Sendov, Blagovest H.

Takev, Milko D.

Bulgarian Digital Mathematics Library at IMI-BAS

PLISKASTUDIA MATHEMATICA BULGARICAПЛИСКАБЪЛГАРСКИ МАТЕМАТИЧЕСКИ СТУДИИProvided for non-commercial research and educational use.Not for reproduction, distribution or commercial use.The attached copy is furnished for non-commercial research and education use only. Authors are permitted to post this version of the article to their personal websites or institutional repositories and to share with other researchers in the form of electronic reprints. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to third party websites are prohibited.For further information on Pliska Studia Mathematica Bulgarica visit the website of the journal http://www.math.bas.bg/-pliska/ or contact: Editorial Office Pliska Studia Mathematica Bulgarica Institute of Mathematics and Informatics Bulgarian Academy of Sciences Telephone: (+359-2)9792818, FAX:(+359-2)971-36-49 e-mail: pliska@math.bas.bgA THEOREM OF W HITNEY ’S TYPE IN /?aBLAGOVEST H. SENDOV, M ILKO D. TAKEVtIn [1,2] H. W h i t n e y  proved the following classical theorem in approxi­mation theory and numerical analysis :T h e o r e m  1. (H. Whitney) For each integer / i≥ l ,  there is a number Wn with the following property. For any interval [a, b] and any bounded function f  on [a, b] there is a polynomial P  o f  a degree at most n — 1 such that(0.1) \ f { x ) -P (x ) \≤ W n<»n{f-, (b—a)/n) ; х Ц а ,  b\.Here со„(/ , 6 ) denotes the modulus of continuity:5): =  sup . {J ∆J/ ( 0 1 : t, t  +  nht[a ,  6 ]};I * l≤S∆" f i x ) : =  ∑o • ( - 1 )”+ ' ( «  ) f i x  +  ih).K. I v a n o v  [3] proved Theorem 1 for functions ∕  integrable in the Lebesgue sense on [a, b\.For а finite interval /, the smallest possible constant Wn in (0.1) is clearly independent of the length of /. Therefore, there are essentially three distinct cases to consider: / =  [0, 1], /* =  [0, oo) and /** =  ( — oo, oo). The smallest possible constants Wn, W*n and W*n* in each of these cases are called the Whitney constants.H. W h i t n e y  [1] gave good es t im a tes  of the constants W*n and U?**. It is proved in [4, 5] that the Whitney constants Wn are bounded by a number which does not depend on n.T h e o r e m  2. (Improved Theorem of H. Whitney [5]) For any function /, defined, bounded and integrable on [0 , 1 ] and fo r  each integer / *≥ 1 , there is a polynomial P  of a degree at most n— 1 such that(0 .2 )  | / ( * ) - P ( * ) | ≤ 6 o „ ( / ;  1 / ( л + 1 ) ) ;  * 6 [0 , 1 ].J. B r u d n y i  [7] proved an analogy of Theorem 1 for approximation of functions of several variables by quasipolynomials.The purpose of this paper is to improve the J. Brudnyi’s theorem in thesense of Theorem 2. 'I.  D e f in i t i o n s  a n d  d e n o t a t i o n s .  W e  shall fix integers n >  1, m >\  andwill introduce the following notations: □  stands for the set( 1 .1 ) ∏ :  = { ( * ,  0 < x ≤ n + \ ,  0 ≤ y ^ m - t - 1 }.Let L (M )  be the set of defined, bounded and integrable in the Lebesgue sense functions on the set M (where M  is a measurable subset of /?a or /?1), equipped with the uniform norm | ∙ ||. >PLISKA Studia mathematita bulgarica, Vol. 11, 1991, p. 78-85.Let Q^m (D ) denote the set of quasipolynomials of the х -th power not exceeding n and of the y-th power not exceeding m :m n(1.2) Q„,m( □ ) :  = P n ,m (x ,y )=  ∑ M x ) y * + ∑  g j ( y)xJt/ = 0  / = 0where f i ( x )^ L { [ 0 , n-\-1 ]) for / =  0 , 1 , . . . ,  m \ £ ) ( . у ) ( Д 1 0 , 1 ]) for y =  0 , 1 ,. . . ,  я.The best uniform approximation of the function /(*, y)£ Ц П )  by means of elements of <?„,„,(□) is(1.3) E(Qn>m ( □  ) ;  /) =  E( f ) : =  inf { || / - P 11: P  6 Qn,m ( □ ) } .As a characteristic of E ( f )  we shall use the following modulus of conti­nuity(1.4) о)л,т (/ ;  □  ) =  ©„,„,(/; 1, 1 )=  sup {\A?'™f(x, y)\:o<t, ∏≤in m , \ $ X ,  y), (x+nt;, y +  mr$(: □  ; A ™ f (x ,  y )=  ∑^ ∑^ ( — 1 )n+m+i+j^X A x  +  i ,^ y+jr\)}.Let х = | Д  +  ст, ц =  0, 1.......... .....  0 ≤ a <  1, j/ =  v  +  t ,  v  =  0, 1 , . . .  , m ,  O ≤ t ≤  1;n t — i(1.5) ln,k(t) ~  ∏ T——. for k =  0, 1,. . . , n, are the basic Lagrange polynomials/=o n~~J jdpkfor the knots 0 , 1 , . . . ,  n.The following operator was introduced in [5] for all functions / f  Z.([0, /z-t-l]):∕  + M 10 ∙ 6) ф„ ( / ;  * ) = ф „ (/ ;  M + g ) ≈  f  A?J(x—\\u)du.( n ) uBy analogy with (1.6), we define in Ц П )  the operator <p„,«(/; x, y ) that will play a very important role in our further work(1-7) <pn i m i f ;  X ,  y) =  4>∏ , m ( f /, Ц+СГ, V +  t ), я+т+ц+v I 1 „ l n \ l m\=  - ( « ) ( " » )  I  I  ) Л й  +  <Г +  ( / - Ц ) Я .  V +  T+  ( j —y)r)dudr:.Obviously, the inequalities( 1 .8 ) |фл,т(/; л:, y)\≤G>„,m( j\  □  ) / ( " ) ( ( ?  )hold for л:([|д, Ц + 1 ] ; y f [ v ,  v + 1 ] and |л =  0 , 1 , . . .  , n, v = 0 , 1 , . .  .,  m.2. P r e l i m i n a r i e s .  We shall use the following lemmas from [5 ]:  L e m m a , ! .  For each natural number n,A theorem of Whitney’s type in R2 7 9where l∏j(x) are as in ( 1 .5 ).L e m m a  2. I f8 0  В .  Sendov, М .  Takev∑  4 - m a x  { ! / „ . , ( * )  I : v ≤ a : ≤ v + 1 } ,  y = °  ( j  )K (2.1) n„,v =then ^ „ v ≤ 1 + 0- t r V+1 ’ where v==0∙ 1. ∙ ∙ ∙ . [ ( л — 1)/2], <ty=  1 +  1/2 +  ∙ ∙ ∙ + l / v ;  ( V Jao =  0 , and ( in particular)(2.2) P „ .o = l+  ∑ t T ^ - m a x {  | / „ , Д х ) |  : 0 ≤ at≤ 1 } ≤ 2 .y_1 \ j  )L e m m a  3. I f  / 6  £([0, /i-+-!)], have the following representa­tion of f(2.3) */ f(ii)dn  =  ∑ /„^(ji +  a) f  f (n )dn -h ∑ Г <рл( / \p +  n) /я.Р(ц 4 -ст—n)rf/z,(J /7 = 0 U /7 = 0 0where |д =  0 , 1 , . . . , л, O ≤ a ≤ l .Now we are ready to formulate and prove an analogous formula to (2.3). T h e o r e m  3. If / £ £ (□ ) ,  then the following equation is trueц -f-o v + T  n m p  q(2.4) ∕  f  f(n, i)dndr  =  ∑ ∑ /„,p(n +  CT)/m,,(v +  T) f  f  f(n, r)dndr0 0 /7 — 0 <7 = 0 0 0n p m  г+  ∑ /„^(m +  ct) ∕  ∑ ∕  <рт (/(я, ∙ ) ;  ^ +  /)/m,9(v +  T — r) dndr/> = 0 0 ? = 0 0m q n n+  ∑ /m,,(v +  t ) ∕  ∑ ∕  <pn(/(∙, r) ; p  +  u)l,t.p (n +  8 — n)dndrq≈z 0 0 /7=0 0n m а я+  ∑  ∑  ∕  ∕  <рл, т ( / ;  /> +  л,  ?  +  r ) / „ , , ( n  +  a  —  « ) / m,9 ( v + T — r)dndr./7 = 0 ?=0 О ОP r o o f .  In (2.3) we substitute /(x) by /(*, j')» where у  is fixed and obtain ∕(2.5) U/  f(n, y)dn-\- ∑ ( ц  +  ct) / ”/ (« ,  ,y)rfrt0 /7 = 0 0Л СГ+  ∑ ∕  <p„(/(∙, y ) ;  P +  n)l„.p(n +  (p—и)^я./7 = 0 0From the definition of the operator <p„(/, x), it is evident that(2.6) <p„(<pm( / ;  ?  +  t)  ;p+o)=<i>-m(<P„(f; p+<*);  ? + * ) = < p p + ° ,  ? + t )for all non-negative integers n, m (See (1.6) and *( 1.7)). In order to complete\ й+аthe proof, we apply (2.3) to the function g (y ) ≈  J  f (u , y)du and from (2.5),(2.6) Theorem 3 follows. ♦If we differentiate (2.4) by ^ and respectively by y, the following basic representation of /(л:, у) is obtained, which holds almost everywhere in □  :(2.7) / (n  +  a , v + t ) =  ∑ ∑ +  a )L ’m (y + t )  ∕  ∕  f ( n ,  r)dndrp ≈ O  g ≈ 0 0 0A theorem of Whitney’s type in R 2 81\ * . _  V ' ∙ . ∙i . ■vn p m+  ∑ lnp(I-1 + a) ∕  ∑ <Pm(/(«. •); q +  x)Lm,g{v)dnp = 0 0 q = 0n p m  r+  Д о  l 'n'p ^  +  a) J Д  ∕  <Р*(ЛЯ’ ') ;  9 +  r ) lm,4( y + x —r)drdnm  q n+  ∑ /m,?(V +  T)  ∕  ∑ 4>a(/(∙ ∙ ' ) ;  P  +  ^ ¥n,p(n)drq = 0  4 0 p =  0m  q n a+  9?o +T) ∕  Д  ∕  (p∏(' ^ '  ’ r ) ; /'-t-«K ,/.^  +  CT_/l) rfMarrл m+  ∑  ∑  < p „ ,m (/; Z^  +  CT, ^  +  / п , Д ц ) / т ,? ( у )  ∙p = 0 q =  0 n m  a+  ∑ ∑ f  4>n,m(f; p + n ,  q -j-x )l ’ (\.i +  a —n) lm,q(v)dnp = 0 q = 0 Ü ИI ∙ .n m  г+  ∑ ∑ ∕  <p„,„,(/; />+cr, 9  +  г )/ „ <1Р ( ц ) Г  _ ( v  +  t — r )d rp =  0 7 =  0 ün m a  T+  ∑ ∑ ∕  ∕  Я>л.т(/‘>/> +  л, q +  r)l 'n ÄV- +  a —n)l'm (y +  x—t) drdn./7 =  0 7 = 0  0  0 1Using (1.7), the right-hand side of (2.7) can be written in the form f (x ,  у ) +  Ф(х, у ), where Ф(лг, 3/) is a continuous function and Ф(лг, y )≈ 0  almost everywhere in □ .  Now it is clear that (2.7) is true for every point of □ .  From (2.7) we have(2.8) /(n +  o, vH-Т) — P * ( f ,  x, y ) =  <pn,m(f; И +  СТ, v +  T)П CT ∙-+- ∑ ∕  <P∏.m(f; Р +  П, v-f-т)/' (|д +  а — ri)dnp ≈ O  0 pm T+  9 - 0  I  <Pn'm ^ ; Ц +  СТ’ 9  +  r ) l m ,q ( V  +  X — r ) d rп m а  г \ f* +  ∑ ∑„ ∕  f  (P«,m(/; Р+П, q +  r ) l '  (\\ +  o — n ) lm A y + T —r)dudr,p = 0  q ≈ O  0 0where P * (/ ;  л:, ^ ) (  Q„_i, m-iC∏).T h e o r e m  4. If / ( Ц П ) ,  tte/i ∙ is a qtiasipolynomial /?(/; л:, у)(Q ^ w (∏ )  satisfying the following conditions'./4-1 / + 1(0 f  f  [f(x, y ) - R ( f ;  X, y )]dxdy  = 0 for i = 0, 1 , . . . , л ; / = 0 ,  1i J - > - ∙ . ∙ ∙/ + 1  *(ii) ∕  [/(jc, y )—R ( f  \ X, y ) ]d y ≈ 0  for y =  0 , 1 , . . . ,  m;j(iii) f  [ f ( x ,  y )—R ( f ;  x, y ) ] d x = 0  for /==0 ,t( i v )  A £ " R ( f ;  X, y ) = — f  f  A"∙f/ (n , r) dndrl,nx\m +  f  A[':™f(n, y)dnr\m+  f  A”f  f (x ,  r)dr^n, O≤C, n ≤ l ∙ÜP r o o f .  Let F (xy y ) ≈  f  f  f{u , v)dudv ; Fx— f  f (x , y )d v  and Fy(x, y)0 0 оX=  ∕  / («, j/)öto. Then we set0(2.9) R (f; Xy y ) ≈  "∑ Fy(p , y)l'n+Up{x )+ X  Fx(x, 9)Гт+идЬ ) ,p = 0 л+ 1  m + 1-  ∑∩ ∑∩ H p , ?)C..,WC+1 ,g(y)∙p  — 0  7 =  0  (We start to prove (ii).j + 1 / + 1 /1+1 y+ 1∕  [/(*. y ) - R U ;  X, y)]dy =  f  f ( x , j ) d y — ∑ ∕  />(/>> у ) 4 К +,,„(*)j  J  P = о УП + 1— ∑ Fx(x, o) [lm+\,q (y-h I) m^-f-1 ,7 ( /)]7  = 0Л+ 1  /Я+ 1+  ∑ ∑ F{p, q)L’ (x )\ lm+\ ,q (j+\ ) — mp ≈  0  7 = 0Я+ 1=  Fx(x, j + 1) -  Fx (x, j )  -  ∑. [/=•(/?. У + 1 ) - Д / 7, />]/„+!.„ ( * )p= 0П+ 1—  Л :  (a t, y - f -  1 )  - f - / 7*  (at, У )  -f- ∑  [/=*(/7, У + 1 )  F (P f  j ) ] l n  + \,p(<x ) —0.P ≈ OBy analogy with (ii), we get (iii). It is easy to see that (i) follows from,(ii) and (iii). In order to complete the proof of theorem 4, we have to prove (iv). From (2.9) we get8 2  В .  Senclov, М .  Takev(2.10) ∆ £ "  /?(/; jc, y )  =  ∑q J- A'" f (n, y )  dn( " + l ) ( −  1 )n+1+' £"+  Д  i  A Z f i x ,  r ) d r ( m+ l ) ( -  1)">+1+>л"'—  "∑M " ∑ ‘ ∕  ∕ / (« .  r ) d n d r [  Л+ Ч ( /Я+ 1 ) ( —  1)»+«+/+/. / =  o / = 0  0 0 \ 1 ) \ J )< 4(2.11) "∑ ∕  ∆* Л/г, д;) rf« ( " j 1) ( −  1 )"+1+'=  T  ∕  ∆»/(/j, j/)rfn[( / " , )  +  ( ”/ ) ] ( - ! ) л+1+'=  ∑ [ / 1 ∆ « / ( / i ,  ^ ) ( ; ) ( - - l ) ' - + ^  -  ∕  ∆ « / ( « ,  v ) ( "  ) ( - i ) " + ^ « ]=  £  ( - 1)n+I ( Г )  T  ∆ ∏ /(«> * )< *«  =  ∑ 0 ∕  ( - i ) " + , ( ■ ) / ( « + ' .  JO*«. = ∕  ∆ 7:“ / ( " '  -v) rf,/∙From (2.11) it is easy to see thatm J ∕ . 1 \ 1(2 .1 2 )  f  ( от+  4 ( —  1 )"> + '+ / ∆ «/ (a t ,  /-)<*•= ∕  f(x ,  у ) dr.Applying (2.11) and (2.12), we obtain (iv).3. Main result. The aim of this section is to prove the following theoremof Whitney’s type /?3.T h e o r e m  5. I f  f  £ Ц П ) ,  then there is a quasipolynomial (O )such that(3.1) II Ax, y ) - P ( f ;  X, y) ||∏≤ 4 9  (on,m (/ ;  □ ) .P r o o f .  By ∏ /t7 = l ,  2, 3, 4, we denote the following four subsets of □  .□  i .* =={(*, y)£ □  : O ≤ x ≤ n ,  O ≤ y ≤ m } ;□  a : = { (x ,  y)£ □  : 1 ≤ x ≤ n  -j-1 , 0 ≤y < m }\□ з :  = { (^ .  ! O ≤ x ≤ n ,  l ≤ j / ≤ m - h l } ;∏ 4: ≈ { (x ,  y ) $ a :  1 ≤ x ≤ r i + l 9 1 ≤ y ≤ m + l } .Obviously,(3.2) . U  - *∕ = 1Let us mention that if we substitute ∕  by f ≈ . f ^ x ,  >')— /?(/; a:, y) (/?(/;л:, 3 ;) is as in Theorem 4), then from (2.7) and (2.8) we get>■(3 .3 ) Я ц  ( g ;  X, y )  =  ( / ;  л:, _y) =  0, f o r  / = 1 ,  2, 3, 4,∕  ∕w h ere  ^ ( л : , у )  = f { n +  1 — дт, m  + 1  — у ) .W e  call the assertion  (3 .3 ) a con d it ion  fo r  s ym m etry .F rom  (2 .8 ), (3 .3 ) and T h eo rem  4 w e  h ave(3 .4 )  f(x ,  y ) = f ( x , y ) - R ( f ∙ , x , y ) ≈  <pn,m (/ ;  p  +  a ,  q + t )n rs ^+  ∑ ∕  4 4 m (/ ;  P + U ,  ? + T ) / '  (n + C T — n ) ∂ np —  0 U  ^w T ^ ∕ '+  ∑ f  <P«.m(/; /> +  <*, tf-fr )/ ' (V +  T—r)rf/<7— 0 О n m a x  ^+  ∑ ∑ ∕  ∕  cp„,m(/ ;  p-t-л, <7 +  ' " ) ^ p(^ +  c —n) l 'ma( v + T — r)dndr,p ≈ о q— 0 U - О 'И ' 'ЧA theorem of Whitney’s type in R2 8 3Applying Lemma i, Lemma 2 and (1.7), we obtain(3.5) I f {x ,  y ) — R ( f ; X, JO I ≤co„,m( f ;  □ ) / ( * ) ( * )+  ∑ W/"hw:J \ -  max 1 ^ ) 1  +  ∑ ' ar « w L ? '  max I lm’9 W' l' - °  ( p ) ( v )  ^ ≤ “ + 1  * - °  ( " ) ( ? )  V≤/≤V+*• ∕Л /Я ®/2,/yi (^> D) I I ∕  «V I I I //\ I+  ∑ ∑ ∏rv∏T\ max \ln∙p^\ max * * . « ( 0  \,  =  0 , - 0  ( " ) ( “ )  * • » » + !  v ≤ / ≤ v + 1, r_l4 l +  H-®M+CTll+1 +  H-tTv+ov+1+(H-tr(l+oll+lXl4-Ov+®v+i)≤  " Щ ( Г )  "From (iv) of Theorem 4, it is obvious that* 1 1 1 1 ) ^ = 4 ( 0 , 1 .m ( f  \ □  ).Therefore,(3.6) II/ (* ,  y )—R (f\  X, j∏ )|b≤4co^ (/ ; ∏ )(1  -+-2-f-2-b4)≤36w„,m(/; □ ) .Now using (2.11), we may represent /?(/; x, y ) £ Qn,m (∏ )  in the form1 ∏  ( x —j )(3.7) R ( f  ; * ,  J / )= P ( / ;  x, y )+  [ ∕  А у Д / г ,  j )  d u - S m^ (  y ) ] ^ -----m. ∏  O '- У )  , ,+  [ ∕  ∆ r / ( x > ---------I  f  A i f A n > r) dudrn th 4∏ (*-/ ) ∏ (y—j )X  — ______ — ______________________________________ .* •  n\  m !where 6 -„_,(x), ^ _ ! ( j / )  are the polynomials from Whitney’s Theorem 2 corres-1 1ponding to the functions J' A™f(x, r )d r , ^ ∆"/(/z, j/)cto. It is evident thatЯ (/ ; j/ )€Q „_ ltm_ i (∏ ) .From (3.6), (3.7) and Theorem 2 we haveIIfix , y)—P(f ;  X, jO||a≤36cD/r,m( / ;  □ )+  11 ∕  K f ( x ,  r ) d r — 5 „ _ 1 (л:)||С]-+  И ∕  ∆J/(«, y)dn—Sm^ ( y )  ||∏+  || £  f  ∆f∙pfCu, /)dndr  ||(3 ≤J<»n,m ( / ;  ∏ X 3 6  +  6 +  6 + l)= 4 9 to „ >m( / ;  □ ) .4. Remarks and generalizations. Theorem 5 may be formulated and pro­ved for integral Lp-norm ( p ≥ l ) .  Whitney’s theorem in R l for integral norm has been proved in [6 ].8 4  B∙ Sendov, M. TakevA theorem of Whitney's type in R2 85Using induction, we may generalize Theorem 5 in RN, where 7V=3, 4, 5 , . . . .  The essential difference between Theorem 5 and the work of J. B r u d n y i[7] is that the constant С is independent of n and m. The constant C =  49 in Theorem 5 obviously is not the best possible.R E F E R E N C E S1 . H. W h i t n e y .  On functions with bounded л -th differences. J. Math. Pures Appl., 36,1957, 67-94.2. H. W h i t n e y .  On hounded functions with bounded n-th differences. Proc. Amer. Math'Soc., 1 0 , 1959, 480-481.3. K. G. I v a n o v .  New estimates of quadrature formulae, formulae of numerical differentia­tion and interpolation. Anal. Math., 6 , 1980, 281-303.4. В I. S e n d o v .  The constants of H. Whitney are bounded. C∙ R. Acad. Bulg . Sei., 38,1985, 10, 1299-1302.5. В 1. S e n d o v .  On the theorem and constants of H. Whitney. Constr. Approx., 1987.6 . В 1. S e n d o v ,  M. D. T a к e v. The theorem of Whitney for integral norm. C∙ R∙ Acad.Bulg. Sei., 39, 1986, 10, 35-38.7. IO. А. Б р у д н ы й .  Приближение функций n переменных квазимногочленами. Изв.АН СССР, Сер. мат., 34, 1970, 564-583.Institute o f Mathematics Bulgarian Academy o f Sciences P. 0. Box 373, Sofia, BulgariaReceived 9. 12. 1986 Revised 14.04.19^7

A theorem of Whitney’s type in R2

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A theorem of Whitney’s type in R2

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