Approximation of functions belonging to the class L p (ω) β by linear operators

Transcription

1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 3, 9, Approximtion of functions belonging to the clss L p ω) β by liner opertors W lodzimierz Lenski nd Bogdn Szl Abstrct. We prove results which correspond to the theorems of M. L. Mittl, B. E. Rhodes, V. N. Mishr Interntionl Journl of Mthemtics nd Mthemticl Sciences, Volume 6 6), Article ID 53538, pges] on the rte of norm nd pointwise pproximtion of conjugte functions by the mtrix summbility mens of their Fourier series.. Introduction Let L p < p < ) be the clss of ll -periodic rel-vlued functions integrble in the Lebesgue sense with p-th power over Q =,] with the norm /p f = f ) L p = ft) ) p. ) Consider its trigonometric Fourier series Sfx) := f) + ν f)cos νx + b ν f)sinνx) nd the conjugte one Sfx) := ν= Q b ν f)cos νx ν f)sin νx) ν= with the prtil sums S k f nd S k f, respectively. We know tht if f L, then f x) := ψ x t) cot t = lim f x,ɛ), ɛ Received September, 8. Mthemtics Subject Clssifiction. 4A4. Key words nd phrses. Degree of pproximtion, conjugte functions.

2 W LODZIMIERZ LENSKI AND BOGDAN SZAL where with f x,ɛ) := ɛ ψ x t) cot t ψ x t) := f x + t) f x t), exists for lmost ll x 7, Th. 3.)IV]. Now, we define two clsses of seuences see ]). A seuence c := c n ) of nonnegtive numbers tending to zero is clled the Rest Bounded Vrition Seuence, or briefly c RBV S, if it hs the property c n c Kc)c m ) k=m for ll nturl numbers m, where Kc) is constnt depending only on c. A seuence c := c n ) of nonnegtive numbers will be clled the Hed Bounded Vrition Seuence, or briefly c HBV S, if it hs the property m c n c Kc)c m 3) for ll nturl numbers m, or only for ll m N if the seuence c hs only finite number of nonzero terms nd the lst nonzero term is c N. We ssume tht the seuence Kα n )) n= is bounded, tht is, there exists constnt K such tht Kα n ) K holds for ll n, where Kα n ) denotes the seuence of constnts ppering in the ineulities ) or 3) for the seuence α n := n,k ) n. Now we cn give the conditions to be used lter on. We ssume tht for ll n nd m n n,k n,k+ K n,m 4) or k=m m n,k n,k+ K n,m 5) holds if α n := n,k ) n belongs to RBV S or HBV S, respectively. Let A := n,k ) be lower tringulr infinite mtrix of rel numbers such tht n,k, n,k = k,n =,,,...),

3 APPROXIMATION OF FUNCTIONS BELONGING TO THE CLASS L p ω) β 3 nd let the A-trnsformtions of S k f) nd Sk f) be given by nd T n,a f x) := T n,a f x) := n,k S k f x) n =,,,...) n,k Sk f x) n =,,,...), respectively. Let for k =,,.., k A n,k = n,i nd A n,k = i= i=n k+ n,i An, = A n, = ). As mesure of pproximtion by the bove untities we use the generlized moduli of continuity of f in the spce L p defined for β by the formuls ω β f δ) L p := sup sin t βp p ψ x t) p dx, where t δ ω β f δ) L p := sup It is cler tht for β > α t δ sin t βp ϕ x t) p dx ϕ x t) := f x + t) + f x t) f x). ω β f δ) L p ω α f δ) L p nd ω β f δ) L p ω α f δ) L p, nd it is esily seen tht ω f ) L p = ωf ) L p, ω f ) L p = ωf ) L p re the clssicl moduli of continuity. The devition T n,a f f ws estimted in the norm of L p by S. Ll nd H. Nigm ]. Their result ws generlized by M. L. Mittl, B. E. Rhodes nd V. N. Mishr 3] in the following form: Let A = n,k ) be n infinite regulr tringulr mtrix with nonnegtive entries stisfying r ) k + ) n,n k n,n k = O n,k, r n. 6) k=n r p,

4 4 W LODZIMIERZ LENSKI AND BOGDAN SZAL Then the degree of pproximtion of function f, conjugte to -periodic function f belonging to the clss W L p,ω ) = f Lp : f x + t) f x)]sin β x p p dx = O ω t)), where p > nd β, is given by T n,a f f )) = O ) β+/p ω provided tht ω stisfies { /) ) } t ψx t) p /p sin βp t = O ω t) nd { /) t γ ) } p /p ψ x t) = O ) γ ) ω t) 7) ) ) 8) uniformly in x, nd ω t)/t is nonincresing in t, in which γ is n rbitrry positive number with γ) >, where p + =, p. The ssumptions of this theorem re not sufficient for the estimtion 7). More precisely, condition 8) leds to the divergent integrl of type /n t +β) ) /. Moreover, condition 6) gives the following estimte { /n t γ β A n, ) } / = O n β γ /) which is incorrect for e.g. β =. Tking β = one hs the bove-mentioned erlier result ]: If A = n,k ) is n infinite regulr tringulr mtrix such tht the elements n,k re nonnegtive nd nondecresing with k, then the degree of pproximtion of function f, conjugte to -periodic function f belonging to Lip ω,p), is given by T n,a f f = O )) ) /p ω provided tht ω stisfies { /n ) } t ψx t) p /p = O ) ) ω t) 9)

5 APPROXIMATION OF FUNCTIONS BELONGING TO THE CLASS L p ω) β 5 nd { /n t γ ) } p /p ψ x t) = O ) γ ) ω t) uniformly in x, where γ is n rbitrry number such tht γ) >, where p + =, p. In ], there re similr mistkes in the proof s in ]. As we noted, the ssumption 8) is, in generl, not proper. In the results formulted below we give, insted of 8), nother condition ) which gurntees the estimte 9). The estimtes of the devition T n,a f f were lso obtined by K. Qureshi 4, 5] in cse β = nd for monotonic seuences n,k ) n. In this note we shll consider the sme devition nd dditionlly the devitions T n,a f ) f, ) nd T n,a f f. In our theorems we formulte the generl nd precise conditions for the functions nd moduli of continuity. Finlly, we lso give some results on the norm pproximtion. We shll write I I if there exists positive constnt K, sometimes depending on some prmeters, such tht I KI.. Sttement of the results Let us consider function ω of modulus of continuity type on the intervl, ], i.e. nondecresing continuous function hving the following properties: ω ) =, ω δ + δ ) ω δ ) + ω δ ) for ny δ δ δ + δ. It is esy to conclude tht the function δ ω δ) is nondecresing function of δ. Let L p ω) β = {f L p : ω β f δ) L p ω δ)}, L p ω) β = {f L p : ω β f δ) L p ω δ)}, where ω nd ω re lso the functions of modulus of continuity type. It is cler tht for β > α L p ω) α L p ω) β nd L p ω) α L p ω) β. We cn now formulte our min results using the following nottion: { n, when n = n,k ) n RBV S, n,n when n,k ) n HBV S. At the beginning, we formulte the results on the degrees of pointwise summbility of conjugte series.

6 6 W LODZIMIERZ LENSKI AND BOGDAN SZAL Theorem. Let f L p ω) β with β < p, n,k) n HBV S or n,k ) n RBV S) nd let ω be such tht { /) ) } t ψx t) p /p sin βp t ω t) = O x ) ) ) nd { /) t γ ψ x t) ω t) hold with < γ < β + p. Then T n,a f x) f x, for considered x. ) } p /p sin βp t = O x ) γ ) ) ) = O x ) β+ p n ) ω )) Theorem. Let f L p ω) β with β < p, n,k) n HBV S or n,k ) n RBV S) nd let ω stisfy ) with < γ < β + p, { /) ) } ψx t) p /p sin βp t ω t) = O x ) /p) ) nd { /) ω t) t sin β t ) } / = O ) β+/p ω where = p p ). Then T n,a f x) f x) = O x ) β+ p n ) ω for considered x such tht f x) exists. )), 3) )) Now we present the pproximtion properties of the opertor T n,a f. Theorem 3. Let f L p ω) β with β < p, n,k) n n,k ) n RBV S) nd let ω stisfy HBV S or { /) with < γ < β + p, { /) t γ ϕ x t) ω t) ) p sin βp t } /p = O x ) γ ) 4) ) } ϕx t) p /p sin βp t ω t) = O x ) /p) 5)

7 APPROXIMATION OF FUNCTIONS BELONGING TO THE CLASS L p ω) β 7 nd { /) ω t) t sin β t ) } / = O ) β+/p ω )), 6) where = p p ). Then Tn,A f x) f x) )) = Ox ) β+ p n ) ω, for considered x. Finlly, we formulte some remrks. Remrk. Considering the L p norms of the devitions from our theorems insted of the pointwise one we cn obtin the sme estimtions without ny dditionl ssumptions like ), ), ) nd 4), 5). Remrk. Under the dditionl ssumptions n,n = O n), β = nd ω t) = O t α ) or ω t) = O t α ) < α ), the degrees of pproximtion in Theorems, or 3, respectively, re O n α) p. We obtin in Theorems, or 3 the sme degrees of pproximtion under the ssumption n, = O n). Remrk 3. Due to the bove remrks, in the specil cse when our seuences n,k ) n re monotonic with respect to k we hve the corrected form of the result of S. Ll nd H. K. Nigm ]. 3. Auxiliry results We begin this section by some nottion following A. Zygmund 7, Section 5 of Chpter II]. It is cler tht nd where S k f x) = S k f x) = T n,a f x) = T n,a f x) = D k t) = f x + t) D k t), f x + t)d k t) f x + t) f x + t) k sin νt = cos t ν= n,k Dk t), n,k D k t), k+)t cos sin t,

8 8 W LODZIMIERZ LENSKI AND BOGDAN SZAL Hence nd where nd T n,a f x) f D k t) = k + cos νt = ) x, ν= T n,a f x) f x) = D k T n,a f x) f x) = = + /) /) ψ x t) k+)t sin sin t. ψ x t) ψ x t) n,k Dk t) n,k D k t) n,k D k t), k+)t cos t) = sin t, ϕ x t) n,k D k t). Now, we formulte some estimtes for the conjugte Dirichlet kernels. Lemm see 7]). If < t /, then D k t) t nd Dk t) t, nd for ny rel number t we hve D k t) k k + ) t nd Dk t) k +. More complicted estimtes we give with proofs. Lemm. If n,k ) n HBV S nd n t, then n,k D k t) = O t ) A n,τ = O t ) n,n, nd if n,k ) n RBV S for < t, then n,k D k t) = O t ) A n,τ = O t ) n,, where τ = mx, t ]).

9 APPROXIMATION OF FUNCTIONS BELONGING TO THE CLASS L p ω) β 9 Proof. Let us consider the sum n,k cos k=m k + ) t k=m+ sin t m + ) t = n,m cos sin t n + sin t k ν + ) t cos + sin t = n,m cos + n k=m+ + n,n sin ν=m+ ν=m+ cos m + ) t ν + ) t n,n sin t n,k n,k+ )sin n m ) t cos k m ) t n,k n,k+ ) n + m + ) t. Hence, for n τ = ] t, k + ) t n,k cos sin t sin t A n,τ + n,τ + or n,k cos k + ) t Since n,k ) n RBV S we hve n,m n,k n,k+ k=m cos k + m + ) t n n,k n,k+ + n,n k=τ sin t sin t n τ A n,n τ+ n, + n,k n,k+ + n,n τ. n,k n,k+ n,r n m r ) k=r nd therefore k + ) t n,k cos sin t sin t A n,τ + n,τ sin t τ A n,τ + t n,τ sin t A n,τ + t τ n,k ta n,τ n,.

10 W LODZIMIERZ LENSKI AND BOGDAN SZAL Anlogously, the reltion n,k ) n HBV S implies nd n,m n,r r k=m r n,k n,k+ n,k n,k+ n,r n r m ) n,m n,r n r m ), whence n,k cos k + ) t sin t sin t A n,n τ + n,n τ ta n,n τ + t n,n τ ta n,n τ + t k=n τ ta n,n τ n,n. k=n τ n,k Next, we present some known estimtes for the Dirichlet kernel. Lemm 3 see 7]). If < t /, then D k t) t nd for ny rel number t we hve D k t) k +. We hve lemm similr to Lemm. Lemm 4 cf., 6]). If n,k ) n HBV S nd n t, then n,k D k t) = O t ) A n,τ = O t ) n,n, nd if n,k ) n RBV S for < t, then n,k D k t) = O t ) A n,τ = O t ) n,, where τ = mx, t ]).

11 APPROXIMATION OF FUNCTIONS BELONGING TO THE CLASS L p ω) β Proof. Similrly s bove, for n τ = ] t, k + ) t n,k sin sin t n A n,τ + n,τ + n,k n,k+ + n,n or n,k sin k + ) t k=τ sin t n τ A n,n τ + n, + n,k n,k+ + n,n τ. Since n,k ) n RBV S we hve n,m n,k n,k+ n,k n,k+ n,r n m r ) k=m nd therefore n,k sin k + ) t k=r sin t ta n,τ + n,τ ta n,τ n,. Anlogously, the reltion n,k ) n HBV S implies nd whence n,k sin n,m n,r k + ) t r k=m r n,k n,k+ n,k n,k+ n,r n r m ) n,m n,r n r m ), sin t ta n,n τ + n,n τ ta n,n τ n,n. 4. Proofs of the results 4.. Proof of Theorem. We strt with the obvious reltions ) T n,a f x) f x, = /) ψ x t) n,k Dk t) + ψ x t) n,k D k t) nd Tn,Af x) f x, =: Ĩ + Ĩ /) ) Ĩ + Ĩ.

12 W LODZIMIERZ LENSKI AND BOGDAN SZAL By Hölder s ineulity Ĩ p + = ), Lemm nd ) /) ) t ψ x t) ) { t ψx t) sin β t p ω t) ] { /) ] ω ) t) } for β < p. t β } p { ω t) sin β t ) ) β+ p ω By Hölder s ineulity p + ), = Lemm nd ) Ĩ ψ x t) n,k D k t) n /) n { t γ ψ x t) sin β t p ω t) ] ω t) } p { { n ) γ t γ sin β t ) { n ) γ+ ω ) β+ p n ) ω /) ] } ) ψ x t) t ω t) t γ sin β t } t γ β ] for < γ < β + p. Collecting these estimtes we obtin the desired result. 4.. Proof of Theorem. We strt with the obvious reltions T n,a f x) f x) = /) ψ x t) n,k D k t) + =: Ĩ + Ĩ /) ψ x t) n,k D k t) nd Tn,A f x) f x) Ĩ + Ĩ. ] } ] }

13 APPROXIMATION OF FUNCTIONS BELONGING TO THE CLASS L p ω) β 3 By Hölder s ineulity p + = ), Lemm, ) nd 3) Ĩ /) { ψ x t) t ψx t) ω t) sin β t ] p } p { ω t) t sin β t ] } ) { ) p ω ] ω t) } t +β By the previous proof Ĩ ) β+ p n ) ω ) ) β ω ) for < γ < β + p. Collecting these estimtes we obtin the desired result Proof of Theorem 3. Let T n,a f x) f x) = + /) =: I + I, /) ϕ x t) ϕ x t) n,k D k t) n,k D k t) then T n,a f x) f x) I + I. By Hölder s ineulity I /) { p + = ), Lemm 3, 5) nd 6), ϕ x t) t ϕx t) ω t) ) { ) p ω sin β t ] p } p { ] ω t) } t +β ω t) t sin β t ] } ) ) β ω

14 4 W LODZIMIERZ LENSKI AND BOGDAN SZAL By Hölder s ineulity I n { p + = ), Lemm 4 nd 4) t γ ϕ x t) sin β t p ω t) ] n ) γ+ ω ) β+ p n ) ω ) { ) } p { } t γ β ] for < γ < β + p. Collecting these estimtes we obtin the desired result. ω t) t γ sin β t ] } Acknowledgement. Authors re grteful to the referee for his vluble suggestions for the improvement of editing of this pper. References ] S. Ll nd H. K. Nigm, Degree of pproximtion of conjugte of function belonging to Lipξ t),p) clss by mtrix summbility mens of conjugte Fourier series, Int. J. Mth. Mth. Sci. 7 ), ] L. Leindler, On the degree of pproximtion of continuous functions, Act Mth. Hungr. 4 4), ] M. L. Mittl, B. E. Rhodes nd V. N. Mishr, Approximtion of signls functions) belonging to the weighted W L p, ξ t))-clss by liner opertors, Int. J. Mth. Mth. Sci. 6 6), Article ID 53538, pp. 4] K. Qureshi, On the degree of pproximtion of functions belonging to the Lipschitz clss by mens of conjugte series, Indin J. Pure Appl. Mth. 98), 3. 5] K. Qureshi, On the degree of pproximtion of functions belonging to the clss Lip α, p) by mens of conjugte series, Indin J. Pure Appl. Mth. 3 98), ] B. Szl, On the strong pproximtion by mtrix mens in the generlized Hlder metric. Rend. Circ. Mt. Plermo ) 56 7), ] A. Zygmund, Trigonometric Series, Cmbridge,. University of Zielon Gr, Fculty of Mthemtics, Computer Science nd Econometrics, ul. Szfrn 4, Zielon Gr, Polnd E-mil ddress: W.Lenski@wmie.uz.zgor.pl E-mil ddress: