A Generalized Modulus of Smoothness

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 5, May 2014, Pages 1577–1590 S 0002-9939(2014)11884-3 Article electronically published on February 6, 2014

A GENERALIZED MODULUS OF SMOOTHNESS

BORISLAV R. DRAGANOV AND KAMEN G. IVANOV

(Communicated by Walter Van Assche)

Abstract. We construct moduli of smoothness that generalize the well-known classical moduli and possess similar properties. They are related to a linear diﬀerential operator L just as the classical moduli are related to the ordinary derivative. The generalized moduli are used to characterize the approximation error of the corresponding L-splines in Lp[a, b], 1 ≤ p ≤∞.

1. Moduli of smoothness and approximation error When using an approximation process it is important to have a practical and computable measure of its error. For such a measure one can use the so-called modulus of smoothness. Loosely speaking, it describes structural properties of the function and, in particular, its smoothness. Then error estimates by means of an appropriate modulus of smoothness state that the smoother a function is, the faster it is approximated. Let us recall the definition of the classical unweighted fixed-step modulus of smoothness for functions on a finite interval. As usual, Lp[a, b], 1 ≤ p ≤∞,arethe Lebesgue spaces of real/complex-valued functions on the interval [a, b] with their standard norm, which we shall denote by ·p. In what follows we can consider C[a, b], the space of continuous real/complex-valued functions on [a, b], in place of L∞[a, b]. The finite difference of order r ∈ N and step h>0 of the function f ∈ Lp[a, b]is defined by ⎧ r ⎪ − r ⎨ (−1)r k f(x + kh)ifx, x + rh ∈ [a, b], r k Δhf(x)=⎪k=0 ⎩⎪ 0, otherwise.

Then the classical modulus of smoothness of order r of f in Lp[a, b], denoted by ωr(f,t)p,isgivenby r ωr(f,t)p =sup Δhf p. 0

Received by the editors February 11, 2012 and, in revised form, May 7, 2012 and May 30, 2012. 2010 Mathematics Subject Classiﬁcation. Primary 41A25; Secondary 41A15, 41A27. Key words and phrases. Modulus of smoothness, K-functional, rate of convergence, L-spline, linear operator. Both authors were supported by grant DDVU 02/30 of the Fund for Scientiﬁc Research of the Bulgarian Ministry of Education and Science.

c 2014 American Mathematical Society Reverts to public domain 28 years from publication 1577

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One of the main properties of ωr(f,t)p is that

ωr(f,t)p → 0ast → 0

for every f ∈ Lp[a, b], as, moreover, the smoother f is, the faster its moduli tend to zero and vice versa (see [1, Ch. 2, §7 and Ch. 6, Theorem 3.1]). A typical application of ωr(f,t)p is an upper estimate of the best approximation of f ∈ Lp[a, b] by algebraic polynomials (see e.g. [1, Ch. 7, Theorem 6.3]):

(1.1) inf f − Pnp ≤ cωr(f,1/n)p, Pn∈Πn

where Πn denotes the set of the algebraic polynomials of degree not exceeding n ∈ N. Above and henceforward c denotes a positive constant independent of the approximated function and the degree of the approximation elements. A shortcom- ing of (1.1) is that it does not reflect the fact, discovered by S. M. Nikolski, that the approximation is better at the ends of the interval (see e.g. [1, Ch. 8]). To correct this, various new moduli were introduced. The most popular is that of Ditzian and Totik [4]. In 1980 the second author defined moduli with which he characterized the best algebraic approximation in Lp [8]. An account of other moduli can be found, for example, in [2], [4, Ch. 13] and [7]. They are all related to differential operators of the type (x−a)α(b−x)β(d/dx)r. Here we define a modulus related to a general linear differential operator with smooth coefficients and a constant leading coefficient. This approach is also applicable in the case when the leading coefficient has singularities at the ends of the interval (finite or infinite) [5, Section 6]. Other finite differences associated with linear differential operators were considered by Sharma and Tzimbalario [15] and Shevaldin [16, 17]. Let us proceed now to the definition of this new generalized modulus. Let 1 ≤ p ≤∞and the differential operator L be defined by

r (r) (r−k) (1.2) Lg(x)=g (x)+ ϕk(x)g (x),x∈ [a, b], k=1

r−k where ϕk ∈ C [a, b], k =1,...,r. We deﬁne the bounded linear operator AL : Lp[a, b] → Lp[a, b]by

− r rk r − k x (x − y)k+j−1 (1.3) (A f)(x)=f(x)+ (−1)j ϕ(j)(y)f(y) dy. L j (k + j − 1)! k k=1 j=0 a

The boundedness of AL can be established by means of the Minkowski integral inequality. We deﬁne the modulus of smoothness ωL(f,t)p by setting for f ∈ Lp[a, b]and t>0

ωL(f,t)p = ωr(ALf,t)p

r and call it the L-modulus of smoothness of f in Lp[a, b]. For L =(d/dx) the operator AL is the identity and ωL(f,t)p coincides with ωr(f,t)p. In the next section we consider the main properties of ωL(f,t)p; they generalize those of the classical modulus ωr(f,t)p. In Section 3 we consider an application of ωL(f,t)p in estimating the approximation error of the L-splines.

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2. Properties of the L-modulus of smoothness

We deﬁne for f ∈ Lp[a, b]andt>0theK-functionals r KL(f,t)p =inf{f − gp + t Lgp} ∈ r g Wp [a,b] and r (r) Kr(f,t)p =inf{f − gp + t g }, ∈ r p g Wp [a,b] r { ∈ r−1 (r) ∈ } where Wp [a, b]= f AC [a, b]:f Lp[a, b] is the Sobolev space as ACk[a, b] denotes the set of the functions on [a, b] whose derivatives up to order k ∈ N0 are absolutely continuous. It is known that the K-functional Kr(f,t)p and the modulus ωr(f,t)p are equivalent in the sense that −1 c ωr(f,t)p ≤ Kr(f,t)p ≤ cωr(f,t)p

for all f ∈ Lp[a, b]andt>0 (see e.g. [1, Ch. 6, Theorem 2.4] or [13, Theorem 2.67]). We shall denote relations like the latter shortly by

(2.1) Kr(f,t)p ∼ ωr(f,t)p.

We shall extend this relation by showing that KL(f,t)p and ωL(f,t)p are equivalent.

r−k Theorem 2.1. Let 1 ≤ p ≤∞, r ∈ N and ϕk ∈ C [a, b], k =1,...,r. Then for f ∈ Lp[a, b] and t>0 there holds

KL(f,t)p ∼ ωL(f,t)p. The main tool in verifying this equivalence relation will be the method formulated in [5] and further generalized in [6]. For an easier reference we shall state it here, particularly for the case under consideration. We denote the kernel of L by ker L.

Proposition 2.2. Let 1 ≤ p ≤∞, r ∈ N and the linear operators A, B : Lp[a, b] → Lp[a, b] satisfy the conditions: ≤ ∈ (a) Af p c f p for every f Lp[a, b]; ∈ r (r) ≤ ∈ r (b) Ag Wp [a, b] and (Ag) p c Lg p for every g Wp [a, b]; ≤ ∈ (c) Bf p c f p for every f Lp[a, b]; ∈ r ≤ (r) ∈ r (d) Bg Wp [a, b] and LBg p c g p for every g Wp [a, b]; (e) f − BAf ∈ ker L for every f ∈ Lp[a, b]; (f) f − ABf ∈ Πr−1 for every f ∈ Lp[a, b].

Then for all f ∈ Lp[a, b] and t>0 there holds

KL(f,t)p ∼ Kr(Af, t)p ∼ ωr(Af, t)p and

Kr(f,t)p ∼ KL(Bf,t)p.

We proceed to check the hypotheses in the proposition above for A = AL. First, we establish the characteristic diﬀerential property of the operator AL,whichac- tually motivated its deﬁnition in (1.3).

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r−k Proposition 2.3. Let 1 ≤ p ≤∞, r ∈ N and ϕk ∈ C [a, b], k =1,...,r.If ∈ r ∈ r (r) ∈ g Wp [a, b],thenALg Wp [a, b] and (ALg) (x)=Lg(x), x [a, b].

Proof. The first assertion follows from the definition of AL and the conditions imposed on g and ϕk, k =1,...,r. The second one is verified by direct computations. We first observe that − r rk r − k x (x − y)k+j−1 (A f)(x)=f(x)+ (−1)j ϕ(j)(y)f(y) dy L j (k + j − 1)! k k=1 j=0 a − r x ∂ r k (x − y)r−1 = f(x)+ (−1)r−k ϕ (y) f(y) dy. ∂y (r − 1)! k k=1 a ∈ r Then integration by parts shows for g Wp [a, b]that r x − r−1 (x y) (r−k) (ALg)(x)=g(x)+ − ϕk(y)g (y) dy + Pr−1g(x) a (r 1)! k=1 x (x − y)r−1 = − Lg(y) dy + Qr−1g(x), a (r 1)! (r) where Pr−1g, Qr−1g ∈ Πr−1.Consequently,(ALg) = Lg.

Now, let us establish the equivalence between KL(f,t)p and ωL(f,t)p. { }r Proof of Theorem 2.1. Let u(x) =1 be a fundamental system of solutions of the homogeneous linear ODE Lg(x)=0,x ∈ [a, b]. We shall show that there exist functions v ∈ C[a, b], =1,...,r, such that the linear operator BL : Lp[a, b] → Lp[a, b], deﬁned by x BLf(x)=f(x)+ B(x, y)f(y) dy, a where r (2.2) B(x, y)= u(x)v(y), =1 has the property that ∈ r (r) ∈ r (2.3) BLg Wp [a, b]andL(BLg)=g for g Wp [a, b].

Then the assertion of the theorem follows from Proposition 2.2 with A = AL and B = BL. Conditions (a) and (c) are veriﬁed by means of the Minkowski integral inequality, as we have noted in Section 1. Condition (b) was established in Proposition 2.3, and (d) is contained in equation (2.3). As for (e) and (f), they ∈ r follow again from Proposition 2.3 and (2.3), since for each g Wp [a, b]wehave (r) (r) (r) L(BL(ALg)) = (ALg) = Lg and (AL(BLg)) = L(BLg)=g . Hence (e) and r r (f) hold in Wp [a, b]. Now, as we take into consideration that Wp [a, b]isdensein Lp[a, b]andkerL and Πr−1 are ﬁnite dimensional, we conclude that (e) and (f) are valid in Lp[a, b]. Thus it remains to construct the operator BL with (2.3). For j =0,...,r− 1weset ∂j B r (2.4) B (x)= (x, x)= u(j)(x)v (x). j ∂xj =1

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≡ ∈ r Set also ϕ0(x) 1. For each g Wp [a, b] we formally have

− − r r k 1 x ∂r−kB L(B g)(x)=Lg(x)+ ϕ (x) (B g)(r−k−j−1)(x)+ (x, y)g(y) dy L k j ∂xr−k k=0 j=0 a r x = Lg(x)+ Lu(x) v(y)g(y) dy =1 a r r−k−1 r−k−j−1 r − k − j − 1 − − − − (2.5) + ϕ (x)B(r k j 1)(x)g()(x). k j k=0 j=0 =0

1 The ﬁrst sum above is equal to 0 since Lu(x) ≡ 0for =1,...,r. Further, we reorder the summands in the second sum as follows: r r−k−1 r−k−j−1 r − k − j − 1 − − − − ϕ (x)B(r k j 1)(x)g()(x) k j k=0 j=0 =0 r −1 −k−1 r − k − j − 1 − − − = g(r−)(x) ϕ (x)B( k j 1)(x) r − k j =1 k=0 j=0 r −1 −j−1 r − k − j − 1 − − − = g(r−)(x) ϕ (x)B( k j 1)(x) r − k j =1 j=0 k=0 r −1 −j−1 − (r−) r + k (k) = g (x) ϕ − − − (x)B (x). k k j 1 j =1 j=0 k=0

Therefore, to establish (2.3) it remains to show that there exist v, =1,...,r, such that −1 −j−1 r − + k (k) (2.6) ϕ (x)+ ϕ − − − (x)B (x) ≡ 0,=1,...,r. k k j 1 j j=0 k=0

These relations show that B−1 is a polynomial function of ϕm, m =1,...,, B(k) − − − and for >1 also of j , k =0,..., j 1, j =0,..., 2. Therefore, by (k) iteration on we conclude that B−1 is a polynomial function of ϕm , m =1,...,, k =0,...,− m. More precisely, for =1,...,r there exist polynomials Q−1 of ( +1)/2 variables such that B (−1) (−2) (2.7) −1(x)=Q−1(ϕ1(x),...,ϕ1 (x),ϕ2(x),...,ϕ2 (x),...,ϕ(x)).

Now, taking into account (2.4), we deduce that the functions v, =1,...,r,arethe solution of a linear system, whose determinant is the Wronskian of the fundamental system of the solution of the homogeneous diﬀerential equation Lg(x)=0,x ∈ [a, b], and hence is non-zero everywhere on [a, b]. It remains to remark that since ϕk ∈ r−k r− C [a, b], k =1,...,r, relation (2.7) implies that B−1 ∈ C [a, b], =1,...,r, and the calculations we have performed above are formally rigorous.

1The necessity to “get rid of” the integral summands motivates us to consider kernels B in the form (2.2).

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Remark 2.4. 1) It can be shown that BL is the inverse of AL.2)Setforeachﬁxed α ∈ [a, b]: (x − α)r−1 R (x)= + ,x∈ [a, b]. α (r − 1)!

Then BLRα(x) is Green’s function associated with the diﬀerential operator L.In- deed, for a ≤ x ≤ α we have B R (x) = 0, whereas for α ≤ x ≤ b there holds L α − r−1 x − r−1 (x α) B (y α) (2.8) BLRα(x)= − + (x, y) − dy. (r 1)! α (r 1)! By (2.3) with a replaced with α we have (r) (2.9) L(BLRα)(x)=(Rα) (x)=0,x∈ [α, b]. Finally, by diﬀerentiating (2.8) in x (cf. (2.5)), we arrive at (j) (BLRα) (α)=0,j=0,...,r− 2, (2.10) (r−1) (BLRα) (α)=1.

Thus BLRα(x) is Green’s function associated with L. This observation enables us to express more clearly the functions v in B.Since BLRα(x) is the solution of the Cauchy problem (2.9)-(2.10), we get that for each ﬁxed α ∈ [a, b] there exist constants c(α), =1,...,r, such that r (2.11) BLRα(x)= c(α)u(x); =1 moreover, in view of the initial value conditions (2.10), we have for each α ∈ [a, b], T T W(α)(c1(α),...,cr(α)) =(0,...,0, 1) , T where W denotes the Wronskian matrix. Thus (c1(α),...,cr(α)) is the last column of W−1(α) (cf. [13, pp. 426–428]). On the other hand, by (2.8) and (2.11), we get r x (y − α)r−1 (x − α)r−1 c (α) − v (y) dy u (x)= . (r − 1)! (r − 1)! =1 α We diﬀerentiate this identity in αrtimes and get r (r) − r−1 c (α)+( 1) v(α) u(x)=0, =1

which, in view of the linear independence of {u}, implies − r (r) ∈ v(α)=( 1) c (α),=1,...,r, α [a, b]. The L-moduli possess properties which generalize the well-known properties of the classical moduli ωr(f,t)p. Below we give a list of the basic ones.

Theorem 2.5. Let 1 ≤ p ≤∞, r, r1,r2 ∈ N and f,g ∈ Lp[a, b].AlsoletL be given r−k by (1.2) as ϕk ∈ C [a, b] for k =1,...,r,andL1 and L2 be given by rj (rj ) (rj −k) Ljg(x)=g (x)+ ϕj,k(x)g (x),x∈ [a, b], k=1 r +r −k as ϕj,k ∈ C 1 2 [a, b] for k =1,...,rj and j =1, 2.Then:

1) ωL(f + g, t)p ≤ ωL(f,t)p + ωL(g, t)p;

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2) ωL(cf, t)p = |c| ωL(f,t)p, c ∈ R; 3) ωL(f,t)p ≤ ωL(f,t )p, t ≤ t ; 4) ωL(f,t)p → 0 as t → 0; r r 5) ωL(f,mt)p ≤ m ωL(f,t)p, m ∈ N,andωL(f,λt)p ≤ (λ +1)ωL(f,t)p, λ>0; ≤ r 6) ωL(f,t)p 2 AL Lp→Lp f p; 7) ωL(f,t)p ≡ 0 iﬀ f ∈ ker L; ≤ r1 ∈ r1 8) ωL2L1 (f,t)p t ωL2 (L1f,t)p, f Wp [a, b]; 9) (Johnen-Scherer) there exists a constant c independent of f and t such that t ωL2L1 (f,u)p ωL (L1f,t)p ≤ c du, 2 r1+1 0 u ∈ r1 as whenever the right side is ﬁnite, then f Wp [a, b]; ≤ 10) there exists a constant c independent of f and t such that ωL2L1 (f,t)p r2 (2 + ct) ωL1 (f,t)p; 11) (Marchaud) there exists a constant c independent of f and t ≤ t0 such that t0 ωL L (f,u)p ≤ r1 2 1 ωL1 (f,t)p ct r +1 du + f p . t u 1

Proof. The conditions imposed on the ϕ’s imply that L, L1,L2 and L2L1 satisfy the conditions of Proposition 2.3. The ﬁrst six properties follow directly from the corresponding well-known properties of the classical moduli of smoothness. Property 7 follows from Theorem 2.1. ∈ r1+r2 To establish property 8, we observe that for g Wp [a, b]wehavebyPropo- sition 2.3 (r1+r2) (r2) (AL2L1 g) = L2L1g =(AL2 (L1g)) . (r ) r +r 1 − ∈ − ∈ 1 2 Consequently, (AL2L1 g) AL2 (L1g) Πr2 1 for all g Wp [a, b]. The space r1+r2 r1 (r1) Wp [a, b]isdenseinWp [a, b], equipped with the norm f p + f p.Further, by means of the well-known inequality for the intermediate derivatives of a function ∈ r1 f Wp [a, b] (see e.g. [1, Ch. 2, Theorem 5.6]), () ≤ (r1) − (2.12) f p c f p + f p ,=1,...,r1 1,

r1 r1 → we establish that the operators D AL2L1 : Wp [a, b] Lp[a, b]andAL2 L1 : r1 → (r1) − Wp [a, b] Lp[a, b] are bounded. Here D = d/dx. Therefore (AL2L1 f) ∈ ∈ r1 ∈ r1 AL2 (L1f) Πr2−1 for all f Wp [a, b]. Then for f Wp [a, b] we have, also using [1, Ch. 2, (7.13)],

≤ r1 (r1) ωL2L1 (f,t)p = ωr1+r2 (AL2L1 f,t)p t ωr2 ((AL2L1 f) ,t)p r1 r1 = t ωr2 (AL2 (L1f),t)p = t ωL2 (L1f,t)p. The next property follows likewise by means of the corresponding Johnen-Scherer inequality for the classical modulus of smoothness (see [1, Ch. 6, Theorem 3.1]). r1 To prove property 10, we set L3 = L2D and observe that, in view of Proposi- ∈ r1+r2 tion 2.3, we have for g Wp [a, b]that

(r1+r2) (r1) (r1+r2) (AL3 AL1 g) = L2(AL1 g) = L2L1g =(AL2L1 g) , which, as in the proof of property 7, implies − ∈ ∈ (2.13) AL2L1 f AL3 AL1 f Πr1+r2−1,f Lp[a, b].

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We put r +r −k−1 1 2 (y − a)m Φ (y)=ϕ (y) − ϕ(m)(a) ,k=1,...,r , k 2,k 2,k m! 2 m=0

and also for f ∈ Lp[a, b]weset

− − − − r2 r1+r2 k r1+r2k j 1 j (m+j) − − Pr1+r2 1f(x)= ( 1) ϕ2,k (a) j=0 m=0 k=1 x − k+j−1 − m × (x y) (y a) − f(y) dy. a (k + j 1)! m! We have Pr +r −1f ∈ Πr +r −1. Further, we deﬁne the bounded linear operator 1 2 1 2 A : Lp[a, b] → Lp[a, b]by − − Af(x)=(AL3 f(x) f(x) Pr1+r2−1f(x)) .

∈ r1 Let us note that thus deﬁned, A has the following property. Let g Wp [a, b]. Then r2 x r1+r2−k − d ∂ (x − y)r1+r2 1 r1+r2−k Ag(x)= (−1) Φk(y) g(y) dy dx ∂y (r1 + r2 − 1)! k=1 a r2 x r2−k − ∂ (x − y)r1+r2 2 r2−k (r1) (2.14) = (−1) Φk(y) g (y) dy. ∂y (r1 + r2 − 2)! k=1 a Now, by means of (2.13) and basic properties of the classical modulus of smoothness, we get

ωL2L1 (f,t)p = ωr1+r2 (AL2L1 f,t)p = ωr1+r2 (AL3 AL1 f,t)p ≤ ωr1+r2 (AL1 f,t)p + tωr1+r2−1(AAL1 f,t)p − ≤ r2 r2 1 2 ωr1 (AL1 f,t)p +2 tωr1 (AAL1 f,t)p. Thus to complete the proof of this property it remains to show that ≤ ∈ ωr1 (Af, t)p cωr1 (f,t)p,f Lp[a, b], or, equivalently, ≤ ∈ (2.15) Kr1 (Af, t)p cKr1 (f,t)p,f Lp[a, b]. The latter follows from ≤ ∈ Af p c f p,f Lp[a, b], and (r1) ≤ (r1) ∈ r1 (2.16) (Ag) p c g p,gWp [a, b].

∈ r1 Indeed, these two relations directly imply that for any g Wp [a, b], r1 (r1) Kr (Af, t)p ≤ Af − Ag + t (Ag) 1 p p ≤ − r1 (r1) c f g p + t g p .

∈ r1 Taking inﬁmum on g Wp [a, b] we get (2.15).

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∈ r1 Let us now establish (2.16). Let g Wp [a, b]. Then by (2.14) we get r r −k 2 2 x r1+k+j−2 r2 − k (x − y) (j) − j (r1) Ag(x)= ( 1) Φk (y)g (y) dy j (r1 + k + j − 2)! k=1 j=0 a r r 2 2 x r1+m−2 − r2 − k (x − y) (m−k) = (−1)m k Φ (y)g(r1)(y) dy − − k m k a (r1 + m 2)! k=1 m=k r m 2 x r1+m−2 (x − y) − r2 − k (m−k) − m k (r1) = ( 1) Φk (y) g (y) dy. (r1 + m − 2)! m − k m=1 a k=1

∈ r1 Consequently, Ag Wp [a, b]as,moreover,

(r1) (r1) (Ag) (x)=Φ1(x)g (x) r2 x m−2 m (x − y) − r2 − k (m−k) + (−1)m k Φ (y) g(r1)(y) dy. (m − 2)! m − k k m=2 a k=1 Hence (2.16) directly follows. Finally, let us establish property 11. We shall show that

≤ r1 ∈ ≤ (2.17) ωr1 (BL3 f,t)p c (ωr1 (f,t)p + t f p),f Lp[a, b], 0

Then, by means of (2.17) with f replaced with AL3 AL1 f, the classical Marchaud inequality (see e.g. [1, Ch. 2, Theorem 8.1]), relation (2.13) and the fact that the

operators AL1 and AL3 are bounded, we arrive at

r1 ωr (BL AL AL f,t)p ≤ c (ωr (AL AL f,t)p + t AL AL fp) 1 3 3 1 1 3 1 3 1 t0 ω (A A f,u) r1 r1+r2 L3 L1 p ≤ ct du + AL AL fp (2.18) ur1+1 3 1 t t0 ωL L (f,u)p ≤ r1 2 1 ct r +1 du + f p . t u 1 On the other hand, as we have established in the beginning of the proof of The- ∈ ∈ orem 2.1, for each F Lp[a, b]thereexistsUF ker L3 such that BL3 AL3 F =

F + UF. The operators AL3 and BL3 are bounded, hence so is U. The latter is of the form r1+r2 UF(x)= c(F ) u(x), =1

where {u(x)} is a fundamental system of solutions of L3g =0andc(F ) are linear functionals, which are bounded since U is. Therefore, for any F ∈ Lp[a, b]there holds r1+r2 (r ) (r1) ≤ | | 1 ≤ (UF) p c(F ) u p c F p. =1 Consequently, ω (F, t) ≤ ω (B A F, t) + ω (UF,t) r1 p r1 L3 L3 p r1 p ≤ r1 (r1) ωr1 (BL3 AL3 F, t)p + t (UF) p ≤ r1 ωr1 (BL3 AL3 F, t)p + ct F p.

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The last estimate with F = AL1 f, (2.18) and the boundedness of AL1 imply property 11. To verify (2.17) we proceed just as in the proof of the previous property. Relation (2.17) is equivalent to

≤ r1 ∈ ≤ (2.19) Kr1 (BL3 f,t)p c (Kr1 (f,t)p + t f p),f Lp[a, b], 0

r1 Indeed, for any g ∈ W [a, b]andt ≤ t0 we have p r (r ) K (B f,t) ≤B f − B g + t 1 (B g) 1 r1 L3 p L3 L3 p L3 r1 r1 (r1) ≤ c f − gp + t gp + t g p r1 r1 r1 (r1) ≤ c f − gp + t f − gp + t fp + t g p ≤ − r1 (r1) r1 c f g p + t g p + t f p ; hence (2.19) follows. Thus to complete the proof of property 11, it remains to establish (2.20). Let ∈ r1 g Wp [a, b]. By (2.5) for k =0andr = r1 we get the following expression for (r1) (BL3 g) : − − − r1 1 r1j 1 r1 − j − 1 (r −j−−1) (B g)(r1)(x)=g(r1)(x)+ B 1 (x)g()(x) L3 j j=0 =0 r1+r2 x (r1) + u (x) v(y)g(y) dy, =1 a where r1+r2 B (j) j (x)= u (x)v(x). =1 Now, by (2.12) we finally derive r1 (r1) ≤ () ≤ (r1) (BL3 g) p c g p c g p + g p . =0 3. L-splines and characterization of their approximation rate The L-moduli are useful for estimating the error of the L-splines. Let us recall their definition. As is known, the kernel of the differential operator (1.2) with ϕk ∈ r C[a, b] forms an rth dimensional linear subspace of C [a, b]. Let M =(m1,...,mn) as mj ∈ N and mj ≤ r for j =1,...,n.AlsoletΔ={a = x0

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where D±s(x) denotes respectively the right and the left th derivative of the function s. S(L, M, Δ) is called the space of L-splines with knots at x1,...,xn of multiplicities m1,...,mn. In [13, §§10.5-10.7] the error of approximation of the L-splines was estimated by means of the K-functionals Kr(f,t)p and KL(f,t)p. They have equivalent big O rates since (cf. [13, Theorem 10.29]) (3.1) K (f,t) ≤ c K (f,t) + trf , L p r p p r Kr(f,t)p ≤ c KL(f,t)p + t fp

for all f ∈ Lp[a, b]andt>0. Let n ∈ N, Mn =(mn,1,...,mn,n), as mn,j ∈ N and mn,j ≤ r for j =1,...,n, and Δn = {a = xn,0

and call it the best L-spline approximation of f in Lp[a, b]. From results of Scherer and Schumaker [14] (see also [13, Theorems 10.19, 10.21 and 10.24]) it follows that r−k if ϕk ∈ C [a, b], then L ≤ (3.2) En (f)p cKL(f,1/n)p. In [13, p. 420] the diﬀerential operator L is given in the form

m−1 m j L = D + aj (x)D j=0

m−j with aj ∈ C [a, b]andD = d/dx. In our opinion, there is a typo here. The j r−k assumptions should be aj ∈ C [a, b], which in our notation are ϕk ∈ C [a, b]. The latter was used in [13, §10.3] to construct an L-spline basis and then establish j the above-mentioned direct estimates. The assumptions aj ∈ C [a, b] were included in many results on L-splines; see e.g. [11, (2.1)-(2.2)] and [18, (1.1.4)]. The estimate (3.2) is best possible in the sense that there exists a function ∈ r f0 Wp [a, b] such that (see [10], [12] and [13, Theorems 10.26 and 10.28]) ≤ L (3.3) 0

with some constant c0 independent of n. However, estimates of the approximation error in terms of K-functionals have little practical value, because it is difficult to calculate the order of a K-functional, as the infimum in its definition is over a space of infinite dimension. So it is useful to have a modulus of smoothness which is equivalent to the K-functional. Thus we are led to derive from (3.1), (3.2) and (2.1) the estimate L ≤ r En (f)p c ωr(f,t)p + t f p . L ∈ But it does not include the trivial fact that En (f)p =0forf ker L.Toovercome that, we can use instead the L-modulus and get directly from (3.2), (3.3) and Theorem 2.1, provided that the coefficients of L are smooth enough, the following characterization, which takes that into account (see Theorem 2.5, property 7).

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Theorem 3.1. Let 1 ≤ p ≤∞and Δn = {a = xn,0

r−k Proposition 3.3. Let r ∈N and ϕk ∈ C [a, b], k =1,...,r.Ifs ∈S(L, Mn, Δn), r then ALs ∈S((d/dx) ,Mn, Δn).

Proof. First, note that AL preserves the smoothness of functions at each point. To see that ALs ∈ Πr−1 on each interval of the partition, we observe that if we change the ﬁxed integration bound a in the deﬁnition of AL to any other real α ∈ [a, b] and denote the new operator by AL,α,thenALf − AL,αf ∈ Πr−1 for every f ∈ (r) Lp[a, b], and hence Proposition 2.3 for the interval [xj ,xj+1] yields (ALs) (x)= (r) ∈ (AL,xj s) (x)=Ls(x)=0,x [xj ,xj+1].

A converse inequality of type C (in the terminology of [3]) for the best approximation by L-splines follows.

Theorem 3.4. Let 1 ≤ p ≤∞, r ∈ N, 0 <δ≤ 1, n := [1/δ], Δm = {0=xm,0 < xm,1 < ···

− c 2n 1 ω (f,δ) ≤ EL (f) . L p n +1 m p m=n−1 ≤ 1 Proof. Fix h,0

Let χS(x) denote the characteristic function of the set S. Using [1, Ch. 12, Lem- ma 2.3] we get

n 2n (3.4) ≤ χ (x),x∈ [0, 1 − rh]. 64 Dm m=n

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∈S L − Let sm (L, Mm, Δm) be such that Em(f)p = f sm p. By (3.4) and Proposi- tion 3.3 we get n 2n 2n Δr A f ≤ χ Δr A f ≤ χ Δr A f 64 h L p Dm h L Dm h L p m=n p m=n 2n 2n r − ≤ r − = χDm ΔhAL(f sm−1) p ΔhAL(f sm−1) p m=n m=n 2n 2n−1 ≤ r − r L 2 AL f sm−1 p =2 AL Em(f)p. m=n m=n−1 Hence − 1 2n 1 ω (f,δ ) ≤ 2r+7A EL (f) . L 1 p L n +1 m p m=n−1 r Finally, the above inequality and ωL(f,δ)p ≤ ωL(f,1/n)p ≤ (16r) ωL(f,δ1)p prove the theorem.

Note that this result is slightly better for p>1 than [1, Ch. 12, Theorem 2.1].

Acknowledgments The authors are thankful to the referees for their prompt and careful reading of the manuscript as well as for their remarks that improved the exposition.

References

[1] Ronald A. DeVore and George G. Lorentz, Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303, Springer-Verlag, Berlin, 1993. MR1261635 (95f:41001) r [2] Z. Ditzian, Polynomial approximation and ωφ(f,t) twenty years later, Surv. Approx. Theory 3 (2007), 106–151. MR2342231 (2008f:41010) [3] Z. Ditzian and K. G. Ivanov, Strong converse inequalities,J.Anal.Math.61 (1993), 61–111, DOI 10.1007/BF02788839. MR1253439 (94m:41038) [4] Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathemat- ics, vol. 9, Springer-Verlag, New York, 1987. MR914149 (89h:41002) [5] Borislav R. Draganov and Kamen G. Ivanov, A new characterization of weighted Peetre K-functionals, Constr. Approx. 21 (2005), no. 1, 113–148, DOI 10.1007/s00365-004-0574-5. MR2105393 (2005i:41028) [6] Borislav R. Draganov and Kamen G. Ivanov, A new characterization of weighted Peetre K-functionals. II, Serdica Math. J. 33 (2007), no. 1, 59–124. MR2313796 (2009d:46041) [7] Borislav R. Draganov and Kamen G. Ivanov, Equivalence between K-functionals based on continuous linear transforms, Serdica Math. J. 33 (2007), no. 4, 475–494. MR2418191 (2010a:46057) [8] K. G. Ivanov, Some characterizations of the best algebraic approximation in Lp[−1, 1] (1 ≤ p ≤∞), C. R. Acad. Bulgare Sci. 34 (1981), no. 9, 1229–1232. MR649151 (83i:41021) [9]K.G.Ivanov,A characterization of weighted Peetre K-functionals, J. Approx. Theory 56 (1989), no. 2, 185–211, DOI 10.1016/0021-9045(89)90109-3. MR982846 (90h:41038) [10] J. W. Jerome and L. L. Schumaker, On the distance to a class of generalized splines,Linear operators and approximation, II (Proc. Conf., Math. Res. Inst., Oberwolfach, 1974), Internat. Ser. Numer. Math., Vol. 25, Birkh¨auser, Basel, 1974, pp. 503–517. MR0393948 (52 #14755) [11] Tom Lyche and Larry L. Schumaker, L-spline wavelets, Wavelets: theory, algorithms, and applications (Taormina, 1993), Wavelet Anal. Appl., vol. 5, Academic Press, San Diego, CA, 1994, pp. 197–212. MR1321430 (96c:42069)

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1590 BORISLAV R. DRAGANOV AND KAMEN G. IVANOV

[12] Larry L. Schumaker, Lower bounds for spline approximation, Approximation theory (Papers, VIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975), Banach Center Publ., vol. 4, PWN, Warsaw, 1979, pp. 213–223. MR553767 (81h:41011) [13] Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, A Wiley- Interscience Publication, John Wiley & Sons Inc., New York, 1981. MR606200 (82j:41001) [14] Karl Scherer and Larry L. Schumaker, A dual basis for L-splines and applications,J.Ap- prox. Theory 29 (1980), no. 2, 151–169, DOI 10.1016/0021-9045(80)90113-6. MR595599 (82i:41015) [15] A. Sarmaˇ and I. Cimbalario, Some linear differential operators, and generalized differences (Russian), Mat. Zametki 21 (1977), no. 2, 161–172. MR0437989 (55 #10910) [16] V. T. Sevaldin,ˇ Extremal interpolation with smallest value of the norm of a linear differential operator (Russian), Mat. Zametki 27 (1980), no. 5, 721–740, 830. MR578257 (81k:41002) [17] V. T. Shevaldin, Some problems of extremal interpolation in the mean for linear differential operators (Russian), Orthogonal series and approximations of functions, Trudy Mat. Inst. Steklov. 164 (1983), 203–240. MR752926 (85m:41038) [18] R. S. Varga, Functional analysis and approximation theory in numerical analysis,Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1971. MR0310504 (46 #9602)

Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria – and – Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria E-mail address: [email protected] Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, bl. 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria E-mail address: [email protected]

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