INVERSE THEOREMS OF APPROXIMATION THEORY IN L p,α (R + )
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1 Electroic Joural of Mathematical Aalysis ad Applicatios, Vol. 3(2) July 2015, pp ISSN: (olie) INVERSE THEOREMS OF APPROXIMATION THEORY IN L p,α (R + ) R. DAHER, S. EL OUADIH Abstract. I this paper, We prove aalogues of some iverse theorems of Stechki,for the Bessel harmoic aalysis, usig the etire fuctios of expoetial type. 1. Itroductio Ad Prelimiaries Yet by the year 1912, S. Berstei obtaied the estimate iverse to Jakso s iequality i the space of cotiuous fuctios for some special cases [2], later S.B.Stechki [4], M.Tima [7], etc, proved such iverse estimates, icludig the case of the space L p, 1 < p <. I this paper, we obtai the estimate iverse to Jakso s iequality i the space L p,α (R + ) (see [1], Theorem 1.1),where the modulus of smoothess is costructed o the basis of the Bessel geeralized traslatio. L p,α (R + ),1 p, is the Baach space of measurable fuctios f(x) o R + with the fiite orm ( f(x) p x 2α+1 dx ) 1 p if 1 p < 0 f p,α = sup f(x) x R + if p = Everywhere α is a real umber, α > 1 2, Let B = d2 (2α + 1) d + dx2 x dx, be the Bessel differetial operator. By j α (x) we deote the Bessel ormalized fuctio of the first kid, i.e, ( 1) ( x ) 2 j α (x) = Γ(α + 1),!Γ( + α + 1) 2 =0 where Γ(x) is the gamma-fuctio (see [3]). The fuctio y = j α (x) satisfies the differetial equatio By + y = 0 with the coditio iitial y(0) = 1 ad y (0) = Mathematics Subject Classificatio. 34A30, 34D20. Key words ad phrases. geeralized cotiuity modulus, etire fuctios of expoetial type, best approximatio. Submitted Dec. 17,
2 EJMAA-2015/3(2) INVERSE THEOREMS OF APPROXIMATION 93 The fuctio j α (x) is the ifiitely differetiable, eve. The Bessel trasform is defied by formula (see [3]) f(t) = 0 f(x)j α (tx)x 2α+1 dx, t R +. The iverse Bessel trasform is give by the formula f(x) = (2 α Γ(α + 1)) 2 f(t)j α (tx)t 2α+1 dt. We ote the importat property of the Bessel trasform 0 Bf(t) = t 2 f(t). I L p,α (R + ),1 p, we cosider the Bessel geeralized traslatio T h f(x) = Γ(α + 1) Γ( 1 2 )Γ(α ) π 0 f( x 2 + h 2 2xh cos x)(si x) 2α xdx. Some of its properties are as follows (see [3] -[8]): (a) T h j α (λx) = j α (λx)j α (λh); x, h, λ R +, (b) T h f(x) = T x f(h), (c) T h f(t) = j α (th) f(t), (d) B(T h f) = T h (Bf), (e) T h f p,α f p,α, (f) T h f f p,α 0, h 0. The fiite differeces of the first ad higher orders are defied as follows: h f(x) = T h f(x) f(x) = (T h I)f(x), where I is the idetity operator i L p,α (R + ), ad k hf(x) = h ( k 1 h f(x)) = (T h 1) k f(x). The k th order geeralized modulus of cotiuity of a fuctio f L p,α (R + ) is defied by ω k (f, t) p,α = sup k hf p,α. 0<h t For ν > 0, we deote by M(ν, p, α) the set of eve etire fuctios of expoetial ν whose restrictios to R + belog to L p,α (R + ). The the fuctios i the space M(ν, p, α) make a atural approximatio tool i L p,α (R + ). The best approximatio of a f L p,α (R + ) by fuctios belogig to M(ν, p, α) is defied as follows E ν (f) p,α := if{ f Φ p,α : Φ M(ν, p, α)} Let W m p,α be the Sobolev space of order m {1, 2,...} costructed from the differetial operator B, that is, W m p,α = {f L p,α (R + ) : B j f L p,α (R + ), j = 1, 2,..., m}, where B j f = B(B j 1 f), ad B 0 f = f. Lemma 1.1 The modulus of smoothess ω k (f, t) has the followig properties. i) ω k (f + g, t) p,α ω k (f, t) p,α + ω k (g, t) p,α, ii) ω k (f, t) p,α 2 k f p,α, iii) if f W m p,α, the ω k (f, t) p,α c 1 t 2m B m f p,α,
3 94 R. DAHER, S. EL OUADIH EJMAA-2015/3(2) where c 1 = c(α, m) is a costat. Proof.(see Propositio 4.1 ad Lemma 4.6 i [1]) Lemma 1.2 For every fuctio f M(ν, p, α), ad ay umbers m N we have where c 2 = c(α, m) is a costat. Proof. (see Theorem 3.4 i [1]) Lemma 1.3 If 1 p < p, the B m f p,α c 2 ν 2m f p,α, f p,α c 3 ν (2α+2)( 1 p 1 p ) f p,α, for all f M(ν, p, α), where c 3 = c(α, p, p ) > 0 is a costat. Proof. (see [1],Theorem 3.5 ) 2. Mai Results Let c 1,c 2,...be positive costats possibly depedig o k, m ad α. Lemma 2.1 For j 1 we have Proof. Note that 2 (j 1) E (f) p,α l= 1 +1 l= 1 +1 l 1 E l (f) p,α. l 1 ( 1 ) 1 1 = 2 (j 1). Sice E l (f) p,α is mootoically decreasig, we coclude that 2 (j 1) E (f) p,α Lemma 2.2 For N we have Proof. Note that (j + 1) 1 2 k E (f) p,α c 4 l= 1 +1 l 1 E l (f) p,α. (j + 1) 1 E j (f) p,α. ( ) 1 (j + 1) = 2. j 2 1 Sice E l (f) p,α is mootoically decreasig, we coclude that 2 k E (f) p,α c 4 (j + 1) 1 E j (f) p,α. Lemma 2.3 If Φ ν M(ν, p, α) such that f Φ ν p,α = E ν (f) p,α For every ν N, the B k Φ +1 B k Φ p,α c 2 2 (j+1)+1 E (f) p,α. I particular B k Φ 1 p,α = B k Φ 1 B k Φ 0 p,α c 2 2 4k+1 E 0 (f) p,α.
4 EJMAA-2015/3(2) INVERSE THEOREMS OF APPROXIMATION 95 Proof. By lemma 1.2 ad the fact that E ν (f) p,α is mootoe decreasig with respect to ν, we obtai B k Φ +1 B k Φ p,α c 2 2 (j+1) Φ +1 Φ p,α ad = c 2 2 (j+1) (f Φ ) (f Φ +1) p,α c 2 2 (j+1) (E (f) p,α + E +1(f) p,α ) p,α c 2 2 (j+1)+1 E (f) p,α B k Φ 1 B k Φ 0 p,α c 2 Φ 1 Φ 0 p,α = c 2 (f Φ 1 ) (f Φ 0 ) p,α c 2 (E 1 (f) p,α + E 0 (f) p,α ) 2c 2 E 0 (f) p,α c 2 2 4k+1 E 0 (f) p,α Theorem 2.4 For ay f L p,α (R + ), 1 p, ν 0, there exists f ν M(ν, p, α), such that f f ν p,α = E ν (f) p,α Proof. If p =, it is the same as the case without weight (see [5], 2.6.2). If p <. Let {f } be a sequece i M(ν, p, α) such that f f p,α E ν (f) p,α as. The ( A > 0)( N) : f p,α A Usig lemma 1.3, we obtai that f,α c 3 Aν 2α+2 p By the compactess theorem (see[6], 3.3.6), there exists a subsequece {f k } such that f k (x) f ν (x) as k, holds uiformly o ay bouded set, for some f ν M(ν,, α). Hece, for ay K > 0, f ν χ {0 x K} p,α = lim k f k χ {0 x K} p,α A Passig to the limit as K, we obtai that f ν p,α A, i.e. f ν M(ν, p, α) Furthermore (f f ν )χ {0 x K} p,α = lim k (f f k )χ {0 x K} p,α lim k f f k p,α = E ν (f) p,α Passig to the limit as K, we obtai that f f ν p,α E ν (f) p,α I view of the defiitio of E ν (f) p,α, we have f f ν p,α = E ν (f) p,α. The followig theorems are aalogues of the classical iverse theorems of approximatio theory due to Stechki i the case p = ad A.F.Tima i the case 1 p < (see [4], [5]). Theorem 2.5 For every fuctio f L p,α (R + ) ad every positive iteger we have ω k (f, 1 ) p,α c (j + 1) 1 E j (f) p,α, where c = c(k, α) is a positive costat. Proof. Let 2 m < 2 m+1 for ay iteger m 0. Accordig to theorem 2.4, for ν 0, there exists Φ ν M(ν, p, α), such that
5 96 R. DAHER, S. EL OUADIH EJMAA-2015/3(2) f Φ ν p,α = E ν (f) p,α. By formulas (i) ad (ii) of lemma 1.1, we obtai ω k (f, 1 ) p,α ω k (f Φ 2 m+1, 1 ) p,α + ω k (Φ 2 m+1, 1 ) p,α 2 k f Φ 2 m+1 p,α + ω k (Φ 2 m+1, 1 ) p,α. Therefore ω k (f, 1 ) p,α 2 k E (f) p,α + ω k (Φ 2 m+1, 1 ) p,α. (1) Now with the aid of lemmas 2.1, 2.3 ad formula (iii) of lemma 1.1, we coclude that ω k (Φ 2 m+1, 1 ) p,α c 1 Bk Φ 2 m+1 p,α Whece c 1 B k Φ 1 B k Φ 0 p,α + 2 4k+1 E 0 (f) p,α + 24k+1 24k+1 24k+1 E 0 (f) p,α + m B k Φ +1 B k Φ p,α m 2 (j+1)+1 E (f) p,α m 2 (j 1) E (f) p,α E 0 (f) p,α + E 1 (f) p,α + ω k (Φ 2 m+1, 1 ) p,α c 5 E 0 (f) p,α + E 1 (f) p,α + 2 m Thus from (1) ad (2) we derive the estimate ω k (f, 1 ) p,α 2 k E (f) p,α + c 5 By lemma 2.2 ad formula (3), we have ω k (f, 1 ) p,α c m j=1 l= m j=2 l 1 E l (f) p,α (j + 1) 1 E j (f) p,α (j + 1) 1 E j (f) p,α. (2) (j + 1) 1 E j (f) p,α. (3) (j + 1) 1 E j (f) p,α. Theorem 3.6 Suppose that f L p,α (R + ) ad j 2m 1 E j (f) p,α <. j=1
6 EJMAA-2015/3(2) INVERSE THEOREMS OF APPROXIMATION 97 The f W m p,α ad,for every positive iteger, we have ω k (B m f, 1 ) p,α C 1 (j + 1) 2(k+m) 1 E j (f) p,α + j=+1 j 2m 1 E j (f) p,α, where C = c(k, m, α) is a positive costat. Proof. Let 2 m < 2 m+1 for ay iteger m 0. For every positive iteger r m, we cosider the series B r Φ 1 + (B r Φ +1 B r Φ ). (4) It follows from lemmas 2.3 ad 2.1 that the series (4) coverges i the orm of L p,α (R + ) because B r Φ +1 B r Φ p,α c 2 2 2r(j+1)+1 E (f) p,α = c 2 2 2r+1 E 1 (f) p,α + c 2 2 4r+1 2 2r(j 1) E (f) p,α j=1 c 2 2 4r+1 E 1 (f) p,α + 2 2r(j 1) E (f) p,α j=1 c 2 2 4r+1 E 1 (f) p,α + j=1 l= 1 +1 c 2 2 4r+1 j 2r 1 E j (f) p,α < j=1 l 2r 1 E l (f) p,α Note that f = Φ 1 + (Φ +1 Φ ). Sice B is a liear cotiuous operator (see [1]), we have B r f = B r Φ 1 + (B r Φ +1 B r Φ ) Whece B r f L p,α (R + ) for r m ad f W m p,α. By formula (i) of lemma 1.1, we obtai ω k (B m f, 1 ) p,α ω k (B m f B m Φ 2 s+1, 1 ) p,α + ω k (B m Φ 2 s+1, 1 ) p,α.
7 98 R. DAHER, S. EL OUADIH EJMAA-2015/3(2) Usig lemmas 1.1, 2.1 ad 2.3, we get Whece ω k (B m f B m Φ 2 s+1, 1 ) p,α 2 k B m f B m Φ 2 s+1 p,α 2 k j=s+1 2 k c 2 j=s+1 c 2 2 k+4m+1 B m Φ +1 B m Φ p,α 2 2m(j+1)+1 E (f) p,α j=s+1 c 2 2 k+4m+1 c 2 2 k+4m+1 ω k (B m f B m Φ 2 s+1, 1 ) p,α c 6 2 2m(j 1) E (f) p,α j=s+1 l= 1 +1 j=2 s +1 j=2 s +1 l 2m 1 E l (f) p,α j 2m 1 E j (f) p,α j 2m 1 E j (f) p,α. (5) Now with the aid of lemmas 2.1, 2.3 ad by formula (iii) of lemma 1.1, we coclude that ω k (B m Φ 2 s+1, 1 ) p,α c 1 Bm+k Φ 2 s+1 p,α c 1 B m+k Φ 1 B m+k Φ 0 p,α + 2 4(k+m)+1 E 0 (f) p,α + 24(k+m)+1 24(k+m)+1 24(k+m)+1 E 0 (f) p,α + s B m+k Φ +1 B m+k Φ p,α s 2 2(k+m)(j+1)+1 E (f) p,α s 2 2(k+m)(j 1) E (f) p,α E 0 (f) p,α + E 1 (f) p,α + s j=1 l= 1 +1 l 2(k+m) 1 E l (f) p,α 2 s E 0 (f) p,α + E 1 (f) p,α + (j + 1) 2(k+m) 1 E j (f) p,α j=2 Whece ω k (B m Φ 2 s+1, 1 ) p,α c 7 (j + 1) 2(k+m) 1 E j (f) p,α. (6) 2 s
8 EJMAA-2015/3(2) INVERSE THEOREMS OF APPROXIMATION 99 Thus from (5) ad (6) we derive the estimate ω k (B m f, 1 ) p,α C j 2m 1 E j (f) p,α + 1 (j + 1) 2(k+m) 1 E j (f) p,α. j=+1 Refereces [1] S.S.Platoov, Bessel harmoic aalysis ad approximatio of fuctios o the half-lie, Izvestiya: Mathematics, 71:5, ,2007. [2] S.Berstei, Sur l ordre de la meilleure approximatio des foctioes cotiues par des polyomes de degré doé, Mém.Cl.Sci.Acad.Roy.Belg.4,1-103, [3] B.M.Levita, Expasio i Fourier series ad itegrals over Bessel fuctios,uspekhi Mat.Nauk, 6, NO.2, , 1951 [4] S.B.Stechki, O the order of best approximatio of cotiuous fuctios, (Russia) Izv.Akad.Nauk.SSR, Ser.Math., 15 (1954), [5] A.F.Tima, Theory of approximatio of fuctios of a real variable, Eglish trasl, Dover publicatios, Ic, New York, [6] S.M. Nikol skii, Approximatio of Fuctios of Several Variables ad Embedig Theorems, Eglish trasl.of 1st ed., Grudlehre Math.Wiss, vol 205, Sprig, New York-Heidelberg [7] M.F.Tima, Best approximatio ad modulus of smoothess of fuctios prescribed o the etier real axis, (Russia), Izv.Vyssh. Ucheb. Zaved Matematika, 25 (1961), [8] I.A. Kipriyaov, Sigular Elliptic Boudary Value Problems, [i Russia], Nauka, Moscow (1997). R. DAHER Faculty of Sciece, Uiversity of Hassa II, Casablaca, Morocco address: rjdaher024@gmail.com S. EL OUADIH Faculty of Sciece, Uiversity of Hassa II, Casablaca, Morocco address: salahwadih@gmail.com
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