F A S C I C U L I M A T H E M A T I C I Nr 55 5 DOI:.55/fascmath-5-7 W. Lenski and B. Szal ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF CONJUGATE FOURIER SERIES Abstract. The results corresponding to some theorems of S. Lal [Tamkang J. Math., 34, 79-88] and the results of the authors [Banach Center Publ. 9, 37-47] are shown. The same degrees of pointwise approximation as in mentioned papers by significantly weaker assumptions on considered functions are obtained. From presented pointwise results the estimation on norm approximation with essentialy better degrees are derived. Some special cases as corollaries for iteration of the Nörlund or the Riesz method with the Euler one are also formulated. Key words: conjugate function, Fourier series, degree of approximation. AMS Mathematics Subject Classification: 4A4.. Introduction Let L p p < [respectively L ] be the class of all periodic real valued functions integrable in the Lebesgue sense with p th power [essentially bounded] over Q [, ] with the norm /p ft p, when p <, Q f : f L p ess sup ft, when p t Q and consider the conjugate trigonometric Fourier series Sfx : a ν f sin νx b ν f cos νx ν with the partial sums S k f. We know that if f L then f x : ψ x t cot t lim f x, ɛ, ɛ +
9 W. Lenski and B. Szal where with f x, ɛ : ɛ ψ x t cot t ψ x t : f x + t f x t, exists for almost all x [, Th.3.IV]. Let A : a n,k and B : b n,k be infinite lower triangular matrices of real numbers such that a n,k and b n,k, when k,,,...n, a n,k and b n,k, when k > n, a n,k and and let, for m,,,.., n, b n,k, where n,,,..., A n,m m a n,k and A n,m Let the AB transformation of S k f be given by T n,a,b f x : r a n,k. km a n,r b r,k Sk f x n,,,.... We define two classes of sequences see [3]. A sequence c : c k of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly c RBV S, if it has the property c k c k+ K c c m, km for all positive integer m, where K c is a constant depending only on c. A sequence c : c k of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly c HBV S, if it has the property m c k c k+ K c c m, for all positive integer m, or only for all m n if the sequence c has only finite nonzero terms and the last nonzero term is c n. Now, we define the another classes of sequences.
On pointwise approximation of... 93 Following by L. Leindler [4], a sequence c : c r of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly c MRBV S, if it has the property c r c r+ K c m + rm m r m/ for all positive integer m. Analogously as in [6], a sequence c : c r of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly c MHBV S, if it has the property n m r c r c r+ K c m + rn m for all positive integers m < n, where the sequence c has only finite nonzero terms and the last nonzero term is c n. Consequently, we assume that the sequence K α n n is bounded, that is, that there exists a constant K such that K α n K holds for all n, where K α n denote the constants appearing in the before inequalities for the sequences α n a n,r n r, n,,,.... Now we can give the conditions to be used later on. We assume that for all n and m < n and n km n m r a n,r a n,r+ K m + a n,r a n,r+ K m + m r m/ rn m hold if a n,r n r belongs to MRBV S and MHBV S, for n,,..., respectively. As a measure of approximation of f by Tn,A,B f we use the pointwise moduli of continuity of f in the space L p defined by the formulas c r, c r, a n,r a n,r { δ w xf p δ ψ x u sin u } p p du, when p <, δ ess sup ψ x u sin u, when p, <u δ
94 W. Lenski and B. Szal { w p sup t <t δ t ψ x u sin u } p p du when p <, xf δ ess sup ψ x u sin u when p, and the classical ones <u δ ω f δ L p sup sin t <t δ ψ t L p. The deviation T n,a f f T n,a,b f f, with b r,r and otherwise, was estimated at the point as well as in the norm of L p by K. Qureshi [7] and S. Lal, H. Nigam []. These results were generalized by K. Qureshi [8]. The next generalization was obtained by S. Lal []. In the case and a n,r n + when r,,,...n and a n,r when r > n r k γ k b r,k + γ r when k,,,...r and b n,r when k > r with γ >, the deviation T n,a,b f f, was estimated by S. Sonker and U. Singh [9] as follows: Theorem. Let fx be a -periodic, Lebesgue integrable function and belongs to the Lipα, r-class with r and αr. Then the degree of approximation of fx, the conjugate of fx by C, E, γ means of its conjugate Fourier series is given by n + provided and r + γ r r γ k Sk f k f L r { /n+ } ψx t r /r O n + { /n+ t α t δ p } /p ψ x t O t α O x n + α+/r n + δ, where δ is an arbitrary positive number with α + δs + <, and r + s, r >. In this paper we shall consider the deviations T n,a,b f f and T n,a,b f f, in general form. In the theorems we formulate the n+
On pointwise approximation of... 95 general conditions for the functions and the modulus of continuity obtaining the same and sometimes essentially better degrees of approximation than the above one. Finally, we also give some results on norm approximation with essentially better degrees of approximation. The obtained results generalize the results from [] and [6]. We shall write I I if there exists a positive constant K, sometimes depending on some parameters, such that I KI.. Statement of the results Let L p w x { } f L p : w xf p δ w x δ, where w x is positive, with w x, and almost nondecreasing continuous function. We can now formulate our main results. At the beginning, we formulate the results on the degrees of pointwise summability of conjugate series. Theorem. Let f L and, ]. If a n,k n MRBV S and ν k + t b r,k cos τ for µ ν, then T n,a,b f x f x, rµ for almost all considered x. O x n + n + a n,k w xf, k + Theorem. Let f L and, ]. If a n,k n MHBV S and the entries of matrix B satisfy the condition for µ ν, then T n,a,b f x f x, for almost all considered x. O x n + n + a n,n k w xf, k + Theorem 3. Let f L p w x with < p <, and let w x satisfy n+ p/p w x t t sin t p /p O n + +/p w x n +
96 W. Lenski and B. Szal and 3 { n+ } ψx t p /p sin p t w x t O x n + p with some. If a n,k n MRBV S and the entries of matrix B satisfy the condition for µ ν, then 4 T n,a,b f x f x O x n + a n,k w x, k + for almost all considered x such that f x exists. Theorem 4. Let f L p w x, with < p <, and and let w x satisfy and 3 with some. If a n,k n MHBV S and the entries of matrix B satisfy the condition for µ ν, then T n,a,b f x f x O x n + a n,n k w x, k + for almost all considered x such that f x exists. Next, we formulate the results on estimates of L p norm of the deviation f, T f n,a,b n+. Theorem 5. Let f L p with < p <. If a n,k n MRBV S and the entries of matrix B satisfy the condition for µ ν, then T n,a,b f f, O n + n + a n,k ω f L k + p L p where <. Theorem 6. Let f L p with < p <. If a n,k n MHBV S and the entries of matrix B satisfy the condition for µ ν, then T n,a,b f f, O n + n + a n,n k ω f L k + p L p where <. In case of the deviation T n,a,b f f let L p ω {f L p : ω f δ L p ω δ}, where ω is positive, with ω, and almost nondecreasing continuous function.
On pointwise approximation of... 97 Theorem 7. Let f L p ω with < p < and < < p, where ω instead of w x satisfy and 3 with some. If a n,k n MRBV S and the entries of matrix B satisfy the condition for µ ν, then T n,a,b f f O n + a L p n,k ω. k + Theorem 8. Let f L p ω with < p < and < < p, where ω instead of w x satisfy and 3 with some. If a n,k n MHBV S and the entries of matrix B satisfy the condition for µ ν, then T n,a,b f f L p O n + a n,n k ω. k + Remark. Under the above remarks we can observe that in the special case, when our sequences a nk are monotonic with respect to k we also have the corrected form of the result of S. Lal and H. K. Nigam []. Finally, we give applications of our results as corollary and some remarks Taking a n,r p n r / n p ν when r,,,..., n and a n,r when ν r > n with p ν >, p ν p ν+ and b r,k r kγ k +γ r when k,,,..., r and b n,r when k > r with γ >, Theorem and Theorem imply: and Corollary. If f L and <, then p r + γ r p ν r ν O x n + p k w xf p ν r ν p ν ν r γ k Sk f x k f x, k + n +, p n r r + γ r γ k Sk f x k f x, n + O x n + p n k w xf k +, p ν ν
98 W. Lenski and B. Szal for almost all considered x, where p ν >, p ν p ν+ and γ >. In special case p r p n r we have estimate n + + γ r O x n + r r γ k Sk f x k f x, [ w xf k + n + ] Analogical corollary we can derive from Theorem 3 and Theorem 4. Remark. If we take in the above considerations then we have to estimate the quantities Ĩ, Ĩ and Ĩ, Ĩ L p using the Hölder L p inequality analogously as in estimate of Ĩ. Thus we obtain in the all above estimates n + /p instead of n +. Remark 3. Analyzing the proofs of Theorem - 4 we can deduce that taking the assumption a n,k n RBV S or a n,k n HBV S instead of a n,k n MRBV S or a n,k n MHBV S, respectively, we obtain the results from the paper [5].. 3. Auxiliary results We begin this section by some notations following A. Zygmund [, Section 5 of Chapter II]. It is clear that and where Hence T n,a,b f x f S k f x T n,a,b f x D k t f x + t f x + t D k t, k sin νt cos t ν x, n + + r n+ n+ a n,r b r,k Dk t, k+t cos sin t. ψ x t ψ x t r r a n,r b r,k Dk t a n,r b r,k D k t
and where On pointwise approximation of... 99 T n,a,b f x f x ψ x t r D k t cot t k sin νt ν a n,r b r,k D k t, k+t cos sin t. Now, we formulate some estimates for the conjugate Dirichlet kernels. Lemma see []. If < t /, then D k t t and for any real t we have D k t k k + t and Dk t k +. Lemma. Let b r,k r be such that the condition holds for µ ν. If a n,k n MRBV S, then a n,r b r,k D k t τa n,τ r and if a n,k n MHBV S, then a n,r b r,k D k t τa n,n τ with τ [/t] and t Proof. Let r [ n+, ], for n,,,... K n t : r a n,r b r,k cos The relation a n,k n MRBV S implies k + t. a n,s a n,m a n,m a n,s kr s km a n,k a n,k+ n a n,k a n,k+ r + a n,k r m < s n k r/
W. Lenski and B. Szal whence a n,s a n,m + r + and thus, by our assumption we obtain K n t lτ+ + r + A n.τ + τ + A n,τ + τ + A n,τ + τ + A n,τ + τ + A n,τ + τ + A n,τ + 3 r a n,r n rτ+ k r/ l b l,k cos rτ+ + a n,n k τ/ k τ/ k τ/ k τ/ k τ/ b r,k a n,r a n,k r m < s n k + t b r,k cos a n,r a n,r+ lτ+ a n,l 4A n,τ, l l b l,k cos a n,k + τ + a n,k + τ + a n,k + τ + a n,k + τ + a n,k + τ + lτ+ τ, k + t k τ/ a n,n k + t l b l,k cos τ k + t a n,k + k a n,l τ k + l k/ a n,k + a n,l τ τ/ + l τ/4 a n,k + a n,l τ τ + l a n,k + a n,l τ τ + k τ/ k τ/ k τ/ k τ/ l
On pointwise approximation of... but the relation a n,k n MHBV S implies a n,m a n,s a n,m a n,s s km r a n,k a n,k+ a n,k a n,k+ n r + a n,k m < s r n kr whence a n,m a n,s + n r + a n,k m < s r n kr and thus, by our assumption l b l,k cos we obtain K n t rn τ [ A n.n τ + τ + A n.n τ + τ + A n.n τ + τ + A n.n τ + l n τ + r a n,r k + t b r,k cos n τ A n.n τ + a n,r a n,r+ r n τ + a n,n τ kn τ kn τ kn τ kn τ a n,k + l l b l,k cos a n,k + a n,n τ ] τ a n,k + n τ/ τ + a n,k + τ + kn τ τ + τ, k + t l k + t kn τ n τ/ kn τ νn τ l b l,k cos a n,n τ τ a n,k + τ + a n,ν A n.n τ. k + t νn τ/ a n,ν τ Now, our proof is complete.
W. Lenski and B. Szal 4.. Proof of Theorem 4. Proofs of the results We start with the obvious relations T n,a,b f x f x, n + + n+ Ĩ + Ĩ n+ ψ x t ψ x t r r a n,r b r,k Dk t a n,r b r,k D k t and T n,a f x f x, n + Ĩ + Ĩ. By Lemma, we have Ĩ n + n + n+ n+ n + w xf for [, ]. Using Lemma we obtain t ψ x t n + n+ ψ x t sin t n + a n,k n + n + k t ψ x t sin t sin t n + w xf n a n,k w xf k + n +, Ĩ n+ m ψ x t t m+ a n,k a n,k m m+ m ψ x t t m m+ ψ x t t a n,k m m k a n,k m a n,k k mk a n,k k mk m+ ψ x t t +a n, + m +a n, + m + m m m a n, + a n,m+ m+ ψ x t t m ψ x t a n,m+ t m+ { [ ] } m ψx t sin t a n,m+ m m+ t +
+ On pointwise approximation of... 3 a n,k k mk m { a n,k { m +a n, m m+ { a n,m+ m + m + k+ n+ + n + a n,m+ w xf m [ ψx t sin t m m+ t + ] [ ψ x t sin t } t d [ t ψ x u sin u ] du m + } ] } { [ a n,k t + + + k+ n+ t ψ x u sin u du ] k+ [ t + t + n + a n,m+ w xf {[ a n,k + + m k + + k+ k+ n+ [ t + w xf t + n + a n,m+ w xf m n+ n+ ψ x u sin u du ] m + ψ x u sin u du ] ] } m + ] a n,k {[k + w xf k + [ ] } k+ + + t + w xf t + n + a n,m+ w xf m { n+ [ a n,k t w xf t k+ + n + a n,m+ w xf m m + ] } m + }
4 W. Lenski and B. Szal Since w p xf δ is nondecreasing majorant of w p xf δ we have with > a n,k w n+ xf k + k+ t + n + a n,m+ w xf m + m n + a n,k w xf. k + Collecting these estimates we obtain the desired result. 4.. Proof of Theorem Let as usual T n,a,b f x f x, n + Ĩ + Ĩ and Tn,A,Bf x f x, n + Ĩ + Ĩ. The term I we can estimate by the same way as in the proof of Theorem. Therefore Ĩ n + w xf n + n + a n,n k w xf n + w xf k + Analogously to the above, by Lemma Ĩ n+ ψ x t t m+ m n + kn τ a n,k m ψ x t t m+ a n,n k w xf k + a n,n k m. m m+. n + ψ x t t a n,n k a n,n k Collecting these estimates we obtain the desired result.
On pointwise approximation of... 5 4.3. Proof of Theorem 3 We start with the obvious relations T n,a f x f x + n+ Ĩ + Ĩ n+ ψ x t ψ x t r a n,r b r,k D k t r a n,r b r,k D k t and Tn,Af x f x Ĩ + Ĩ. By the Hölder inequality p + q, Lemma, 3 and, for < p n+ ψ x t Ĩ t { [ n+ ψx t t ] p } { p [ ] q q n+ w x t } w x t sin t sin t n + w x n + w x a n,k n + n + n + a n,k w x. k + The term Ĩ we can estimate by the same way like in the proof of Theorem. So Ĩ n + a n,k w x k +. Collecting these estimates we obtain the desired result. 4.4. Proof of Theorem 4 Let as usual T n,a f x f x Ĩ + Ĩ and Tn,Af x f x Ĩ + Ĩ.
6 W. Lenski and B. Szal For the first term, by the proof of Theorem 3, we have Ĩ n + w x n + n + a n,n k w x k + where < p, and for the second one, by the proof of Theorem, we can write Ĩ n + a n,n k w x k +. Collecting these estimates we obtain the desired result., 4.5. Proofs of Theorems 5-8 The proofs are similar to these above and follows from the evident inequality w p L f δ ω f δ p L p immediately. 4.6. Proof of corollary We have to verify the assumptions of Lemma only. For the first one we note that any non decreasing sequence belongs to the class MRBV S and any non increasing sequence belongs to the class M HBV S. The second one follows from the following calculations ν + γ r r γ k cos k rµ ν Re + γ r rµ k + t r γ k exp k k + it ν Re + γ r exp it r γ k exp ikt k rµ ν Re + γ r exp it + γ exp itr rµ [ Re exp it ν ] + γ exp it r + γ rµ
On pointwise approximation of... 7 Re exp it + γ exp it + γ ν µ+ +γ exp it +γ +γ exp it +γ +γ exp it +γ µ +γ exp it +γ + γ + γ γ exp it + γ γ exp it + γ γ cos t + sin t + γ γ cos t +γ exp it +γ ν µ+ + γ γ + γ sin t γ sin t + γ γ t τ. Thus proof is complete. References [] Lal S., Nigam H.K., Degree of approximation of conjugate of a function belonging to Lip ξ t, p class by matrix summability means of conjugate Fourier series, Int. J. Math. Math. Sci., 79, 555-563. [] Lal S., On the degree of approximation of conjugate of a function belonging to weighted W L p, ξ t class by matrix summability means of conjugate series of a Fourier series, Tamkang J. Math., 34, 79-88. [3] Leindler L., On the degree of approximation of continuous functions, Acta Math. Hungar., 4-4, 5-3. [4] Leindler L., Integrability conditions pertaining to Orlicz space, J. Inequal. Pure and Appl. Math., 87, Art. 38, 6 pp. [5] Lenski W., Szal B., Approximation of functions belonging to the class L p ω by linear operators, Acta Comment. Univ. Tartu. Math., 39, -4. [6] Lenski W., Szal B., Approximation of functions from L p w by linear operators of conjugate Fourier series, Banach Center Publ., 9, 37-47. [7] Qureshi K., On the degree of approximation of functions belonging to the Lipschitz class by means of a conjugate series, Indian J. Pure Appl. Math., 998, -3. [8] Qureshi K., On the degree of approximation of functions belonging to the class Lip α, p by means of a conjugate series, Indian J. Pure Appl. Math., 3598, 56-563. [9] Sonker S., Singh U., Degree of approximation of the conjugate of signals functions belonging to Lip α, r - class by C, E, q means of conjugate trigonometric Fourier series, J. Inequal. Appl.,, :78.
8 W. Lenski and B. Szal [] Szal B., A note on the uniform convergence and boundedness a generalized class of sine series, Commentat. Math., 488, 85-94. [] Zygmund A., Trigonometric series, Cambridge,. W. Lenski University of Zielona Góra Faculty of Mathematics Computer Science and Econometrics 65-56 Zielona Góra, ul. Szafrana 4a, Poland e-mail: W.Lenski@wmie.uz.zgora.pl B. Szal University of Zielona Góra Faculty of Mathematics Computer Science and Econometrics 65-56 Zielona Góra, ul. Szafrana 4a, Poland e-mail: B.Szal @wmie.uz.zgora.pl Received on 6.3.4 and, in revised form, on.6.5.