaer cardial sic series.

10 350 Sori G. Gal Therefore, applyig the max-product method, the correspodig maxproduct Whittaer operator will be give by ϕw t f S M W,ϕ ft = = = ϕw t W, t R. Theorem 4.2. Coroiau-Gal [3] If ϕt = sict = siπt πt ad f : R R + is bouded ad cotiuous o R, the S M W,ϕ ft ft 2ω f;, for all t R, W R ω f; δ R = sup{ fu fv ; u, v R, u v δ}. Remars. If f Lipα, α 0, ], the i Theorem 4.2 we get S M W,ϕ f f = O W, while it is well-ow that for the usual Whitaer cardial α series, we have the worst estimate logw S W,ϕ f f = O W α. 2 We get similar results for other erels ϕt too. The Cardaliaguet-Euvrard eural etwor is defied by C,α fx = 2 = 2 f/ I α b α x, 0 < α <, N ad f : R R is cotiuous ad bouded or uiformly cotiuous o R. The correspodig max-product Cardaliaguet-Euvrard etwor operator is formally give by C M,α fx = 2 b = 2 2 b = 2 [ ] α x f [ α x ], x R. Theorem 4.3. Aastassiou-Coroiau-Gal [] Let bx be a cetered bellshaped fuctio, cotiuous ad with compact support [ T, T ], T > 0 ad 0 < α <. I additio, suppose that the followig requiremets are fulfilled: i There exist 0 < m M < such that m T x bx M T x for all x [0, T ]; ii There exist 0 < m 2 M 2 < such that m 2 x + T bx M 2 x + T for all x [ T, 0].

11 Approximatio by max-product type oliear operators 35 The for all f CB + R, x R ad for all N satisfyig > max{t + x, 2/T /α }, we have the estimate fx C M,α fx cω f; α R, { T M2 c = 2 max, T M } +. 2m 2 2m Remar. Let f Lipα. For 2 α <, we get the same order of approximatio O for both operators α C,α fx ad C,α M fx, while for 0 < α < 2, the approximatio order obtaied by the max-product operator C,α M fx is essetially better tha that obtaied by the liear operator C,α fx. Refereces [] Aastassiou, G.A., Coroiau, L., Gal, S.G., Approximatio by a oliear Cardaliaguet-Euvrard eural etwor operator of max-product id, J. Comp. Aalysis ad Applicatios, 2200, o. 2, [2] Bardaro, C., Butzer, P.L., Stes, R.L., Viti, G., Predictio by samples from the past with error estimates coverig discotiuous sigals, IEEE Trasactio o Iformatio Theory, 56200, o., [3] Bede, B., Gal, S.G., Approximatio by oliear Berstei ad Favard-Szász- Miraja operators of max-product id, J. Cocrete ad Applicable Mathematics, 8200, o. 2, [4] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the Berstei operator of max-product id, Iter. J. Math. ad Math. Sci., volume 2009, Article ID , 26 pages, doi:0.55/2009/ [5] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the oliear Meyer-Koig ad Zeller operator of max-product id, Numerical Fuctioal Aalysis ad Optimizatio, 3200, o. 3, [6] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the oliear Basaov operator of max-product id, Stud. Uiv. Babeş-Bolyai Math., 200 accepted for publicatio. [7] Bede, B., Coroiau, L., Gal, S.G., Approximatio by trucated Favard-Szász- Miraja operator of max-product id, Demostratio Mathematica accepted for publicatio. [8] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the oliear Bleima-Butzer-Hah operators of max-product id, Commet. Math. Uiv. Carol. Praga, 5200, o. 3, [9] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the trucated Basaov operator of max-product id, submitted for publicatio. [0] Bede, B., Coroiau, L., Gal, S.G., Approximatio ad shape preservig properties of the oliear Favard-Szász-Miraja operator of max-product id, Filomat Nis, 24200, o. 3,

12 352 Sori G. Gal [] Bede, B., Nobuhara, H., Fodor, J., Hirota, K., Max-product Shepard approximatio operators, Joural of Advaced Computatioal Itelligece ad Itelliget Iformatics, 02006, [2] Coroiau, L., Gal, S.G., Approximatio by oliear Lagrage iterpolatio operators of max-product id o Chebyshev ots of secod id, Joural Comp. Aalysis ad Applicatios accepted for publicatio. [3] Coroiau, L., Gal, S.G., Approximatio by oliear geeralized samplig operators of max-product id, Samplig Theory i Sigal ad Image Processig accepted for publicatio. [4] Coroiau, L., Gal, S.G., Approximatio by oliear Hermite-Fejér iterpolatio operators of max-product id o Chebyshev odes, Rev. d Aal. Numér. Théor. Approx. accepted for publicatio. [5] Coroiau, L., Gal, S.G., Classes of fuctios with improved estimates i approximatio by the max-product Berstei operator, Aalysis ad Applicatios accepted for publicatio. [6] Coroiau, L., Gal, S.G., Approximatio by max-product samplig operators based o sic-type erels, i preparatio. [7] Coroiau, L., Gal, S.G., Approximatio by max-product Lagrage iterpolatio operators, Stud. Uiv. Babeş-Bolyai Math., this volume. [8] Gal, S.G., Shape-Preservig Approximatio by Real ad Complex Polyomials, Birhauser Publ. Co., Bosto, Basel, Berli, Sori G. Gal Uiversity of Oradea Departmet of Mathematics ad Computer Sciece, Uiversitătii Street Oradea Romaia galso@uoradea.ro