Constr. Approx. (2002) 18: 443–466 DOI: 10.1007/s00365-001-0021-9

CONSTRUCTIVE APPROXIMATION © 2002 Springer-Verlag New York Inc.

Differential Operators with Second-Order Degeneracy and Positive Approximation Processes E. M. Mangino Abstract. A sequence of integral operators acting on continuous functions on [0, +∞[ is introduced in order to study the semigroup of operators generated by a differential operator with second-order degeneracy at the origin.

1. Introduction In this paper we apply methods of positive approximation operators to the investigation of the degenerate differential operators Au := qu  , where q is a continuous function on [0, +∞[ satisfying lim+

x→0

q(x) ∈R x2

and

0 < q0 ≤

q(x) ≤ q1 x2

(x > 0)

for some q0 , q1 ∈ R. The operator A is defined on its maximal domain in the space C([0, +∞]) or in the spaces E m0 := { f ∈ C([0, +∞[) : limx→+∞ f (x)/(1 + x m ) = 0}, with m a natural number. At least, in the case in which A acts on C([0, +∞]), the operator has been widely investigated from the point of view of semigroup theory (see [13] or, e.g., [8]). Here we extend the results of generation to the case in which A acts on E m0 and we study the shape-preserving properties and saturation properties of the semigroup generated by A both on C([0, +∞]) and on E m0 . The approach that we adopt was first introduced by F. Altomare and has been successfully applied both in the framework of bounded closed intervals or, more generally, convex compact sets of Rn (see, e.g., [1], [3] and the references quoted therein), and on unbounded intervals. Indeed, in [4], [5], and [6] these methods have been applied to the study of the operator Bu = r u  , for functions r which are differentiable at 0 and satisfy the assumption 0 < r0 ≤ r (x)/x ≤ r1 (x > 0) or 0 < r0 ≤ r (x)/x(x + 1) ≤ r1 (x > 0) for some r0 , r1 ∈ R. There B was defined on a suitable domain of E m0 with m ≥ 2. Date received: November 14, 2000. Date revised: July 17, 2001. Date accepted: September 17, 2001. Communicated by Vilmos Totik. AMS classification: 41A36, 34A45, 47D06, 35K65. Key words and phrases: Positive approximation process, Degenerate differential operator, C0 -Semigroup, Weighted function space, Saturation. 443

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The basic idea is to construct a sequence (L n )n∈N of positive linear operators defined on E = E m0 or on E = C([0, +∞]) such that for a large class of functions f ∈ C 2 ([0, +∞[) ∩ E with q f  ∈ E the equality lim n(L n ( f ) − f ) = q f 

(1.1)

n→∞

in E

holds. The asymptotic relation (1.1) is called a Voronovskaja-type formula and is the link, via Trotter’s theorem (see [17]), between C0 -semigroups theory and positive approximation processes. In order to carry out the study of A, we introduce and study a sequence of integraltype positive operators (Pα,n )n∈N depending on a continuous bounded positive function α : [0, +∞[ → R and acting on E m0 . The function α will be related to q by the relation q(x) = α(x)x 2 (x ≥ 0). In particular, when α(x) = 1 for every x ∈ [0, +∞[, the operators Pα,n coincide with the operators that have been studied in [10] or, formally, with Post-Widder operators with parameter n/2 (see [11]). In the first sections of this paper we study the approximation properties of the sequence (Pα,n )n∈N , by estimating the rate of convergence, and we give a probabilistic interpretation of the operators Pα,n . The middle sections are devoted to the study of the operator A. We establish a generation result in E m0 for every m ≥ 1 and we represent the semigroup generated by A in terms of iterates of the operators Pα,n on C([0, +∞]) and on E m0 . Moreover, we consider the behavior of the operators Pα,n and of the semigroup generated by A when acting on convex, monotone, or H¨older continuous functions. Finally, we find the saturation class and the trivial class of the sequence (Pα,n )n∈N and the Favard class of the semigroup in C([0, +∞]) and E m0 . Throughout the paper, let C([0, +∞[) denote the space of all real-valued continuous functions on [0, +∞[ and let Cb ([0, +∞[) (resp., UCb ([0, +∞[), C0 ([0, +∞[), C([0, +∞])) be the subspace of all bounded continuous functions (resp., bounded uniformly continuous functions, continuous functions that vanish at infinity, continuous functions that converge at infinity) on [0, +∞[. The previous spaces of bounded functions, endowed with the natural order and the norm

f ∞ := sup | f (x)|,

f ∈ Cb ([0, +∞[),

x≥0

are Banach lattices. For every m ∈ N, set wm (x) := (1 + x m )−1 for all x ∈ [0, +∞[ and consider the following weighted spaces:
 E m := f ∈ C([0, +∞[) : f m := sup wm (x)| f (x)| < +∞ , x≥0


E m0 E m∗

f ∈ C([0, +∞[) : lim wm (x) f (x) = 0 ,

:=

x→+∞


:=

 

f ∈ C([0, +∞[) : ∃ lim wm (x) f (x) ∈ R . x→+∞

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For every m ∈ N, the spaces E m , E m0 , and E m∗ , endowed with the norm · m and the natural  order, are Banach lattices and the inclusions E m → E m+1 are continuous. Set E = m Em . For all t ∈ [0, +∞[ set 1(t) := 1 and, for every r ∈ ]0, +∞[, set er (t) := t r . Moreover, for every x ≥ 0, let ψx be the function defined by ψx (t) = t − x

(t ≥ 0).

If X is a Banach space and T : X → X is a continuous linear operator, let T X denote the operator norm of T . 2. The Operators Pα,n Let α ∈ Cb ([0, +∞[) be a nonnull positive function and let c := (2 supx≥0 α(x))−1 . For every x ∈ [0, +∞[, let εx be the unit mass at x. Set v0,α := ε0 , and vx,α := 2cα(x)vx m1 + (1 − 2cα(x))ε0

(x > 0),

where m1 is the one-dimensional Lebesgue measure, and for every x > 0:  c−1 −ct/x t e   , t > 0,   x c (c) vx (t) := (2.1)  c   0, t ≤ 0. Then vx,α is a distribution on R with supp(vx,α ) ⊆ [0, +∞[. A function g : [0, +∞ [ → R is vx,α -integrable if and only if g is vx -integrable. In this case, for every x > 0,

+∞

2cα(x) +∞ g(t)t c−1 e−ct/x (2.2) g dvx,α = dt + (1 − 2cα(x))g(x).  x c (c) 0 0 c Note also that, if f ∈ E m , then, for every n ∈ N and x ∈ [0, +∞[, by using Fubini’s theorem, the convexity and the vx -integrability of 1/wm , we get



+∞

+∞     f x1 + · · · + xn  dvx,α (x1 ) · · · dvx,α (xn ) ···   n 0 0



+∞

+∞ x1 + · · · + xn m ≤ f m 1+ dvx,α (x1 ) · · · dvx,α (xn ) ··· n 0 0

+∞ 

n

f m +∞ ≤ ··· (1 + xim ) dvx,α (x1 ) · · · dvx,α (xn ) n 0 0 i=1

+∞ = f m (1 + t m ) dvx,α (t) 0





+∞

= f m 2cα(x) 0

 (1 + t )vx (t) dt + (1 − 2cα(x))(1 + x ) m

m

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2cα(x) +∞ t m+c−1 e−tc/x m = f m 1 + dt + (1 − 2cα(x))x (c) 0 (x/c)c

 x m  (m + c) = f m 1 + 2cα(x) + (1 − 2cα(x))x m ≤ M(1 + x m ). c (c) We can now introduce the positive operators that will be studied in this paper. For every n ∈ N, f ∈ E, and x ∈ [0, +∞[, set (2.3) Pα,n ( f )(x) :=



 

+∞ 0

···



+∞

f 0

x1 + · · · + xn n

dvx,α (x1 ) · · · dvx,α (xn ),

f (0),

x > 0, x = 0.

By applying some classical results about Dirichlet’s integral (see, e.g., [18, 12.5, p. 258]), we derive for every x > 0 an explicit representation of Pα,n ( f )(x); indeed, we have n  n (2cα(x)) p (1 − 2cα(x))n− p (2.4) Pα,n ( f )(x) = (1 − 2cα(x))n f (x) + p p=1

+∞ +∞ x1 + · · · + x p + (n − p)x × ··· f n 0 0 × vx (x1 ) · · · vx (x p ) d x1 · · · d x p n  n (2cα(x)) p (1 − 2cα(x))n− p = (1 − 2cα(x))n f (x) + p p=1  c pc 1 +∞ × d x1 · · · x (c) p 0

+∞ x1 + · · · + x p + (n − p)x × f n 0 × (x1 · . . . · x p )c−1 e−c[(x1 +···+x p )/x] d x p = (1 − 2cα(x))n f (x) n  c pc 1  n + (2cα(x)) p (1 − 2cα(x))n− p p x ( pc) p=1

+∞ t + (n − p)x × t pc−1 f e−tc/x dt n 0 = (1 − 2cα(x))n f (x) n  1 n + (2cα(x)) p (1 − 2cα(x))n− p p ( pc) p=1

+∞  xs  p × e−s s pc−1 f + 1− x ds. cn n 0

Differential Operators with Second-Order Degeneracy

447

By observing this explicit representation of Pα,n ( f )(x) and by applying Lebesgue’s dominated convergence theorem, we get that Pα,n ( f ) is a continuous function on [0, +∞[. When α = 1 the operators Pα,n are the so-called Post-Widder operators (see, e.g., [11]) that have also been studied by Cismasiu (see [10], [3]):

+∞ 1 2xt n/2−1 −t  Pn ( f )(x) := t e f dt (x ≥ 0). n n 0  2 More generally, if α = k1, with k ∈ ]0, +∞[, then

+∞ 1 2kxt n/2k−1 −t Pk1,n ( f )(x) =  n t e f dt n 0  2k By setting

(2.5)

p

 p t + 1− x (t ≥ 0), n n denote the identity operator on E, we can represent the operators Pα,n f n, p,x (t) := f

and letting Pα,0 as follows:

(x ≥ 0).

Pα,n ( f )(x) =

n  n p=0

p

(2cα(x)) p (1 − 2cα(x))n− p P(1/2c)1, p ( f n, p,x )(x).

The following lemma and proposition are the key steps in order to estimate the operator norm of Pα,n in the spaces E m and to study the convergence of the sequence (Pα,n )n∈N . Proposition 2.1. (1) For every n, m ∈ N and x ≥ 0: (2.6)

Pα,n (em )(x) = x m + x m  × 1−

n m    n m (2cα(x)) p (1 − 2cα(x))n− p p k p=1 k=2 p m−k cp(cp + 1) · . . . · (cp + k − 1) − ck p k

n

ck n k

.

In particular, Pα,n (e1 ) = e1 and Pα,n (e2 ) = e2 +(2α/n)e2 . Moreover, Pα,n (1) = 1. (2) For every m ∈ N, there exists Cm ≥ 0 such that, for every x ≥ 0, (2.7)

Pα,n (em )(x) xm α(x) ≤ + Cm . m 1+x 1 + xm n In particular, it is possible to choose C1 = 0 and C2 = 2.

Proof.

(1) Let m, n ∈ N and fix x ≥ 0. Then

Pα,n (em )(x) = (1 − 2cα(x))n x m n  1 n (2cα(x)) p (1 − 2cα(x))n− p + p ( pc) p=1

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E. M. Mangino



+∞

×

e−s s pc−1

0

 xs

 p m ds + 1− x cn n

= (1 − 2cα(x))n x m n m    1 n m p m−k m p n− p (2cα(x)) (1 − 2cα(x)) 1− +x k n k ( pc) p k n c p=1 k=0

+∞ × e−s s pc+k−1 ds 0

= (1 − 2cα(x))n x m n m    n m p m−k ( pc + k) (2cα(x)) p (1 − 2cα(x))n− p 1− + xm p k n ck n k ( pc) p=1 k=0 n  n m m = (1 − 2cα(x)) + x (2cα(x)) p (1 − 2cα(x))n− p p p=1  

m   m p m−k cp(cp + 1) · . . . · (cp + k − 1) − ck p k 1− × 1+ k n ck n k k=1 n  n = xm + xm (2cα(x)) p (1 − 2cα(x))n− p p p=1 m   m p m−k cp(cp + 1) · . . . · (cp + k − 1) − ck p k 1− × . k n ck n k k=2 Moreover, Pα,n (e1 )(x) = x and n 2 2  n 2 2 p n− p cp(cp + 1) − c p Pα,n (e2 )(x) = x + x (2cα(x)) (1 − 2cα(x)) p c2 n 2 p=1 2α(x) 2 x . n Finally, it is immediate that Pα,n (1) = 1. (2) If m = 1 (resp., m = 2), then the assertion clearly holds with C1 = 0 (resp., C2 = 2). Fix m ≥ 3. For every k ∈ N, let qk (t) := (t + 1)(t + 2) · . . . · (t + k − 1) − t k−1 and choose K m > 0 such that 0 ≤ qk (t) ≤ K m t k−2 for every t ≥ 0 and for every k = 1, . . . , m. By (1), n  n p Pα,n (em )(x) = x m + x m (2cα(x)) p (1 − 2cα(x))n− p 2 p cn p=1 m   m p m−k qk ( pc) 1− × k n ck−2 n k−2 k=2

n n−1 2K m α(x)  m (2cα(x)) p−1 (1 − 2cα(x))n− p (2m − 1) ≤ x + p−1 n p=1 = x2 +

Cm α(x) , n where Cm := 2K m (2m − 1). = xm + xm

Differential Operators with Second-Order Degeneracy

Theorem 2.2.

449

Let m, n ∈ N. The following statements hold true:

(1) Pα,n is a positive continuous linear operator from E m into itself and

Pα,n Em ≤ 1 + α ∞

Cm , n

in particular, sup Pα,n Em ≤ 1 + Cm α ∞ ; n∈N

(2) every operator Pα,n maps continuously Cb ([0, +∞[) into itself and Pα,n Cb ([0,+∞[) = 1; (3) Pα,n (C0 ([0, +∞[)) ⊆ C0 ([0, +∞[) and Pα,n (C([0, +∞])) ⊆ C([0, +∞]); (4) Pα,n (E m0 ) ⊆ E m0 ; and (5) if α ∈ C([0, +∞]), then Pα,n (E m∗ ) ⊆ E m∗ . Proof. (1) is an immediate consequence of Proposition 2.1, while (2) follows immediately from the equality Pα,n (1) = 1. (3) Since Pα,n (1) = 1, it is clearly enough to show that Pα,n (C0 ([0, +∞[)) ⊆ C0 ([0, +∞[). Let f ∈ C0 ([0, +∞[) and let (xk )k∈N be a divergent sequence of positive numbers. For every n ∈ N:  x s  p  k  e−s s pc−1  f + 1− xk  ≤ f ∞ e−s s pc−1 , cn n and limk→∞ f (xk s/cn + (1 − p/n)xk ) = 0. By Lebesgue’s theorem, we get that limk→∞ Pα,n ( f )(xk ) = 0. (4) For every β > 0, let gβ (x) := exp{−βx}. The subspace D := lin{gβ : β > 0} is dense in E m0 by [7, Lemma 4.9] and Pα,n (D) ⊆ Pα,n (C0 ([0, +∞[)) ⊆ C0 ([0, +∞[) ⊆ E m0 , hence Pα,n (E m0 ) = Pα,n (D) ⊆ Pα,n (D) ⊆ E m0 , where the closures are taken with respect to the norm · m . (5) Let l := limx→+∞ α(x). Since E m∗ = {awm−1 : a ∈ R}+ E m0 , and Pα,n maps E m0 into itself by (4), it is enough to show that Pα,n (wm−1 ) = Pα,n (1 + em ) = 1 + Pα,n (em ) ∈ E m∗ , i.e., that Pα,n (em ) ∈ E m∗ . This is clear by Proposition 2.1. 3. Probabilistic Interpretation By a theorem due to Kolmogorov (see, e.g., [3, p. 44, Remark after (1.3.41)]), there exist a probability space (, F, P) and a family (Y (n, x))n∈N,x≥0 of real independent random variables on  such that for every n ∈ N and x ≥ 0, PY (n,x) = vx,α , where PY (n,x) denotes the distribution of Y (n, x) with respect to P. Then the operators  (Pα,n )n∈N are the Feller operators associated with the random scheme X (n, x) = (1/n) nk=1 Y (k, x) (see [10] and [3, (5.2.6)]), i.e.,

+∞ Pα,n ( f ) = (3.1) f d PX (n,x) . 0

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Denote by E(Y (n, x)) and Var(Y (n, x)), respectively, the expected value and the variance of Y (n, x). Then we have

+∞

+∞ E(Y (n, x)) = t dvx,α (t) = 2cα(x) tvx (t) dt + (1 − 2cα(x))x −∞

∞

= 2cα(x)x + (1 − 2cα(x))x = x and

Var(Y (n, x)) =

+∞

−∞



t dvx,α (t) − 2

= 2cα(x)



+∞ −∞

+∞ −∞

2 t dvx,α (t)

t 2 vx (t) dt + (1 − 2cα(x))x 2 − x 2 = 2α(x)x 2 .

2 Therefore αn,x := E(X (n, x)) = x and σn,x = Var(X (n, x)) = (2/n)α(x)x 2 (see [3, 2 5.2.1, 5.2.2]). The determination of the parameters αn,x and σn,x , together with formula (3.1), will be useful to estimate the rate of convergence of the operators Pα,n . Formula (3.1) can also be used to easily deduce the following monotonicity property.

Proposition 3.1.

For every convex function f ∈ E and for every n ∈ N:

f ≤ Pα,n+1 f ≤ Pα,n ( f ) ≤ 2cα P(1/2c)1,1 ( f ) + (1 − 2cα) f ≤ P(1/2c)1,1 ( f ). Proof. The first two inequalities can be proved by using the representation (3.1) and by adapting the proof of Theorem 3 in [14] to functions f ∈ E. For an alternative proof one can use the same method for the proof of Theorem 6.1.14 in [3]. To show the last inequalities note that

+∞

+∞ x1 + · · · + xn Pα,n ( f )(x) = ··· f dvx,α (x1 ) . . . dvx,α (xn ) n 0 0 n +∞ 1 ≤ f (xi ) dvx,α (xi ) n i=1 0

+∞ = 2cα(x) f (t)vx (t) dt + (1 − 2cα(x)) f (x) 0

= 2cα(x)P(1/2c)1,1 ( f )(x) + (1 − 2cα(x)) f (x) and, finally, that f ≤ P(1/2c)1,1 ( f ). 4. Convergence of the Sequence (Pα,n )n∈N We recall that a subset H of a Banach lattice E is called a Korovkin subset in E with respect to positive operators (briefly, a K + -subset) if for every equicontinuous net (L i )i∈I of positive linear operators on E satisfying limi∈I L i (h) = h for every h ∈ H , one has limi∈I L i ( f ) = f for every f ∈ E (see [3] for more information).

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The main approximation properties of the sequence (Pα,n )n∈N are listed below. Theorem 4.1. Let m ∈ N. The following statements hold true: limn→∞ Pα,n (em ) = em in E m ; for every f ∈ E m0 , limn→∞ Pα,n ( f ) = f in E m0 ; for every f ∈ E m , limn→∞ Pα,n ( f ) = f in E m+1 ; for every f ∈ E m∗ , limn→∞ Pα,n ( f ) = f in E m ; for every f ∈ E, limn→∞ Pα,n ( f ) = f uniformly on the compact subsets of [0, +∞[; (6) if f ∈ C([0, +∞]), then limn→∞ Pα,n ( f ) = f uniformly on [0, +∞[; and (7) if f ∈ U Cb ([0, +∞[) and supx≥0 x 2 α(x) < +∞, then limn→∞ Pα,n ( f ) = f uniformly on [0, +∞[.

(1) (2) (3) (4) (5)

Proof. (1) This is an immediate consequence of Proposition 2.1(2) and of the inequality em ≤ Pα,n (em ) (see Proposition 3.1). (2) If m ≥ 3, the claim follows since, by Theorem 2.2, the sequence (Pα,n )n∈N is equicontinuous on E m0 and, by (1), converges to the identity on the K + -subset {1, e1 , em−1 } in E m−1 , hence in E m0 (see Lemma 4.1 in [5]). Let m ≤ 2. Choose 0 < λ1 < λ2 < 1. Then the functions eλi , i = 1, 2, are concave, hence, by Proposition 3.1, (Pα,n (eλi ))n∈N is an increasing sequence of functions that converges to eλi in E 30 . Therefore (wm Pα,n (eλi ))n is a bounded increasing sequence of functions in C0 ([0, +∞[) that converges pointwise to wm eλi . By Dini’s theorem, the convergence is uniform, hence limn→∞ Pα,n (eλi ) = eλi in E m0 . Since {1, eλ1 , eλ2 } is a K + -subset in E m0 by Lemma 4.1 in [5], the assertion follows. 0 (3) Only observe that E m → E m+1 continuously. (4) The assertion follows by (1) and (2), since E m∗ = {αwm−1 : α ∈ R} + E m0 . (5) This is an easy consequence of the boundedness of 1/wm on the compact subsets of [0, +∞[ and statement (3). (6) Choose 0 < λ1 < λ2 and set gi (x) := exp(−λi x) for x ≥ 0 and i = 1, 2. By [3, 4.2.5(7)], the set {1, g1 , g2 } is a K + -subset in C([0, +∞]), hence it is enough to show that limn→∞ Pα,n (gi ) = gi in C([0, +∞]). The functions gi are convex functions, then (Pα,n (gi ))n∈N is a decreasing sequence of functions in C0 ([0, +∞[) that converges pointwise to gi by (2). By Dini’s theorem, the convergence is uniform on [0, +∞[. (7) Apply Theorem 5.2.2 in [3] and the results in Section 2. We conclude this section with some estimates about the rate of convergence in terms of the first and second modulus of smoothness (see, e.g., [3, Section 5.1]) and of a particular modulus of smoothness that has been first introduced by Totik in [16] and then generalized by Ditzian and Totik [11]. Theorem 4.2.

Let n ∈ N. The following statements hold true:

(1) If f ∈ Cb ([0, +∞[), for every x ∈ [0, +∞[:  |Pα,n ( f )(x) − f (x)| ≤ (1 + x 2α(x))ω( f, n −1/2 ).

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(2) If f ∈ Cb ([0, +∞[) is differentiable and f  ∈ Cb ([0, +∞[), then for every x ∈ [0, +∞[):   2α(x) |Pα,n ( f )(x) − f (x)| ≤ x (1 + x 2α(x))ω( f  , n −1/2 ). n (3) If f ∈ UCb ([0, +∞[), then for every x ≥ 0: |Pα,n ( f )(x) − f (x)| ≤ Mω2



 f, x

 2α(x) , n

where M neither depends on f nor on n; in particular, if K := supx≥0 α(x)x 2 < +∞, then    2K |Pα,n ( f )(x) − f (x)| ≤ Mω2 f, . n (4) For every f ∈ Cb ([0, +∞[):

Pα,n ( f ) − f ∞ ≤ K ωϕ2 ( f, n −1/2 )∞ , where K does not depend on n and ωϕ2 ( f, δ)∞ :=

sup 0≤h≤δ,x±hϕ(x)≥0

|2hϕ(x) f (x)|

(δ > 0),

with 2hϕ(x) = f (x − hϕ(x)) − 2 f (x) + f (x + hϕ(x))

and ϕ(x) =

√ c−1 x (see [11], [16]).

Proof. Statements (1), (2), and (3) are immediate consequences of the probabilistic interpretation of the operators Pα,n and of Theorem 5.2.4 and (5.2.31) of [3]. (4) By Proposition 2.1, we have 2α(x)x 2 x2 ≤ n cn and so the estimate follows from Theorem 1 of [16]. Pα,n (ψx2 )(x) =

5. A Voronovskaja-Type Formula In this section we study the asymptotic behavior of the remainders Pα,n ( f ) − f . In this way, we establish the link between the sequence (Pα,n )n∈N and the problem that has been considered in the Introduction. Theorem 5.1. Let α ∈ Cb ([0, +∞[) be a nonnull positive function. Let f ∈ C 2 ([0, +∞[) such that f  ∈ Cb ([0, +∞[) (resp., f  ∈ C0 ([0, +∞[)). Then (5.1)

lim n(Pα,n ( f ) − f ) = αe2 f 

n→∞

in E 3 (resp., E 2 ) and hence in E m for every m ≥ 3 (resp., m ≥ 2).

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Proof. Simply observe that Pα,n (1) = 1 and for every x ≥ 0, Pα,n (ψx ) = 0, Pα,n (ψx2 ) (x) = (2/n)α(x)x 2 , Pα,n (ψx3 )(x) = (2α(x)/cn 2 )x 3 ,

6α(x) 4 4 12α(x) 2n − 1 Pα,n (ψx )(x) = x + 12α(x) + 2 3 . c2 n 3 n3 c n So the result follows from Proposition 5.1 in [5] applied to H = E. Proposition 5.2. Assume that α(x) ≥ α0 > 0 for every x ≥ 0. Let f ∈ E ∩ C 2 ([0, +∞[) such that f  vanishes in a neighborhood of +∞. Then lim n(Pα,n ( f ) − f ) = αe2 f 

n→∞

uniformly on [0, +∞[. Proof. Let a > 0 be such that f (x) = c1 x + c2 for x ≥ a, with c1 = 0. Then g = f − c1 e1 − c2 vanishes on [a, +∞[ and n(Pα,n ( f ) − f ) = n(Pα,n (g) − g). Therefore w.l.o.g. we can assume that f vanishes on [a, +∞[. By the previous theorem, limn→∞ n(Pα,n ( f ) − f ) = e2 α f  uniformly on [0, ae + 1]. For every x ≥ ae + 1, we have |n(Pα,n ( f )(x) − f (x))| = |n Pα,n ( f )(x)|  n   1 n  = n (2cα(x)) p (1 − 2cα(x))n− p  p=1 p ( pc) 

+∞  xs  p  −s pc−1 × + 1− x ds  e s f cn n 0 n  n 1 ≤ n f ∞ (2cα(x)) p (1 − 2cα(x))n− p p ( pc) p=1

×

max(0,(nc/x)(a−(1− p/n)x))

s pc−1 ds. 0

Note that, since x > a, then (nc/x)(a − (1 − p/n)x) < apc/x. Moreover, by the asymptotic relation (see, e.g., [18, 12.33, p. 253]): (t) = 1, √ t→+∞ t t 2π e−t lim

there exists C > 0 such that |n Pα,n ( f )(x)| ≤ n f ∞

n  n p=1

= n f ∞

(2cα(x)) (1 − 2cα(x))

p

p

n  n p=1

p

n− p

(2cα(x)) p (1 − 2cα(x))n− p

1 ( pc)



apc/x

s pc−1 ds 0

 a pc ( pc) pc−1 x ( pc)

454

E. M. Mangino n  a pc  n 1 (2cα(x)) p (1 − 2cα(x))n− p √ p x pc 2π e− pc p=1 n  ae c n f ∞ C  ≤ + 1 − 2cα(x) 2cα(x) √ x c 2π

c

n ae n f ∞ C 1 − 2cα0 1 − ≤ . √ ae + 1 c 2π

≤ n f ∞ C

Therefore limn→∞ n(Pα,n ( f )(x) − f (x)) = 0 = α(x)x 2 f  (x) uniformly on [ae + 1, +∞[. 6. The Operator Aα on C([0, +∞]) Let α ∈ Cb ([0, +∞[) such that 0 < α0 ≤ α(x) ≤ α1 for every x ≥ 0. W.l.o.g. assume α1 > 1. Consider the differential operator (Aα , D(Aα )) on C([0, +∞]) defined by (6.1)

D(Aα ) := {u ∈ C([0, +∞]) ∩ C 2 (]0, +∞[) : αe2 u  ∈ C([0, +∞])},

(6.2)

Aα u := αe2 u 

(u ∈ D(Aα )).

D(Aα ) is clearly the maximal domain of the differential operator Aα on C([0, +∞]). By using Feller’s classification (see [13], see also [8]), we have that 0 and +∞ are natural points for Aα , hence (Aα , D(Aα )) generates a contraction semigroup on C([0, +∞]). Remark 6.1. D(Aα ): (6.3)

By [13, Corollary after Theorem 13.1], we have that for every u ∈ lim x 2 u  (x) = lim x 2 u  (x) = 0. x→+∞

x→0+

Moreover, for every u ∈ D(Aα ), limx→0+ xu  (x) = 0. Indeed, given ε > 0 there exists δ > 0 such that |u  (x)| < εx −2 for every x ∈ ]0, δ[. By integrating, one gets

1 1 ε |u  (x)| ≤ |u  (δ)| + ε − ≤ |u  (δ)| + (0 < x < δ), x δ x hence limx→0+ xu  (x) = 0. Analogously we can prove that limx→+∞ xu  (x) = 0. Indeed, assume w.l.o.g. that limx→+∞ u(x) = 0. Fix ε > 0. Then there exists M > 0 such that |u(x)| < ε,

|x 2 u  (x)| < ε

(x > M).

Hence, for every x > M, 

 2  x u (x) − M 2 u  (M) − 2xu(x) + 2Mu(M) + 2  

 = 

M x



 2  t v (t) dt  ≤

x M

x M

  u(t) dt 

|t 2 u  (t)| dt ≤ ε(x − M).

Differential Operators with Second-Order Degeneracy

455

Therefore |xu  (x)| ≤ M 2

|u  (M)| |u(M)| 2ε(x − M) ε(x − M) + 2|u(x)| + 2M + + . x x x x

Then lim supx→+∞ |xu  (x)| ≤ 3ε. Proposition 6.2. The space D0 := {u ∈ C 2 ([0, +∞[) : u  vanishes in a neighborhood of +∞} is a core for (Aα , D(Aα )), i.e., D0 is dense in D(Aα ) with respect to the graph norm. Proof. set (6.4)

Let u ∈ D(Aα ) and assume w.l.o.g. that limx→+∞ u(x) = 0. For every n ∈ N



 1 1 1 1 1 1 2  u , + u x− + u  x− u n (x) := n n n 2 n n  u(x),

0≤x ≤

1 , n

x > 1/n.

Then u n ∈ C([0, +∞])∩C 2 ([0, +∞[) for every n ∈ N. Given ε > 0, there exists ν ∈ N such that for every n > ν, one gets     ε u 1 − u(x) < ε , (6.5) |x 2 u  (x)| < ,   3 n α1 3  

1  1  ε 1  u < 0 < x ≤ . n n  3 n Hence, for every x ∈ [0, 1/n]:        1  1   1  1   1   u (6.6) |u(x) − u n (x)| ≤ u(x) − u + u + <ε n  n n  2n 2  n  and (6.7)

|Aα u(x) − Aα u n (x)| ≤ |Aα u(x)| + α1

  1   1  u < ε. n2  n 

Therefore, for every n > ν, u n − u < ε and Aα u n − Aα u < ε. Fix n > ν. By Remark 6.1, there exists M > 1 such that (6.8) |u(x)| <

ε , 80α1

|xu  (x)| <

ε , 80α1

|x 2 u  (x)| <

ε 80α1

Set v(x) = u n (x) if x ∈ [0, M]: v(x) =

2u  (M) + u  (M)M (x − M)4 4M 3 3u  (M) + 2u  (M) u  (M) 3 (x − M)2 + (x − M) + 3M 2 2 + u  (M)(x − M) + u(M)

if x ∈ [M, 2M] and v(x) = 12 u  (M)M +

1 M 2 u  (M) 12

+ u(M) if x ≥ 2M.

(x ≥ M).

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A straightforward calculation shows that v ∈ C 2 ([0, +∞]) and is constant in a neighborhood of +∞. Hence v ∈ D0 . Moreover, for every x ∈ [0, M]: (6.9)

|v(x) − u(x)| = |u n (x) − u(x)| < ε,

|Aα v(x) − Aα u(x)| < ε.

If x ∈ [M, 2M], then M  M 2  2M 2  |u (M)| + |u (M)| + M|u  (M)| + |u (M)| 2 4 3 + M|u  (M)| + M|u  (M)| + |u(M)| < ε,

|v(x) − u(x)| ≤ |u(x)| +

and, analogously, (6.10)

|Aα v(x) − Aα u(x)| ≤ 4α1 M 2 |v  (x)| + α1 |x 2 u  (x)| < ε.

Finally, if x ≥ 2M, then |v(x) − u(x)| ≤ |v(x)| + |u(x)| < 4ε

(6.11) and

|Aα v(x) − Aα u(x)| = |Aα u(x)| < ε.

(6.12)

Therefore, v − u ∞ < ε and Aα v − Aα u ∞ < ε. Theorem 6.3. Let α ∈ Cb ([0, +∞[) such that infx≥0 α(x) > 0. Let (Tα (t))t≥0 be the contraction C0 -semigroup of operators on C([0, +∞]) generated by (Aα , D(Aα )). Then for every f ∈ C([0, +∞]) and t ≥ 0 and for every sequence (k(n))n of positive integer numbers such that limn→∞ k(n)/n = t: k(n) Tα (t) f = lim Pα,n f

(6.13)

n→∞

uniformly on [0, +∞[. Proof. By Theorem 4.15 in [8], Aα generates a contraction semigroup in C([0, +∞]). Thus, since D0 is a core for Aα , we have that D0 and (λI − A)(D0 ) (λ > 0) are dense in C([0, +∞]). Consider the operator (Z α , D(Z α )), where D(Z α ) := { f ∈ C([0, +∞]) : lim n(Pα,n ( f ) − f ) ∈ C([0, +∞])} n→∞

and Z α ( f ) = limn→∞ n(Pα,n ( f ) − f ). Clearly D0 ⊆ D(Z α ), hence by Propositions 5.2 and 6.2, Z α satisfies all the assumptions of Trotter’s theorem (see [17] or, e.g., [15]). Therefore it is closable and its closure (A1 , D(A1 )) generates a contraction C0 -semigroup of operators on C([0, +∞]) which verifies (6.13). Since D0 is a core for (Aα , D(Aα )) and Aα = Z α on D0 , we have that D(A1 ) = D(Aα ) and Aα = A1 .

Differential Operators with Second-Order Degeneracy

457

7. The Operator Aα,m on E m0 In this section we consider the same operator Aα acting on its maximal domain in the spaces E m0 . Throughout this section let α ∈ Cb ([0, +∞[ such that infx≥0 α(x) > 0. For every m ∈ N, consider the differential operator (Aα,m , D(Aα,m )), where D(Aα,m ) := {u ∈ E m0 ∩ C 2 (]0, +∞[) : αe2 u  ∈ E m0 }, Aα,m u(x) := αe2 u 

(u ∈ D(Aα,m )).

Remark 7.1. Also in this case, by [13, Corollary after Theorem 13.1], we have that, for every u ∈ D(Aα,m ), limx→0+ x 2 u  (x) = 0. Set D1 := {u ∈ C 2 ([0, +∞[) : u is constant in a neighborhood of + ∞},
 D2 := u ∈ C 2 ([0, +∞[) : lim u  (x) = 0 , x→+∞

Dm := {u ∈ C 2 ([0, +∞[) : u  ∈ Cb ([0, +∞[)}

(m ≥ 3).

Observe that Dm ⊆ E m0 for every m ≥ 1 (see, e.g., [6, Remarks after Proposition 2.1]). Proposition 7.2. (1) (2) (3) (4)

For every m ≥ 1:

(Aα,m , D(Aα,m )) is closed; Dm is dense in E m0 ; Dm is a core for (Aα,m , D(Aα,m )); and (λI − Aα,m )(Dm ) is dense in E m0 for every λ > 0.

Proof. (1) Analogously to the proof of [6, Proposition 2.1]. (2) If m ≥ 2, observe that Dm contains the linear hull of the set {exp(−a·) : a > 0}, hence it is dense in E m0 by [7, Lemma 4.9]. If m = 1, observe that D1 = D0 is dense in C([0, +∞]), hence it is dense in E 10 . If m ≥ 2, the proof of (3) and (4) follows, with minor modifications, as in [6, Proposition 2.2 (2), (3)]. Let us prove that D1 is a core for (Aα,1 , D(Aα,1 )). Assume w.l.o.g. that α ∞ ≥ 1. Fix u ∈ D(Aα,1 ). Then limx→+∞ xu  (x) = 0. Moreover, limx→+∞ u  (x) = 0. Indeed, fix ε > 0. Then there exists M > 0 such that |x 2 u  (x)| < ε for every x > M. Hence, for every x > M, |xu  (x) − Mu  (M) − u(x) + u(M)| < ε(x − M). Therefore |u  (x)| ≤ ε + Then lim supx→+∞ |u  (x)| ≤ ε.

M|u  (M)| |u(M)| u(x) + + . x x x

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E. M. Mangino

Given ε > 0 choose M > 1 such that for every x ≥ M: 1 |u (x)| < ε, 80 α ∞ 

1 |xu (x)| < ε, 80 α ∞ 

   u(x)  1    1 + x  < 80 α ε, ∞

and define v as in the proof of Proposition 6.2. Then v ∈ D1 and v − u 1 < ε and

Aα,1 v − Aα,1 u 1 < ε. Finally, observe that for every λ > 0, (λI − Aα )(D1 ) is dense in C([0, +∞]), hence it is dense in E m0 . Theorem 7.3. Let α ∈ Cb ([0, +∞[) such that infx≥0 α(x) > 0 and let m ≥ 1. Then (Aα,m , D(Aα,m )) generates a positive C0 -semigroup (Tα,m (t))t≥0 of operators on E m0 such that: (1) Tα,m (t) ≤ exp( α ∞ Cm t) for every t ≥ 0, where Cm is the constant appearing in (2.7); in particular, (Tα,1 )t≥0 is a contraction semigroup; and (2) for every f ∈ E m0 and t ≥ 0 and for every sequence (k(n))n of positive integer numbers such that limn→∞ k(n)/n = t: k(n) Tα,m (t) f = lim Pα,n f

(7.1)

n→∞

in E m0 .

In particular, the limit holds uniformly on the compact subsets of [0, +∞[. Proof. 2.2]).

The proof is similar to that of Theorem 6.3 (see also the proof of [6, Theorem

Remark 7.4. We point out that the generation result in Theorem 7.3 can also be achieved by applying Theorem 3.2 in [2]. Here we also present a representation of the semigroup generated by Aα,m . Remark 7.5. It is worth noting that by the representation (7.1), we have that Tα,m+1 (t) ( f ) = Tα,m (t)( f ) for every t ≥ 0 and for every f ∈ E m0 . This helps in calculating more carefully the norm of T1,m (t) when m ≥ 2. For every n ∈ N, we have n

m−1 + m m   2h m2 m 2 n = x Pn (em )(x) = x m . 1+ n n  h=1 2 Let (k(n))n be a sequence of positive integer numbers such that limn→∞ k(n)/n = t. Then

m−1  2h k(n) Pnk(n) (em )(x) = x m 1+ n h=1 and lim Pnk(n) (em )(x) = x m

n→∞

m−1  h=2

e2ht = x m em(m−1)t .

Differential Operators with Second-Order Degeneracy

459

Therefore, for every f ∈ E m0 : (7.2)

|T1,m (t)( f )(x)| = |T1,m+1 ( f )(x)| ≤ f m |T1,m+1 (t)(1 + em )(x)| = f m (1 + x m em(m−1)t ),

hence

T1,m (t) m ≤ em(m−1)t .

(7.3)

In the next sections, for the sake of brevity, we denote by E 00 the space C([0, +∞]), by Aα,0 the operator Aα , by Tα,0 the semigroup generated by Aα on C([0, +∞]), and we set C0 := 0. 8. Some Shape-Preserving Properties This section is devoted to the study of the behavior of the operators Pα,n and of the semigroups of operators (Tα,m (t))t≥0 when acting on convex, monotone, or H¨older continuous functions. Proposition 8.1.

Let k ∈ ]0, +∞[ and f ∈ E.

(1) The following conditions are equivalent: (i) f is increasing; (ii) Pk1,n ( f ) is increasing for every n ∈ N; and (iii) Tk1,m (t)( f ) is increasing for every t ≥ 0. (2) The following conditions are equivalent: (i) f is convex; (ii) Pk1,n ( f ) is convex for every n ∈ N; and (iii) Tk1,m (t)( f ) is convex for every t ≥ 0. Proof. This follows immediately from the representation of the operators Pα,n and from (6.13), (7.1). When f is convex, something more can be said, even in the case in which α is not constant. Proposition 8.2. Let α ∈ Cb ([0, +∞[) such that infx≥0 α(x) > 0. Let m ≥ 0 and let f ∈ E m0 . The following conditions are equivalent: (i) (ii) (iii) (iv) (v) (vi) Proof.

f is convex; Pα,n+1 ( f ) ≤ Pα,n ( f ) for every n ∈ N; f ≤ Pα,n ( f ) for every n ∈ N; f ≤ Tα,m (t)( f ) for every t ≥ 0; Tα,m (t)( f ) is convex for every t ≥ 0; and Tα,m (s)( f ) ≤ Tα,m (t)( f ) for every 0 ≤ s ≤ t. The proof runs analogously to that of Theorem 3.1 in [6].

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E. M. Mangino

Finally, consider the space Lip K (r ) (K > 0, r ∈ ]0, 1]) of all functions f ∈ C([0, +∞[) such that | f (x) − f (y)| ≤ K |x − y|r Proposition 8.3.

(x, y ≥ 0).

Let k ∈ ]0, +∞[. Then:

(1) For every n ∈ N, Pk1,n (Lip K (r ) ∩ E m0 ) ⊆ Lip Hn (r ) where n +r (2k)r  2kn Hn = K . r n  2k In particular, Pk1,n (Lip K (1)) ⊆ Lip K (1). (2) For every t ≥ 0, Tk1,m (t)(Lip K (r )) ∩ E m0 ) ⊆ Lip H (r ), where H = supn∈N Hn . In particular, Tk1,m (t)(Lip K (1) ∩ E m0 ) ⊆ Lip K (1). Proof. The assertion follows immediately from the explicit representation of the operators Pα,n . 9. Saturation Properties Throughout this section assume that α ∈ Cb ([0, +∞[) with α0 := infx≥0 α(x) > 0. Proposition 9.1.

Let m ≥ 0 and let f ∈ E m0 . The following conditions are equivalent:

(i) Pα,n ( f ) − f m = o(1/n) as n → ∞; (ii) Tα,m (t)( f ) − f m = o(t) as t → 0+ ; (iii) f ∈ C 2 ([0, +∞[) and f  = 0. If m = 0, 1 then (i) is equivalent to: (iii) f is constant. If m ≥ 2, then (iii) is equivalent to: (iii) There exist a, b > 0 such that f = ae1 + b. Proof. The implications (iii) ⇒ (i) and (iii) ⇒ (ii) are clear. (i) ⇒ (ii) Let t ≥ 0 and let (k(n))n∈N be an increasing sequence of natural numbers such that limn→∞ k(n)/n = t. Let βn := n Pα,n ( f ) − f m for every n ∈ N. Then k(n)

Pα,n ( f ) − f m ≤

k(n)−1 

h

Pα,n

Em0 · Pα,n ( f ) − f m

h=0

≤ βn

k(n)−1  h=0

α ∞ Cm 1+ n

h ≤ βn

k(n) Cm k(n)/n . e n

Therefore k(n)

Tα,m ( f ) − f m = lim Pα,n ( f ) − f m = 0, n→∞

Differential Operators with Second-Order Degeneracy

461

hence Tα,m (t) f = f , f ∈ D(Aα,m ), and u(t, x) := Tα,m (t) f (x) = f (x) is a solution of u t = α(x)x 2 u x x (t, x), that is, f  (x) = 0 for every x > 0. Since f is continuous at 0, we get f (x) = ax + b for some a, b ∈ R. (ii) ⇒ (iii) Simply observe that (ii) is equivalent to, say, that f ∈ D(Aα,m ) and Aα,m ( f ) = 0. The assertion follows as in the previous implication. Lemma 9.2. For every m ∈ N0 , there exists Mm > 0 such that for every x ≥ 0 and for every n ∈ N: Pα,n (ψx2 (1 + em ))(x) x2 ≤ Mm . m 1+x n Proof. If m = 0 the assertion is clear. Let m > 0. Fix x ≥ 0 and n ∈ N. By applying H¨older’s inequality to the positive operator µx : E → R, µx ( f ) = Pα,n ( f )(x), we get 2α(x)x 2 √ 2α(x)x 2 + Pα,n (ψx4 )(x)Pα,n (e2m )(x) ≤ n n 



n − 1 6α(x) α(x) 12α(x) 2 2m + C 2m ) x + x2 + (1 + x + 12α(x) 2m n 3 c2 n cn 3 n

Pα,n (ψx2 (1 + em ))(x) ≤

≤ Mm

x2 (1 + x m ). n

Lemma 9.3. For every m ≥ 0, there exists Hm > 0 such that for every f ∈ D(Am ) and for every n ∈ N, we have

n(Pα,n ( f ) − f ) m ≤ Hm ( f m + Aα,m f m ). Proof. If m = 0, the assertion has been proved in [16, Proof of Theorem 1, Part 3]. For m ≥ 1, we adopt a suitable modification of the idea of Totik. First of all, observe that the operators Pα,n ( f ) are also well-defined for functions g : [0, +∞[ → R such that supx≥0 |g(x)|/1 + x m < +∞ and continuous with the exception of a finite number of points. Step 1: For every x > 0, set Ix := ]x/2, 3x/2[ and let  t v (1 + u m )  du dv, u2 h x (t) := (9.1) x x  0

t ∈ Ix , t∈ / Ix .

Then, for every t ≥ 0, (9.2)

4 0 ≤ h x (t) ≤ 2 x ≤ 4( 32 )m

t

x

v x

4 (1 + u ) du dv ≤ 2 x

1 + xm 2 ψx (t), x2

m



1+

3x 2

m ψx2 (t)

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E. M. Mangino

hence 0 ≤ Pα,n (h x )(x) ≤ 4( 32 )m

(9.3)

1 + xm Pα,n (ψx2 )(x) x2

1 ≤ 8 α ∞ ( 32 )m (1 + x m ). n Moreover,

h x m = sup

(9.4)

t∈I x

m 2 h x (t) 1 3 m1+x x ≤ 4( ) ≤ 3m . 2 m 2 1+t x 4 1 + (x/2)m

For every ϕ : [0, +∞[ → R, set ϕ(x, t) :=




ϕ(t),

t ∈ Ix ,

0,

t∈ / Ix .

Since |t − x| > 12 x for every t ∈ / Ix , we have |ϕ(x, t) − ϕ(t)| ≤ |ϕ(t)|

4ψx2 (t) , x2

hence |Pα,n (ϕ(x, ·) − ϕ)(x)| ≤

(9.5)

4 Pα,n (|ϕ|ψx2 )(x). x2

Step 2: Fix x > 0. Let g : [0, +∞[ → R such that supx≥0 |g(x)|/(1 + x m ) < +∞, continuous with the exception of a finite number of points and convex in Ix . Then there exists γ ∈ R such that g(t) ≥ g(x) + γ (t − x)

(9.6)

(t ∈ Ix ).

Set g1 (t) := g(t) − g(x) for every t ≥ 0. Then g1 (x, ·) ≥ γ ψx (x, ·) in [0, +∞[. We have 4 4

g m Pα,n (ψx2 (1 + em ))(x) ≥ Pα,n (g)(x) + 2 Pα,n (|g|ψx2 )(x) x2 x 4 = Pα,n (g)(x) + 2 Pα,n (|g1 + g(x)|ψx2 )(x) x 4 4 ≥ Pα,n (g1 )(x) + g(x) + 2 Pα,n (|g1 |ψx2 )(x) − 2 |g(x)|Pα,n (ψx2 )(x) x x 8α(x) ≥ Pα,n (g1 )(x) + g(x) + Pα,n (−g1 + g1 (x, ·))(x) − |g(x)| n 8α(x) = Pα,n (g1 (x, ·))(x) + g(x) − |g(x)| n 8α(x) ≥ γ Pα,n (ψx (x, ·))(x) + g(x) − |g(x)|. n

Pα,n (g)(x) +

Note that −2 g m ≤

g( 32 x) − g(x) g(x) − g( 12 x) γx ≤ ≤ ≤ 2 g m , 1 + xm 2(1 + x m ) 1 + xm

Differential Operators with Second-Order Degeneracy

463

hence |γ | ≤ 4 g m [(1 + x m )/x]. Therefore |γPα,n (ψx (x, ·))(x)| ≤ 4 g m

1 + xm |Pα,n (ψx (x, ·))(x)| x

1 + xm |Pα,n (ψx (x, ·))(x) − Pα,n (ψx )(x)| x 1 + xm 1 + xm ≤ 4 g m Pα,n (|ψx (x, ·) − ψx |)(x) ≤ 8 g m Pα,n (ψx2 )(x) x x2 1 + xm = 16 g m α(x). n

= 4 g m

Then 4

g m Pα,n (ψx2 (1 + em ))(x) x2 2cα(x) (1 + x m )α(x) ≥ g(x) − 8 |g(x)| − 16 g m n n α(x) ≥ g(x) − 24 g m (1 + x m ) , n

Pα,n (g)(x) +

and we get, finally, 4

g m Pα,n (ψx2 (1 + em ))(x) x2 α(x)(1 + x m ) − 24 g m . n

Pα,n (g)(x) ≥ g(x) −

(9.7)

Step 3: Set M := Aα,m f m and define f±

(9.8)

  M h (t) ± f (t), t ∈ I , x x := α0  ± f (t), t∈ / Ix .

Then

f ± m ≤ f m +

M M3m

h x m ≤ f m + =: K f α0 α0

and f ± is convex in Ix . Indeed, for every t ∈ Ix : ( f ± ) (t) =

M(1 + t m ) α(t)| f  (t)|  ± f (t) ≥ ± f  (t) ≥ 0. t 2 2cα0 α0

By (9.7), we get 4 α(x) K f Pα,n (ψx (1 + em ))(x) − 24K f (1 + x m ) , 2 x n 4 α(x) Pα,n ( f − )(x) ≥ − f (x) − 2 K f Pα,n (ψx2 (1 + em ))(x) − 24K f (1 + x m ) . x n

Pα,n ( f + )(x) ≥ f (x) −

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Hence Pα,n ( f )(x) − f (x) ≤ Pα,n ( f )(x) + Pα,n ( f − )(x) +

4 K f Pα,n (ψx2 (1 + em ))(x) x2

α(x) n M 4 ≤ Pα,n (h x )(x) + 2 K f Pα,n (ψx2 (1 + em ))(x) α0 x m α(x) + 24K f (1 + x ) , n + 24K f (1 + x m )t

and Pα,n ( f )(x) − f (x) ≥ Pα,n ( f )(x) − Pα,n ( f + )(x) 4 α(x) − 2 K f Pα,n (ψx2 (1 + em ))(x) − 24K f (1 + x m ) x n 4 M 2 ≥ − Pα,n (h x )(x) − 2 K f Pα,n (ψx (1 + em ))(x) α0 x α(x) − 24K f (1 + x m ) . n Therefore, |Pα,n ( f )(x) − f (x)| Pα,n (ψx2 (1 + em ))(x) α(x) M Pα,n (h x )(x) 4 ≤ + 2 Kf + 24K f m m m 1+x α0 1 + x x 1+x n



m 1 M 3 24 α m ∞ 3 m ≤ (2)

Aα,m f m + f m +

Aα,m f m + nα0 2nα0 n n ≤ Hm ( f m + Aα,m f m ), where Hm depends only on α and m. We are now ready to characterize the saturation class of the sequence (Pα,n )n∈N and the Favard class of the semigroups (Tα,m (t))t≥0 . Proposition 9.4.

Let m ≥ 0 and f ∈ E m0 . The following conditions are equivalent:

(i) Pα,n ( f ) − f m = O(1/n) as n → ∞; (ii) Tα,m (t) f − f m = O(t) as t → 0+ ; (iii) there exists a sequence (u n )n∈N in D(Aα,m ) such that supn∈N Aα,m u n m < +∞ and limn→∞ u n − f m = 0; and (iv) ( f (· + 2h) − 2 f (· + h) + f )e2 m = O(h 2 ) as h → 0+ . Proof. (i) ⇒ (ii) See the proof of (i) ⇒ (iii) in Proposition 9.1. (ii) ⇒ (iii) Assume that there exists M > 0 such that

Tα,m (t)( f ) − f m ≤ Mt

(t ∈ [0, δ]).

Differential Operators with Second-Order Degeneracy

Set u t :=

1 t



t

465

Tα,m (s) f ds.

0

Then u t ∈ D(Aα,m ), Aα,m u t m = (1/t) Tα,m (t) f − f m ≤ M, and limt→0+ u t = f in E m0 . (iii) ⇒ (iv) Observe that for every n ∈ N and for every x ≥ 0: x2 |u n (x + 2h) − 2u n (x + h) + u n (x)| 1 + xm

h h x2 = u  (x + s + t) ds dt 1 + xm 0 0 n

h h x2 1 + (x + s + t)m ≤M ds dt 1 + xm 0 0 (x + s + t)2 ≤ Mh 2

1 + (x + 2)m ≤ M1 h 2 . 1 + xm

We get the result by passing to the limit as n tends to infinity. (iv) ⇒ (iii) For every h ∈ ]0, 1], set 1 f h (x) := 2 h



h 0



h

f (x + s + t) ds dt

(x ≥ 0).

0

If f ∈ C([0, +∞]) and limx→+∞ f (x) = l, then it is easy to prove that f h is continuous on [0, +∞[ and converges to l as x tends to +∞. If f ∈ E m0 , with m ≥ 1, then f h ∈ C([0, +∞[) and limx→+∞ f h (x)/(1 + x m ) = 0. Indeed, given ε > 0, there exists M > 0 such that | f (x)/(1+ x m )| < ε for every x > M. Hence,  

h h  f h (x)  ε 1 + (x + 2)m  ≤ (1 + (x + s + t)m ) ds dt < ε ≤ 2m ε.  1 + xm  1 + xm m 1 + x 0 0 Therefore for every m ≥ 0, if f ∈ E m0 , then f h ∈ E m0 . Moreover, f h ∈ C 2 ([0, +∞[), = f (x + 2h) − 2 f (x + h) + f (x), hence f h ∈ D(Aα,m ). Finally, observe that limh→0+ f h = f in E m0 and that Aα,m f h m ≤ M by assumption. (iii) ⇒ (i) Let f ∈ E m0 such that (iii) holds. Then, by Lemma 9.3,

f h (x)

n(Pα,n (u k ) − u k ) m ≤ Hm Aα,m u k m + u k m . By taking the limit as k → ∞, we get n(Pα,n ( f ) − f ) m ≤ M + f m . We remember that the saturation problem in C([0, +∞]) for a large class of operators, among which the operators Pn , has been already solved by Totik in [16]. Acknowledgments. The author thanks Professor F. Altomare for turning the author’s attention to Post-Widder operators and for his encouragement.

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E. M. Mangino Dipartimento di Matematica “E. De Giorgi” Universit`a degli Studi di Lecce I-73100 Lecce Italy [email protected]