In this paper, several direct and inverse theorems are proved concerningthe approximation of one-variable functions from the space L2, by partialsums of Fourier-Jacobi series

Daher, Radouan

el Ouadih, Salah

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NISˇ)Ser. Math. Inform. Vol. 31, No 1 (2016), 1–10SOME NEW ESTIMTES OF APPROXIMATION OF FUNCTIONS BYFOURIER-JACOBI SUMSRadouan Daher and Salah El ouadihAbstract. In this paper, several direct and inverse theorems are proved concerning theapproximation of one-variable functions from the space L(α,β)2 by partial sums of Fourier-Jacobi series.Keywords: Partial sums of Fourier-Jacobi series; Generalized translation operator; Gen-eralized modulus of continuity; Estimate of approximation.1. Introduction and PreliminariesIt is well known that many problems for partial diﬀerential equations are re-duced to a power series expansion of the desired solution in terms of specialfunctions or orthogonal polynomials (such as Laguerre, Hermite, Jacobi, etc., poly-nomials). In particular, this is associatedwith the separation of variables as appliedto problems in mathematical physics (see, e.g.,[2]-[3]). In [8], Abilov et al. provedseveral estimates for the Fourier-Bessel series in the space L2([0, 1], x2p+1), p > − 12 ,on certain classes of functions characterized by the generalized continuitymodulus,using a translation operator.In this paper, we also discuss this subject. More specifically, several direct andinverse theorems are proved concerning the approximation of functions form thespace L(α,β)2 by partial sums of Fourier-Jacobi series, analogous of the statementsproved in [8]. For this purpose, we use a generalized translation operator whichwas defined by Flensted-Jensen and Koornwinder (see [5]).Throughout the paper, α and β are arbitrary real numberswithα ≥ β ≥ −1/2 andα  −1/2. Weputw(x) = (1−x)α(1+x)β and consider problems of the approximationof functions in the Hilbert spaces L2([−1, 1],w(x)dx).Let P(α,β)n (x) be the Jacobi orthogonal polynomials, n ∈ N0 := {0, 1, 2, .....} (see [4] or[1]). The polynomials P(α,β)n (x), n ∈ N0, form a complete orthogonal system in theReceived May 21, 2014; Accepted December 27, 20152010 Mathematics Subject Classification. Primary 42B37; Secondary 47B4812 R. Daher and S. El ouadihHilbert space L2([−1, 1],w(x)dx).It is known (see [4], Ch. IV) thatmax−1≤x≤1|P(α,β)n (x)| = P(α,β)n (1) = (n+αα ) = Γ(α + n + 1)n!Γ(α + 1) .The polynomialsR(α,β)n (x) :=P(α,β)n (x)P(α,β)n (1),are called normalized Jacobi polynomials.In what follows it is convenient to change the variable by the formula x = cos t, t ∈I := [0, π]. We use the notationρ(t) = w(cos t) sin t = 2α+β+1(sint2)2α+1 (cost2)2β+1,ϕn(t) = ϕ(α,β)n (t) := R(α,β)n (cos t), n ∈N0.LetL(α,β)2 denote the space of square integrable functions f (t) on the closed intervalI with the weight function ρ(t) and the norm‖ f ‖ =√∫ π0| f (t)|2ρ(t)dt.The Jacobi diﬀerential operator is defined asB := d2dt2+((α +12)cott2−(β +12)tant2) ddt.The function ϕn(t) satisfies the diﬀerential equationBϕn = −λnϕn, λn = n(n + α + β + 1), n ∈N0,with the initial conditions ϕn(0) = 1 and ϕ′n(0) = 0.Lemma 1.1. The following inequalities are valid for Jacobi functions ϕn(t)1. For t ∈ (0, π/2] we have|ϕn(t)| < 1.2. For t ∈ [0, π/2] we have1 − ϕn(t) ≤ c1λnt2.3. For every γ there is a number c2 = c2(γ, α, β) > 0 such that for all n and t withγ ≤ nt ≤ πn2 we have|ϕn(t)| ≤ c2(nt)−α−1/2.Proof. (See [7], Proposition 3.5.)Some New Estimtes of Approximation of Functions... 3Recall from [7], the Fourier-Jacobi series of a function f ∈ L(α,β)2 is defined byf (t) =∞∑n=0an( f )ϕ˜n(t),(1.1)whereϕ˜n =ϕn‖ϕn‖ , an( f ) = 〈 f , ϕ˜n〉 =∫ π0f (t)ϕ˜n(t)ρ(t)dt.LetSm f (t) =m−1∑n=0an( f )ϕ˜n(t),be a partial sum of series (1.1), and letEm( f ) = infPm‖ f − Pm‖,denote the best approximation of f ∈ L(α,β)2 by polynomials of the formPm(t) =m−1∑n=0cnϕ˜n(t), cn ∈ R.It is well known that‖ f ‖ =√∞∑n=0|an( f )|2,Em( f ) = ‖ f − Sm f ‖ =√ ∞∑n=m|an( f )|2.The Jacobi generalized translation is defined by the formulaTh f (t) =∫ π0f (θ)K(t, h, θ)ρ(θ)dθ, 0 < t, h < π,where K(t, s, θ) is a certain function (see [6]).Below are some properties (see [7]):(i) Th : L(α,β)2 → L(α,β)2 is a continuous linear operator,(ii) ‖Th f ‖ ≤ ‖ f ‖,(iii) Th(ϕn(t)) = ϕn(h)ϕn(t),(iv) an(Th f ) = ϕn(h)an( f ),(v) ‖Th f − f ‖ → 0, h→ 0,(vi) B(Th f ) = Th(B f ).For every function f ∈ L(α,β)2 we define the diﬀerences Δkh f of order, k ∈ N ={1, 2, 3, ...}, with step h, 0 < h < π, and the modulus of smoothness Ωk( f , δ) by the4 R. Daher and S. El ouadihformulaeΔ1h f (t) = Δh f (t) = (Th − I) f (t),where I is the identity operator in L(α,β)2 .Δkh f (t) = Δh(Δk−1h f (t)) = (Th − I)k f (t) =k∑i=0(−1)k−1(ki )Tih f (t), k > 1,Ωk( f , δ) = sup0<h≤δ‖Δkh f ‖, δ > 0,whereT0h f (t) = f (t), Tih f (t) = Th(Ti−1h f (t)), i = 1, 2, ..., k.Let Wr2, r ∈ N0, denote the class of functions f ∈ L(α,β)2 that have generalizedderivatives satisfying Br f ∈ L(α,β)2 ,i.e.,Wr2 := { f ∈ L(α,β)2 : Br f ∈ L(α,β)2 },where B0 f = f , Br f = B(Br−1 f ), r = 1, 2, ...Lemma 1.2. If f ∈Wr2, thenan( f ) = (−1)r 1λrn an(Br f ), r ∈N0.Proof. Since B is self-adjoint (see [7]), we havean( f ) = 〈 f , ϕ˜n〉 = − 1λn 〈 f ,Bϕ˜n〉= − 1λn〈B f , ϕ˜n〉 = − 1λn an(B f ).This completes the proof.Lemma 1.3. Iff (t) =∞∑n=0an( f )ϕ˜n(t),thenTh f (t) =∞∑n=0ϕn(h)an( f )ϕ˜n(t).Here, the convergence of the series on the right-hand side is understood in the sense ofL(α,β)2 .Some New Estimtes of Approximation of Functions... 5Proof. By the definition of the operator Th,Th(ϕ˜n(t)) = ϕn(h)ϕ˜n(t).Therefore, for any polynomialQN(t) =N∑n=0an( f )ϕ˜n(t),since Th is linear, we haveThQN(t) =N∑n=0ϕn(h)an( f )ϕ˜n(t).(1.2)Since Th is a linear bounded operator in L(α,β)2 and the set of all polynomials QN(t)is everywhere dense in L(α,β)2 , passage to the limit in (1.2) gives the required equal-ity.Remark. SinceTh f (t) − f (t) =∞∑n=0(ϕn(h) − 1)an( f )ϕ˜n(t),the Parseval’s identity gives‖Th f − f ‖2 =∞∑n=0(1 − ϕn(h))2|an( f )|2.If f ∈Wr2, from Lemma 1.2, we have‖Δkh(Br f )‖2 =∞∑n=0(1 − ϕn(h))2kλ2rn |an( f )|2.(1.3)2. Estimate of Best ApproximationThe goal of this work is to prove several estimates for Em( f ) in certain classes offunctions in L(α,β)2 .Theorem 2.1. Let r ∈ N0, k ∈ N. Then there is a constant c > 0 such that, for everyf ∈Wr2,Em( f ) = O(λ−rmΩk(Br f , cm−1)),when m→ +∞.6 R. Daher and S. El ouadihProof. Let f ∈Wr2. By the Ho¨lder inequality, we haveE2m( f )−∞∑n=mϕn(h)|an( f )|2 =∞∑n=m(1 − ϕn(h))|an( f )|2=∞∑n=m|an( f )|2− 1k (1 − ϕn(h))|an( f )| 1k≤⎛⎜⎜⎜⎜⎝ ∞∑n=m|an( f )|2⎞⎟⎟⎟⎟⎠2k−12k⎛⎜⎜⎜⎜⎝ ∞∑n=m|an( f )|2(1 − ϕn(h))2k⎞⎟⎟⎟⎟⎠12k≤(E2m( f )) 2k−12k⎛⎜⎜⎜⎜⎝λ−2rm ∞∑n=mλ2rn |an( f )|2(1 − ϕn(h))2k⎞⎟⎟⎟⎟⎠12k.From (1.3), we have∞∑n=m(1 − ϕn(h))2kλ2rn |an( f )|2 ≤ ‖Δkh(Br f )‖2.ThereforeE2m( f ) ≤∞∑n=mϕn(h)|an( f )|2 +(E2m( f )) 2k−12k(λ−2rm ‖Δkh(Br f )‖2) 12k .(2.1)From Lemma 1.1, we have∞∑n=mϕn(h)|an( f )|2 ≤ c2(mh)−α−1/2E2m( f ).Choose a constant c3 such that the number c4 = 1 − c2c−α−1/23 is positive.Setting h = c3/m in the inequality (2.1), we havec4E2m( f ) ≤(Em( f ))2− 1k (λ−rm ‖Δkc3m−1 (Br f )‖) 1k .By raising both sides to the power k and simplifying by (Em( f ))2k−1 we finally obtainck4Em( f ) ≤ λ−rm Ωk(Br f , c3m−1),for all m > 0. The theorem is proved with c = c3.Theorem 2.2. Let f ∈ L(α,β)2 . Then, for each k ∈N,Ω2k( f ,m−1) = O⎛⎜⎜⎜⎜⎜⎝m−4km∑n=1λ2k−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠ .Some New Estimtes of Approximation of Functions... 7Proof. From (1.3), one can veify‖Δkh f ‖2 =∞∑n=0(1 − ϕn(h))2k|an( f )|2.Let m ∈N and 0 < h ≤ 1/m. An application of Lemma 1.1 imples∞∑n=0(1 − ϕn(h))2k|an( f )|2 =m−1∑n=0(1 − ϕn(h))2k|an( f )|2 +∞∑n=m(1 − ϕn(h))2k|an( f )|2= O⎛⎜⎜⎜⎜⎜⎝h4km−1∑n=1λ2kn |an( f )|2 +∞∑n=m|an( f )|2⎞⎟⎟⎟⎟⎟⎠= O(m−4k)⎛⎜⎜⎜⎜⎜⎝m−1∑n=1λ2kn |an( f )|2 +m4k∞∑n=m|an( f )|2⎞⎟⎟⎟⎟⎟⎠= O(m−4k)⎛⎜⎜⎜⎜⎜⎝m−1∑n=1λ2kn⎛⎜⎜⎜⎜⎜⎝∞∑l=n|an( f )|2 −∞∑l=n+1|an( f )|2⎞⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎠+ O(m−4k)m4k∞∑n=m|an( f )|2= O(m−4k)m∑n=1(λ2kn − λ2kn−1)∞∑l=n|an( f )|2.By the equality λ2kn − λ2kn−1 = O(λ2k−1n ), we obtain‖Δkh f ‖2 = O⎛⎜⎜⎜⎜⎜⎝m−4km∑n=1λ2k−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠ ,which impliesΩ2k( f ,m−1) = O⎛⎜⎜⎜⎜⎜⎝m−4km∑n=1λ2k−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠ .This completes the proof of the theorem.Theorem 2.3. Let f ∈ L(α,β)2 . If the series∞∑n=1n2r−1En( f ), r ∈N,converges, then f ∈Wr2 andΩk(Br f ,m−1) = O⎛⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎝m−4km∑n=1λ2r+2k−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠12+∞∑n=mn2r−1En( f )⎞⎟⎟⎟⎟⎟⎟⎟⎠ ,where r, k ∈N.8 R. Daher and S. El ouadihProof. Let f ∈ L(α,β)2 . By the equality λ2rn − λ2rn−1 = O(λ2r−1n ), we obtain‖Br f ‖ =⎛⎜⎜⎜⎜⎜⎝∞∑n=1λ2rn |an( f )|2⎞⎟⎟⎟⎟⎟⎠1/2=⎛⎜⎜⎜⎜⎜⎝∞∑n=1λ2rn⎛⎜⎜⎜⎜⎜⎝∞∑i=n|an( f )|2 −∞∑i=n+1|an( f )|2⎞⎟⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎟⎠1/2=⎛⎜⎜⎜⎜⎜⎝∞∑n=1(λ2rn − λ2rn−1)∞∑i=n|an( f )|2⎞⎟⎟⎟⎟⎟⎠1/2= O⎛⎜⎜⎜⎜⎜⎝∞∑n=1λ2r−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠1/2= O⎛⎜⎜⎜⎜⎜⎝∞∑n=1λr−1/2n En( f )⎞⎟⎟⎟⎟⎟⎠ = O⎛⎜⎜⎜⎜⎜⎝∞∑n=1n2r−1En( f )⎞⎟⎟⎟⎟⎟⎠ .Since the series ∞∑n=1n2r−1En( f ), r ∈N,converges, we see that f ∈Wr2.Let j,m ∈N be such that12 j+1< h ≤ 12 j, m = [h−1].From (1.3), we have‖Δkh(Br f )‖2 =m∑n=0(1 − ϕn(h))2kλ2rn |an( f )|2 +∞∑n=m+1(1 − ϕn(h))2kλ2rn |an( f )|2 = I1 + I2.Estimate the summands I1 and I2.By Lemma 1.1, it is easy to see thatI1 =m∑n=0(1 − ϕn(h))2kλ2rn |an( f )|2= O(h4k)m∑n=1λ2kn λ2rn |an( f )|2 = O(h4k)m∑n=1λ2r+2kn |an( f )|2= O(h4k)m∑n=1λ2r+2kn⎛⎜⎜⎜⎜⎜⎝∞∑i=n|an( f )|2 −∞∑i=n+1|an( f )|2⎞⎟⎟⎟⎟⎟⎠= O(m−4k)m∑n=1(λ2k+2rn − λ2k+2rn−1 )∞∑i=n|an( f )|2= O⎛⎜⎜⎜⎜⎜⎝m−4km∑n=1λ2r+2k−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠ .Some New Estimtes of Approximation of Functions... 9By (1.3), we haveI2 =∞∑n=m+1(1 − ϕn(h))2kλ2rn |an( f )|2= O⎛⎜⎜⎜⎜⎜⎜⎝∞∑n= j+12n−1∑l=2n−1(1 − ϕl(h))2kλ2rl |an( f )|2⎞⎟⎟⎟⎟⎟⎟⎠= O⎛⎜⎜⎜⎜⎜⎜⎝∞∑n= j+1‖ΔkhBr(S2n ( f ) − S2n−1 ( f ))‖2⎞⎟⎟⎟⎟⎟⎟⎠ ,i.e.,I1/22 = O⎛⎜⎜⎜⎜⎜⎜⎝∞∑n= j+1‖ΔkhBr(S2n( f ) − S2n−1 ( f ))‖⎞⎟⎟⎟⎟⎟⎟⎠ .In view of‖Δkh f ‖ ≤ 2k‖ f ‖, ‖Br(S2n f − S2n−1 f )‖ ≤ λr2n‖S2n f − S2n−1 f ‖.we obtain‖ΔkhBr(S2n ( f ) − S2n−1 ( f ))‖ ≤ 2kλr2n‖S2n ( f ) − S2n−1 ( f )‖≤ 2kλr2n+1 (‖ f − S2n ( f )‖ + ‖ f − S2n−1 ( f )‖)≤ 2kλr2n+1 (E2n( f ) + E2n−1 ( f ))≤ 2k2λr2n+1E2n−1 ( f ).ThereforeI1/22 = O⎛⎜⎜⎜⎜⎜⎜⎝∞∑n= j+122r(n+1)+1E2n−1( f )⎞⎟⎟⎟⎟⎟⎟⎠ = O⎛⎜⎜⎜⎜⎜⎜⎝22r+1∞∑n= j22r(n+1)E2n( f )⎞⎟⎟⎟⎟⎟⎟⎠ .Note that for n ∈N we derive24r2n∑l=2n−1l2r−1 ≥ 24r(2n−1)2r−12n−1 = 22r(n+1).(2.2)Using (2.2) and the fact that El( f ) is monotone decreasing with respect to l , we getthe following inequality for n ∈N:22r(n+1)E2n( f ) ≤ 24r2n∑l=2n−1l2r−1El( f ).Consequently, we find thatI1/22 = O⎛⎜⎜⎜⎜⎝ ∞∑n=mn2r−1En( f )⎞⎟⎟⎟⎟⎠ .10 R. Daher and S. El ouadihCombining the estimates for I1 and I2 gives‖Δkh(Br f )‖ = O⎛⎜⎜⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎜⎝m−4km∑n=1λ2r+2k−1n E2n( f )⎞⎟⎟⎟⎟⎟⎠12+∞∑n=mn2r−1En( f )⎞⎟⎟⎟⎟⎟⎟⎟⎠ .Theorem 2.3 is proved.AcknowledgementsThe authors would like to thank the referee for his valuable comments and sugges-tions.REFERENCES1. A. Erde´lyi, W. Magnus, F. Oberttinger and F. G. Tricomi: Higher transcendentalfunctions, vol.II, McGraw-Hill, New York-Toronto-London 1953; Russian transl.,Nauka, Moscow, 1974.2. A.N.Tikhonov and A. A.Samarskii: Equations of mathematical physics, (Gostekhte-orizdat, Moscow, 1953; Pergamon Press, Oxford, 1964).3. A. Sveshnikov, A. N. Bogolyubov and V. V. Kravtsov: Lectures on mathematicalphysics, (Nauka, Moscow, 2004) [in Russian].4. G.Szego¨: Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol.23, Amer.Math.Soc., Providence, RI 1959; Russian transl., Fizmatgiz, Moscow 1962.5. M. Flensted-Jensen andT.Koornwinder: The convolution structure for Jacobi functionexpansions, Ark.Mat. 11, 245-262 (1973).6. R. Askey and S. Wainger: A convolution structure for Jacobi series, Amer. J.Math. 91,463-485 (1969).7. S. S. Platonov: Fourier-Jacobi harmonic analysis and approximation of functions,Izvestiya RAN: Ser.Mat. 78:1, 117-166 (2014).8. V. A.Abilov, F. V. Abilova andM.K. Kerimov: Some issues concerning approximationsof functions by Fourier-Bessel sums, Comput. Math. Math. Phys, Vol. 53, No.7, 867-873(2013).RADOUAN DAHERDepartment of MathematicsFaculty of Sciences Aı¨n ChockUniversity Hassan II, Casablanca, Moroccorjdaher024@gmail.comSALAH EL OUADIHDepartment of MathematicsFaculty of Sciences Aı¨n ChockUniversity Hassan II, Casablanca, Moroccosalahwadih@gmail.com

SOME NEWESTIMTES OF APPROXIMATION OF FUNCTIONS BY FOURIER-JACOBI SUMS

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SOME NEWESTIMTES OF APPROXIMATION OF FUNCTIONS BY FOURIER-JACOBI SUMS

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University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу