ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER

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1 ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER S. TIKHONOV Abstract. In this aer we consider the roerties of moduli of smoothness of fractional order. The main result of the aer describes the equivalence of the modulus of smoothness and a function from some class.. Introduction In 977 P.L. Butzer, H. Dyckhoff, E. Goerlich, R.L. Stens see [] and R.Tabersky see [4] introduced the modulus of smoothness of fractional order. This notion could be considered as a direct generalization of the classical modulus of smoothness, and it is more natural to use it for a number of roblems of harmonic analysis see, for examle, [], [5], [7], [0]. The imortant roblem of aroximation theory and theory of Fourier series is the roblem of descrition of moduli of smoothness see [], [4], [8], []. One can consider this roblem from the viewoint of descrition of majorant of smoothness moduli. Recently, A. Medvedev see [6] has roved that for any modulus of continuity on [0, there exists a concave majorant that is infinitely differentiable. In this aer, we obtain the descrition of the modulus of smoothness of fractional order from the viewoint of the order of decreasing to zero of the modulus of smoothness. Let us introduce several definitions. If <, let L be the sace of -eriodic, measurable functions fx such that f = fx dx <. 0 Similarly, let L be the sace of -eriodic, continuous functions fx with f = fx. We will define the difference of fractional order max x [0,] > 0 of function fx at the oint x x R with increment h h R 0 I would like to thank the Centre de Recerca Matemàtica for their suort.

2 S. TIKHONOV by h fx = ν fx + νh, ν where ν = ν+ ν! for ν >, ν = for ν =, ν = for ν = 0. The modulus of smoothness of order > 0 of function f L,, is given by ω f, t = su h f see [],[4]. h t Let Φ γ γ R be the set of nonnegative, bounded functions δ on 0, such that a: δ 0 as δ 0, b: δ is nondecreasing, c: δδ γ is nonincreasing. If for f L there exists g L such that lim h h f g = 0 inf g W h 0+ then g is called the Liouville-Grunwald-Letnikov derivative of order > 0 of a function f in the L -norm, denoted by g = D f see [], []. Set W := { } f L : D f exists as element in L. The K-functional is given by Kf, t, L, W := f g + t D g.. Results Let fx L, [, ] and > 0. It is clear that see [] = ν + ν ν! C ν +, ν N imlies C := < and the fractional difference fx is defined almost everywhere and belongs to L : ν h f C f. It is easy to write the following reresentation for C see [4]: k C ν, if k < k + k = 0,,,, = k ν+, if k + < k + k = 0,,,. The fractional differences and moduli of smoothness have some useful roerties and we shall establish some of them in the following lemmas. Lemma.. [], [4] Let f L, [, ], α, > 0; h R. Then a: α h hfx = α+ h fx for almost every x; h

3 ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER 3 b: c: α+ h f C α h f ; lim h 0+ α h f = 0. Lemma.. Let f, f, f L, [, ], α, > 0; x, h R. Then a: ω f, δ is nondecreasing nonnegative function of δ on 0, with lim ω f, δ = 0; δ 0+ b: ω f + f, δ ω f, δ + ω f, δ ; c: ω α+ f, δ C αω f, δ ; d: if λ, then ω f, λδ Cλ ω f, δ ; e: if 0 < t δ, then ω f, δ δ Cω f, t t. Indeed, we immediately have a c from Lemma., d was roved in [], and d imlies e: ω f, δ = ω f, δ δ t t C ω f, t. t Lemma.3. Let f L, [, ], > 0. a: If N, then f [ + ] f. b: If / N, then f [ + ]+ f. Corollary.4. For a function t = t α 0 t to be a modulus of smoothness of order > ] 0 of a function f L, it is necessary to have α +. [ + Theorem.5. Let [, ], > 0. A: If f L, then there exists a function Φ such that t ω f, t Ct 0 < t <, where C is a ositive constant deending only on. B: If Φ, then there exist a function f L and a constant t > 0 such that C ω f, t t C ω f, t 0 < t < t, where C, C are ositive constants deending only on. Corollary.6. Let [, ], > 0. A: If f L, then there exists a function Φ such that is true. C t Kf, t, L, W C t. 3

4 4 S. TIKHONOV B: If Φ, then there exists a function f L such that 3 is true. Remark.7.. We can relace the condition f L by condition f L in the art B of Theorem.5.. Note that theorem.5 for N was roved in []. Also, for H -saces the analogue of Corollary.6 for R + and the analogue of theorem.5 for N were roved in [5]. 3. Proofs Proof of Lemma.3. The first inequality was roved in [3]. Let >, / N. We shall use the following reresentation see [4] h fx h = hfx h νh for almost every x 4 ν By Lemma.a and art a of this Lemma, it follows that f = [] [] f [ []+ ] [] [] f. Here we use 4 for h =. We have f { [ []+ ] [] [] ν = [ []+ ] [] [] ν [] [] Thus, by Lemma.a and inequality, we get f C [] [ []+ ] } f ν f. f. If we combine this result with C [] = see and [ []+ ] = [ + ], we obtain the required inequality. If 0 < <, then we use and 4. This comletes the roof of Lemma. We will need the following lemma. Lemma 3.. Let > 0, n N, δ > 0. a: If fx = sin x and [, ], then there exist t > 0 and C, C > 0 such that for any δ 0, t we have C δ ω f, δ C δ. 5

5 ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER 5 b: If fx = sin nx and [, ], then for any δ 0, ] we have δ f nδ. c: If fx = sin nx, then /n f = +. d: If fx = sin nx, then for any δ ] 0, n we have δ f 4 δn. Proof of Lemma 3. Let T n x = n δ T nx δ = n ν= n ν= n Thus, for fx = sin nx, n N, we get δ δ fx = sin nδ c ν e iνx, then i siδ c ν e iνx. sin nx +. 6 For n = we obviously have C sin δ δ sin C sin δ. If we combine this inequality with sin t t t 0 and sin t t 0 t, then we obtain 5. In the same way, by 6, we shall have the roofs of b d. This comletes the roof of Lemma. Proof of Theorem.5. A. Let us define t := t inf 0<ξt {ξ ω f, ξ }. We immediately have t Φ from [3, ]. It is trivial, that t ω f, t. By Lemma.e, we have ω f, t Ct: ω f, t = t t ω f, t Ct { inf ξ } ω f, ξ 0<ξt = Ct. Therefore, for any t > 0 the following inequality t ω f, t Ct holds and A follows. t B. case. Let lim = C 0 C <. Then, by virtue of t 0 t monotonicity of t, we write t t Ct for 0 < t ; there exists t > 0 such that t Ct for 0 < t t. t Indeed, is trivial like for C = 0. If C > 0 and lim = C, then t 0 t for any ε > 0 there exists t > 0 such that C t ε for 0 < t t t. Then t C ε, and choosing small ε we have. t Here = 0.

6 6 S. TIKHONOV Define fx = C sin x. By Lemma 3.a, we have ω f, δ CC δ C δ for 0 < δ, ω f, δ CC 3 δ C 4 δ for 0 < δ t, comleting the roof in this case. t t case. Let lim = +. Then lim t = 0 and lim t 0 t t 0 t 0 t = 0. We fix a. Then, following Oskolkov [9], we define the sequence { } ν=, where = m ν are the numbers m ν such that m =, m ν+ = min { m m N : max m, mν mν ν m m } a ν N. From the definition of { } ν= it follows that m ν+ > m ν, + and for any ν N we have a n ; 7 ν+ nν n ν a n ν Let us fix κ = d d N such that κ >. Note that 7 imlies n a ν <, and, therefore, we can define the function fx = ν= ν= sinκ x. ν= First, we shall estimate ω f, δ from above. By the inequality f f f, [,, it is enough to rove ω f, δ Cδ. Let δ 0, n ]. For all h 0, n ] we can find the number N N such that n N+ < h n N. Then h fx N n ν= ν + =: I + I. ν=n+ h sinκx + h n sinκx ν

7 ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER 7 Combining Lemma 3.b, inequality 8, and condition c in the definition of Φ, we get I N h n sinκn νx ν= ν C κh N n N n N C n N h n N C h. ν= a N ν Inequalities and 7 yield that I h sinκn νx ν=n+ C ν=n+ C n N+ C n N+ C h. ν=n+ a N+ ν Therefore, if h n N+, n N ], N N, then h fx C h, which imlies ω f, δ Cδ. Now we shall obtain the inequality δ Cω f, δ. From the inequality f f, [, ] it is sufficient to rove δ Cω f, δ. Also, we note that if the last inequality holds for δ =, k = k N, N +, N +,, where N N, thet holds for δ, k. Indeed, from the monotonicity of t t, we see that the estimate δ k+ C is true. By Lemma.a, we get k k δ C To go further, we suose that δ = k. Cω f, k Cω f, δ.

8 8 S. TIKHONOV Let M be the integer, M >, and, let h = κ. We shall show that h fx a For this urose, we shall use the following reresentation of a function fx: fx = M ν= + ν=m+ =: f + f + f 3. sinκ x + sinκ x sinκ x + Note, that sinκ x + = sinκ x for ν > M, and f 3 x has the eriod T = h = κ. We therefore obtain h f 3 x = fx + h By Lemma 3.b and 8, we have ξ = 0. ξ ξ=0 h f x M ν= M ν= = h sinκ x κ h M ν= n ν M n M a M ν. ν= Using M a M ν and 8, we obtain h f x 4+ a ν= By Lemma 3.c, h f x = n M h sinκ x = +..

9 ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER 9 Therefore, for h = κ, the inequality f f f f 3 imlies h fx h f x h f x h f 3 x = h f x h f x 4 +, a i.e. we obtain 9. Further, we choose the integer i such that = + m < δ i+ m =. i Note, that, by definition of m i, at the least one of the following inequalities is true: mi+ mi+ < a m i m, 0 i mi+ > a mi Case a. Let 0 be true. Using the monotonicity of t and 0, we get n i+ mi+ + mi+ < a n i. We write fx = i ν= + ν=i+ =: f + f + f 3. sinκ x + sinκ x sinκ x + It is clear, that the function f 3 has a eriod T = κ+. Then, for κ = d > we have δ = > r + > T, therefore, f 3 has a eriod δ and δ f 3x = 0.

10 0 S. TIKHONOV For 0 < δ κ, by Lemma 3.d, we have δ f x = δ sinκn ix 4 κ δ. Using Lemma 3.b and inequality 8, we estimate f : i h f x δ sinκn νx ν= ν= i κ δ 4 κ δ 4 κ δ a. For + < δ κ we obtain δ fx δ f x δ f x { } κ δ 4 4. a Now we choose a such that + a = γ > 0 then 4 4 a = γ > 0. From and the condition c in the definition of Φ, we have δ δni+ a a δ. + Thus, the inequality ω f, δ Cδ holds for + < δ κ < δ, then 9 imlies ω f, δ ω f, κ C C δ. κ. If

11 ON MODULI OF SMOOTHNESS OF FRACTIONAL ORDER The theorem has been roved in case a. Case b. Let be true. By virtue of monotonicity of write m i+ m i+. Hence, = + m i+ > a = a mi+ mi It follows from 9 and 3 that ω f, δ ω f, + ω f, κ+ C + C C δ. t, we t. 3 This comletes the roof of case b and Theorem.5. Proof of Corollary.6 follows from the following estimates see []: C ω f, t Kf, t, L, W C ω f, t. References [] O. V. Besov; S. B. Stečkin A descrition of the moduli of continuity in L // Theory of functions and its alications collection of articles dedicated to S. M. Nikol skii on the occasion of his seventieth birthday. Trudy Mat. Inst. Steklov , 3 5, 407. [] P.L. Butzer, H. Dyckhoff, E. Goerlich, R.L. Stens Best trigonometric aroximation, fractional order derivatives and Lischitz classes // Can. J. Math., 9, 977,

12 S. TIKHONOV [3] V. I. Ivanov; S. A. Pichugov Aroximation of eriodic functions in L by linear ositive methods, and multile moduli of continuity. Mat. Zametki 4 987, no. 6, , 909. [4] V.I. Koljada Imbedding in the classes L // Izv. Akad. Nauk SSSR Ser. Mat , no., , 47. [5] Y. Kryakin, W. Trebels q-moduli of continuity in H D, > 0, and anequality of Hardy and Littlewood // J. Arox. Theory 5 00, no., [6] A.V. Medvedev On a concave differentiable majorant of a modulus of continuity // Real Anal. Exchange 7 00/0, no., 3 9. [7] C. J. Neugebauer The L modulus of continuity and Fourier series of Lischitz functions // Proc. Amer. Math. Soc , no., [8] S. M. Nikol skii The Fourier series with given modulus of continuity // Doklady Acad. Sci. URSS N.S. 5, [9] K.I. Oskolkov Estimation of the rate of aroximation of a continuous function and its conjugate by Fourier sums on a set of full measure // Izv. Akad. Nauk SSSR Ser. Mat , [0] S. G. Pribegin On a method for aroximation H, 0 < // Mat. Sb. 9 00, no., 3 36; translation Sb. Math. 9 00, no. -, [] T.V. Radoslavova Decrease orders of the L -moduli of continuity 0 < // Analysis Mathematica, V.5, 979, [] S. Samko; A. Kilbas; O. Marichev Fractional integrals and derivatives. Theory and alications. Gordon and Breach Science Publishers, Yverdon, 993. [3] S.B. Stechkin On absolute convergence of Fourier series II // Izv. Akad. Nauk SSSR, Ser. Mat. 9, No. 4, [4] R. Taberski Differences, moduli and derivatives of fractional orders. // Commentat. Math. V. 9, , Sergey Tikhonov Centre de Recerca Matemàtica CRM Bellaterra Barcelona E-0893, Sain stikhonov@crm.es