Journal of Mathematical Analysis and Applications

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1 J Math Anal Appl 388 0) Contents lsts avalable at ScVerse ScenceDrect Journal of Mathematcal Analyss an Applcatons wwwelsevercom/locate/jmaa Multvarate nequaltes of Chernoff type for classcal orthogonal polynomals Przemysław Rutka Ryszar Smarzewsk Insttute of Mathematcs an Computer Scence The John Paul II Catholc Unversty of Lubln ul Konstantynów H Lubln Polan artcle nfo abstract Artcle hstory: Receve May 00 Avalable onlne 4 November 0 Submtte by BC Bernt Keywors: Chernoff-type nequaltes Multvarate classcal orthogonal polynomals Optmal constants Generc fferental equatons In ths paper we present two-se Chernoff-type nequaltes for the error of the best approxmaton of a smooth -varate functon by polynomals of total egree less than k n the L w -norm It s suppose that the -varate weght functon w has one-mensonal classcal component weghts whch satsfy Pearson fferental equatons Smlarly as n the unvarate case the leang coeffcents of the multvarate classcal polynomals orthonormal wth respect to the weght functon w play an mportant role n the presente estmates 0 Elsever Inc All rghts reserve Introucton an prelmnares Let S k be the set of all orere -tuples α = α α ) of nonnegatve ntegers α such that α < k where k s a postve nteger an α =α + +α Then the space P of all polynomals k px) = c α x α x α = x α xα α S k wth real coeffcents c α = c α α R s sa to be the space of polynomals of total egree less than k n the varable x = x x ) R The menson of ths space s equal to ) ) + k m P k = Throughout ths paper t wll be assume that P k = P ) k = 0 s a subspace of the real Hlbert space k ) L w ) = cl P k ) k=0 wth the followng nner prouct an the norm * Corresponng author E-mal aresses: rootus@kullublnpl P Rutka) rsmax@kullublnpl R Smarzewsk) 00-47X/$ see front matter 0 Elsever Inc All rghts reserve o:006/jjmaa00

2 f g) w = P Rutka R Smarzewsk / J Math Anal Appl 388 0) f x)gx)wx) x an f w = f f ) w where s the Cartesan prouct of copes of a fnte or nfnte nterval a b) an the symbol x means the -mensonal Lebesgue measure on R Moreover we suppose that the weght functon wx) = w x ) w x ) x = x x ) has postve components w x ) on a b) whch are normalze by the conton b a w x ) x = Hence the weght functon wx) x fullfes wx) x = The weght functon wx) x s calle classcal f ts components w x ) x a b) are classcal e satsfy the Pearson fferental equaton [ A x )w x ) ] = B x )w x ) x an bounary contons of the form lm x a A x )w x ) = lm A x )w x ) = 0 x b where the polynomals A x ) = a 0 + a x + a x an B x ) = b 0 + b x are such that A x )>0ona b) an b 0 Let us recall that the fnte or nfnte sequence of polynomals Q n x )0 n < n w ) orthonormal wth respect to the nner prouct b f g) w = a f x )gx )w x ) x s sa to be a fnte or nfnte sequence of classcal orthonormal polynomals f the weght functon w x ) s classcal In aton these polynomals Q n x ) solve the followng generc fferental equaton [ A x )w x ) ] Q x ) = λ n w x )Q x ) x x wth the coeffcent λ n equal to λ n = n [ n )a + b ] ; cf Bochner [4] Krall [ 3] Agarwal an Mlovanovć [] Anrews et al [3] Chhara [6] Lesky [4] an Nkforov an Uvarov [7] By the efnton of the weght functon wx) x t s clear that the -mensonal polynomals Q α x) = Q α x ) Q α x ) 0 α < n w are orthonormal wth respect to the nner prouct ) w where n w s the least lower boun of all n w n w Inthe followng they are also calle classcal In ths paper we apply the one-mensonal results from the papers [589] to prove multmensonal estmates of Chernoff type for the best approxmaton error E wk f ) = nf p P k f p w f L k w ) whenever wx) s a classcal -varate weght functon k < n w an L k w ) enotes the space of all functons f n L w ) such that all partal ervatves

3 80 P Rutka R Smarzewsk / J Math Anal Appl 388 0) ) α ) α D α f x) = f x) x x x are absolutely contnuous functons of each separate varable x for α < k an that the partal ervatve D α f x) belongs to the space L w ) for For ths purpose we shall use the fact that the classcal orthonormal polynomals Q αx) α S k form an orthonormal bass n the space P k ) of all polynomals of total egree less than k < n w The man result If Q α x) α = α α ) s an orthonormal classcal polynomal wth respect to the weght functon wx) = w x ) w x ) then t can be wrtten n the form Q α x) = c α x α + c β x β k = α β S k The leang coeffcent of ths polynomal wll be enote by c α = c α Q α ) It s clear that ths coeffcent s fferent from 0 Moreover we have c α = α! Dα Q α x) = c α α!=α! α! ) where c α = c α Q α ) s the leang coeffcent of the corresponng unvarate classcal orthonormal polynomal Q α of egree α wth respect to the weght w x ) Now we prove the followng -mensonal generalzaton of the one-mensonal result from the paper [8] whch s the man result of ths note For ths purpose we enote A α x )w x ) g α whenever α = α α ) s a mult-nex an b g α = a A α x )w x ) x Theorem Let w be a normalze classcal weght functon on Then the nequaltes Dα ) f )x)w α x) x E wk α!c f ) D α ) f w α 0 < k < n w ) α α!c α hol whenever f belongs to L k w ) or P n w for n w = or n w < respectvely Atonally both these nequaltes become equaltes for every polynomal f of total egree less or equal to k Proof For the smplcty n the followng proof t s always assume that orers of all mult-nces are less than n w Snce the sequence of one-mensonal polynomals Q n x ) n = 0 n w s classcal t s known [ 36] that the sequence of ervatves r x Q n x ) n = r r + n w s also orthogonal wth respect to the weght functon ν r x ) = A r x )w x )/g r Moreover these ervatves solve the fferental equaton [ A x )ν r x ) ] Q x ) = λ nr ν r x )Q x ) x x wth the constant λ nr = n r) [ n + r )a + b ] The sequence of partal ervatves D α Q β x) β α of the multmensonal polynomals

4 P Rutka R Smarzewsk / J Math Anal Appl 388 0) Q β x) = Q β x ) Q β x ) s also orthogonal wth respect to the weght functon ν α x ) ν α x ) where β β ) α α ) means that β α for all = Inee f β α an μ α thenwehave D α Q β x)d α Q μ x)w α x) x = b a = c α x Q β x ) α x Q μ x )ν α x ) x { f β = μ 0 otherwse where c s a postve constant In vew of the well-known formula [7] for the best approxmaton error n Hlbert spaces we get E wk f ) = β/ S k f Q β ) w Moreover f α = α α ) s a mult-nex of orer Parseval s entty yels α f = f Q wα β ) w D α Q β = f Q β ) w D α Q β β α wα β α On the other han the one-mensonal classcal orthonormal polynomals satsfy the entty nf D α Q να β = α Q να α α β <n w whch has been prove n [8] formula 3) Hence t follows from the efnton of w α that α Q β = α Q β wα ν α wα α Q α ν = α Q α α = wα α!c α) for all mult-nces β α Therefore we get α f = f Q wα β ) β α w D α Q β wα α!c α) f Q β ) β α w an D α ) f w α f Q β ) w α!c E wk f ) ) α β α In orer to prove the converse nequalty we apply orthogonalty of classcal polynomals D α Q β to obtan an D α ) f x)wα x) x = f Q β ) w D α ) Q β x)wα x) x = f Q α ) w α!c α β α Dα f )x)w α x) x α!c α ) = f Q α ) w E wk f ) 3) It s easy to notce that E f ) = 0 whenever f sapolynomalofthetotalegreelessthank Clearly the left-han wk ses of nequaltes ) an 3) are also equal to 0 n ths case Moreover f f s a polynomal of the total egree equal to k then f = r β Q β + h β =k where h s a polynomal of the total egree less than k an r β are some coeffcents Ths mples that

5 8 P Rutka R Smarzewsk / J Math Anal Appl 388 0) E wk f ) = f Q α ) w = rα an D α f = r α D α Q α = r α α!c α Thus nequaltes ) an 3) agan become equaltes In applcaton of Theorem t s necessary to compute the constants c α For ths purpose t s recommene to use the followng remark In vew of the formula ) t follows rectly from Remark presente n the paper [8] Remark The constants α!c α ) n the Chernoff-type nequaltes can be evaluate from the formula α!c α ) = α! 3 Applcatons g α α j=0 [ α + j )a b ] 0 < α < n w In the unvarate case the results presente n ths secton agree wth the results from the paper [8] for all sx classes of classcal orthogonal polynomals On the other han the results presente below are base on the corresponng unvarate case For the smplcty we o not gve the etals about unvarate classcal orthogonal polynomals [8 058] Instea we present formulae for the constants g α an for the multvarate weghts w α x) whch conce wth the weghts wx) for mult-nex α = 0 3 Hermte weght functons Let = R A x ) = B x ) = x an e x g α g α = π By applyng Remark we get α!c α ) = π ) α! α = π ) j=0 α! k Hence Theorem gves the nequaltes π ) k Dα f )x)w α x) x α! E wk f ) π ) k D α f wα α! for all f L k w ) an k = Both these nequaltes are attane whenever f s a -varate polynomal of the total egree k 3 Laguerre weght functons Let = R + γ > A x ) = x B x ) = γ x an x γ +α e x g α = Γγ + α + ) g α It follows from Remark that ρ α = α!c α ) = Γγ + α + ) α! Hence Theorem yels the followng Chernoff-type nequaltes ρ α D α ) f x)wα x) x wk E f ) ρ α D α f wα for all f L k w ) an k = whch become equaltes whenever f s a -varate polynomal of the total egree k

6 P Rutka R Smarzewsk / J Math Anal Appl 388 0) Jacob weght functons where Let ={x R : < x < } γ > δ > A x ) = x B x ) = δ γ γ + δ + )x an x ) γ +α + x ) δ +α g α g α = γ +δ +α Γγ + + α + )Γ δ + α + ) Γγ + δ + α + ) Usng Remark an Theorem we obtan the followng nequaltes σ α D α ) f x)wα x) x wk E f ) σ α D α f wα for all f L k w ) an k = where σ α = γ +δ +)+k α! Γγ + α + )Γ δ + α + ) Γγ + δ + α + )γ + δ + α + ) α an x ) α = x x + ) x + α ) enotes the Pochhammer symbol They become enttes for every -mensonal polynomal of the total egree k Note that for specal cases of Jacob weghts e for Legenre an Chebyshev weghts the Chernoff-type nequaltes are slghtly smplfe ) For the Legenre weght functon γ = δ = 0) wehave +k α Dα f )x)w α x) x ) [α )!] α + ) E wk f ) ) For the Chebyshev weght functon of the frst kn γ = δ = )wehave π Dα f )x)w α x) x [α α )!!] E wk f ) π +k α! D α f w α ) [α )!] α + ) D α f wα [α α )!!] ) For the Chebyshev weght functon of the secon kn γ = δ = )wehave π Dα f )x)w α x) x [α )!!] 34 The generalze Bessel weght functons where E wk f ) π D α f wα [α )!!] Let = R + A x ) = x B x ) = γ x + δ γ / {0 } δ 0 an g α = + 0 x γ +α e δ /x g α In ths case we have γ n w = mn x γ +α e δ /x x = δ γ +α Γ γ α ) where x s the floor functon By applyng Remark an Theorem we get

7 84 P Rutka R Smarzewsk / J Math Anal Appl 388 0) ρ α = α!c α ) = α! δ γ +α Γ γ α ) γ α ) α an the followng Chernoff-type nequaltes ρ α D α ) p x)wα x) x wk E p) ρ α D α p wα 3) whch hol for all polynomals p P n w an become equaltes whenever p s a polynomal of total egree k 35 Jacob weght functons on R + where Let = R + A x ) = x + x B x ) = γ )x + δ + δ > an g α = + 0 x δ +α g α + x ) γ +δ α x δ +α + x ) γ x +δ α = Γγ α )Γ δ + α + ) Γγ + δ α ) an n w = mn γ It follows from Remark an Theorem that the Chernoff-type nequaltes 3) hol wth the constants ρ α = α!c α ) = α! Γγ α )Γ δ + α + ) Γγ + δ α )γ α ) α for all polynomals p P n w an become equaltes for every -varate polynomal of total egree k 36 Pseuo-Jacob weght functons Let = R an a = a = A B + C D )/ ) A + C a0 = ) ) B + D / A + C b = γ ) b 0 = δ A D B C )/ ) A + C + γ )a wth A D B C > 0 an A + C n w = mn γ > 0 In ths case we have Then t follows from Remark that the constants α!c α ) n Theorem are equal to α!c α ) = α! A + C π )γ α π A cos θ C sn θ) γ α e δθ θ γ α ) α A D B C ) γ α where the normalzng factors g α are efne as n [8] Fnally note that t s possble to combne together fferent kns of classcal orthogonal polynomals n the multvarate Chernoff-type nequaltes For example the component weght functons w x ) an w j x j ) can be ether Laguerre an generalze Bessel or Hermte an pseuo-jacob weght functons etc Snce t oes not requre any new eas we omt the etals

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