2 THE DIVERGENCE PHENOMENA OF INTERPOLATION TYPE OPERATORS IN Lp SPACE BY T

VOL. LXIII 1992 FASC. 2

THE DIVERGENCE PHENOMENA OF

INTERPOLATION TYPE OPERATORS IN L^p SPACE

BY

T. F. X I E (HANGZHOU) AND

S. P. Z H O U (HALIFAX, NOVA SCOTIA)

Let L^p_[−1,1], 1 ≤ p < ∞, be the class of real p-integrable functions on [−1, 1], L^∞_[−1,1]= C_[−1,1] the class of all real continuous functions on [−1, 1].

Denote by C_[−1,1]^r the space of real functions on [−1, 1] which have r contin- uous derivatives, and by C_[−1,1]^∞ the space of real functions on [−1, 1] which are infinitely differentiable.

For f ∈ L^p_[−1,1], let En(f )p be the best approximation to f by polyno- mials of degree n in L^p space.

Our works [1], [5] concern the divergence phenomena of trigonometric Lagrange interpolation approximations in comparison with best approxima- tions in L^p space; the paper [1] contains the following theorem:

Let 1 ≤ p < ∞. Suppose that {Xn}, Xn = {xn,j}²ⁿ_j=0, is a given sequence of real distinct (by a 6= b we mean that a 6≡ b (mod 2π)) nodes and {λn} is any given positive decreasing sequence. Then there exists an infinitely differentiable function f with period 2π such that

lim sup

n→∞

kf − L^X_n(f )kL^p_[0,2π]

λ⁻¹n E_n^∗(f )p

> 0 ,

where L^X_n(f, x) is the n-th trigonometric Lagrange interpolating polynomial of f (x) with nodes Xn and E^∗_n(f )pis the best approximation to f by trigono- metric polynomials of degree n.

Here and throughout, we write kf k_L^p

[a,b] =

 R^b

a

|f (x)|^pdx

1/p

, 1 ≤ p < ∞ , kf k_[a,b] = kf kL^∞_[a,b] = max

a≤x≤b|f (x)| ,

1991 Mathematics Subject Classification: Primary 41A05.

Key words and phrases: approximation, interpolation type operator, divergence phe- nomenon, real distinct nodes, L^p space.

kf k_L^p = kf k_L^p

[−1,1], 1 ≤ p < ∞ .

In spite of this counterexample, there do exist several positive results in this direction. For example, in [2], V. P. Motorny˘ı discussed the rate of convergence of the Ln(f, x) to f (x) in L¹, expressed in terms of the sequence of best approximations of the function in L¹; he proved that if f is absolutely continuous with period 2π, f⁰ ∈ L¹_[0,2π], and E_n⁰(f⁰)1 is the best approximation to f⁰ by trigonometric polynomials of degree n with mean value zero in L¹, then

kf − L_n(f )k_L¹

[0,2π]= O(n⁻¹log nE_n⁰(f⁰)1) ,

where Ln(f, x) is the nth trigonometric Lagrange interpolating polynomial to f with nodes xn,j = 2jπ/(2n + 1) for j = 0, 1, . . . , 2n.

In L^pspace for 1 < p < ∞, K. I. Oskolkov [3] showed the following better estimate. Let f be absolutely continuous with period 2π, and f⁰ ∈ L^p_[0,2π]

for 1 < p < ∞; then

kf − L_n(f )k_L^p

[0,2π]= O(n⁻¹E^∗_n(f⁰)p) .

One might ask what happens to other interpolation operators? More generally, to “interpolation type” operators? In this paper, by “interpolation type” operators we mean operators I_n^r(f, X, x) of the form

I_n^r(f, X, x) =

r

X

k=0 nk

X

j=1

f^(k)(x^k_n,j)l_n,j^k (x) for f ∈ C_[−1,1]^r , where Xn = Sr

k=0{x^k_n,j}ⁿ_j=1^k is a sequence of real nodes within [−1, 1], {x^r_n,j} 6⊆ {−1, 1},

r

X

k=0

nj = n + 1 ,

and l^k_n,j(x), j = 1, . . . , nk, k = 0, 1, . . . , r, are polynomials of degree not greater than n. Furthermore, if f is a polynomial of degree ≤ n, then I_n^r(f, X, x) = f (x). In particular, if r = 0,

l_n,j⁰ (x) = Ωn(x)

Ω⁰_n(xn,j)(x − xn,j), Ωn(x) =

n+1

Y

k=1

(x − xn,k) ,

then I_n^r(f, X, x) becomes the nth Lagrange interpolating polynomial with nodes {xn,j}ⁿ⁺¹_j=1; if r = 1,

l⁰_m,j(x) =



1 −Ω_n⁰⁰(xn,j)

Ω_n⁰(xn,j)(x − xn,j)

 Ωn(x) Ω_n⁰(xn,j)(x − xn,j)

2

,

l_m,j¹ (x) = (x − xn,j)

 Ωn(x) Ω⁰_n(xn,j)(x − xn,j)

2

,

then I_n^r(f, X, x) becomes the Hermite–Fej´er interpolating polynomial of de- gree m = 2n + 1 with nodes {xn,j}ⁿ⁺¹_j=1; and so on.

In the present paper we refine the idea used in [1] and prove the following Theorem. Let 1 ≤ p < ∞. Suppose that {Xn} is a given sequence of real distinct nodes within [−1, 1], and {λn} is any given positive decreasing sequence. Then there exists a function f ∈ C_[−1,1]^∞ such that

lim sup

n→∞

kf − I_n^r(f, X)kL^p

λ⁻¹n En(f^(r))p

> 0 .

P r o o f. Without loss of generality assume that −1 < x^r_n,1 < 1. Fix n.

Considering the nonnegative function

gn(x) = (1 + x)(1 − x)^{(1−xrn,1)/(1+xrn,1)},

we note that gn(x) strictly increases on [−1, x^r_n,1] and strictly decreases on [x^r_n,1, 1]; accordingly we can choose a sufficiently large natural number Tn

such that for all x in [−1, 1] \ (x^r_n,1− δ_n, x^r_n,1+ δn) and all m ≥ Tn, g^m_n(x) ≤ 1

2n max

1≤j≤nr

kl^r_n,jk⁻¹_Lpηng_n^m(x^r_n,1) , where

δn := min

2≤j≤nr

|x^r_n,j − x^r_n,1| , ηn:= kl^r_n,1k_L^p. In particular, for all 2 ≤ j ≤ nr,

(1) g_n^m(x^r_n,j) ≤ 1 2n max

1≤j≤nr

kl_n,j^r k⁻¹_Lpηng^m_n(x^r_n,1) . Let Nn be a natural number not less than Tn, and

N_n^∗= 1 − x^r_n,1 1 + x^r_n,1Nn. Write

hn(x) = g_n^−Nn(x^r_n,1)

x

R

−1

dt1 t1

R

−1

dt2. . .

tr−1

R

−1

g_n^Nn(tr) dtr. Then hn ∈ C_[−1,1]^r and we clearly have

(2) kh^(r)_n k = h^(r)_n (x^r_n,1) = 1 , and for 2 ≤ j ≤ nr, by (1),

(3) 0 ≤ h^(r)_n (x^r_n,j) ≤ 1

2nηn max

1≤j≤nr

kl_n,j^r k⁻¹_Lp .

On the other hand, a calculation gives

kg^N_nⁿk_L^p = 2^Nn+Nn∗+1/p Γ (Nnp + 1)Γ (N_n^∗p + 1) Γ (Nnp + N_n^∗p + 2)

1/p

≤ Cg^N_nⁿ(x^r_n,1)N_n^−1/(2p),

where here and throughout the paper, C always indicates a positive constant independent of n which may have different values in different places. So (4) kh^(r)_n k_L^p ≤ CN_n^−1/(2p),

and for 0 ≤ s ≤ r − 1,

(5) kh^(s)_n k ≤ 2^r−1kh^(r)_n k_L¹ ≤ CN_n^−1/2. We now establish that

(6) kh_n− I_n^r(hn, X)kL^p ≥ ¹₂ηn− Cn%_nNn^−1/(2p), where

%n:= max

1≤j≤nk, 0≤k≤r−1{1, kl_n,j^k k_L^p} . In fact, from the definition,

I_n^r(hn, X, x) =

r

X

k=0 nk

X

j=1

h^(k)_n (x^k_n,j)l_n,j^k (x) .

By (2), (3) and (5),

kh_n− I_n^r(hn, X)kL^p ≥ η_n−

nr

X

j=2

h^(r)_n (x^r_n,j)kl^r_n,jk_L^p

−

r−1

X

k=0 nk

X

j=1

h^(k)_n (x^k_n,j)kl^k_n,jk_L^p− kh_nk_L^p

≥ ¹₂ηn− Cn%_nNn^−1/(2p),

thus (6) is proved. Without loss suppose that λn ≤ 1. Now choose Nn = [λ^−2p_n n^4p(4%^2p_n η^−2p_n + 1) + Tn] .

Then for sufficiently large n, (6) becomes

(7) khn− I_n^r(hn, X)kL^p ≥ ¹₄ηn, and (4) becomes

(8) kh^(r)_n k_L^p ≤ Cλ_nηn.

Because hn ∈ C_[−1,1]^r , select an algebraic polynomial f_n^∗with sufficiently large degree Mn ≥ n such that (cf., for example, A. F. Timan [4]) for

0 ≤ s ≤ r,

(9) kh^(s)_n − (f_n^∗)^(s)k ≤ n⁻¹ηnλn(1 + kI_n^rk)⁻¹, where for bounded operators B on C[−1,1],

kBk := sup

f ∈C_[−1,1], kf k=1

{kBf k} . Hence by (8) and (9),

k(f_n^∗)^(r)kL^p ≤ k(f_n^∗)^(r)− h^(r)_n k + kh^(r)_n kL^p

≤ n⁻¹ηnλn+ Cηnλn ≤ Cη_nλn, and similarly, from (7) and (9),

kf_n^∗− I_n^r(f_n^∗, X)kL^p ≥ khn− I_n^r(hn, X)kL^p− kf_n^∗− hnk

− kI_n^r(hn, X) − I_n^r(f_n^∗, X)k

≥ Cηn− n⁻¹ηnλn(kI_n^rk + 1)⁻¹(1 + kI_n^rk) ≥ Cηn

for large enough n. Set fn(x) = η_n⁻¹f_n^∗(x); we thus have kf_n^(r)k_L^p = O(λn) ,

(10)

kfn− I_n^r(fn, X)kL^p ≥ C . (11)

Select a sequence {mj} by induction. Let m1= 4r. After mj, choose (12) mj+1= [(M_m^∗_j)²λ^−1/m_mj ^j(kI_m^r_jk + 1) + m_j + 1] ,

where M_n^∗= Mn(η^2/nn + 1). Define f (x) =

∞

X

j=1

(M_m^∗_j)^−mjfmj(x) .

Clearly f ∈ C_[−1,1]^∞ (since fmj is a polynomial of degree Mmj) in view of (2) and (9). Together with (12), (11) implies that

kf − I_m^r_j(f, X)kL^p ≥ (M_m^∗_j)^−mjkf_mj − I_m^r_j(fmj, X)kL^p

− C(kI_m^r_jk + 1)

∞

X

k=j+1

(M_m^∗_k)^−mkkf_mkk

≥ C(M_m^∗

j)^−mj − C(M_m^∗

j+1)^−mj+1/2 ≥ C(M_m^∗

j)^−mj. At the same time, by (10) and (12),

Emj(f^(r))p= O



(M_m^∗_j)^−mjkf_m^(r)_jk_L^p +

∞

X

k=j+1

(M_m^∗_k)^−mkkf_m^(r)_kk

= O((M_m^∗_j)^−mjλmj + (M_m^∗_j+1)^−mj+1/2) = O((M_m^∗_j)^−mjλmj) .

Altogether,

kf − I_m^r_j(f, X)kL^p

λ⁻¹mjEmj(f^(r))p

≥ C > 0 , which is the required result.

R e m a r k. Considering the Theorem together with Motorny˘ı’s and Os- kolkov’s results, we might have reasons to guess that there might be some connections between the interpolation approximation rate of a given func- tion with some kinds of nodes in L^p space and the best approximation rate of a higher derivative of that function in L^p.

REFERENCES

[1] P. B. B o r w e i n, T. F. X i e and S. P. Z h o u, On approximation by trigonometric Lagrange interpolating polynomials II , Bull. Austral. Math. Soc. 45 (2) (1992), in print.

[2] V. P. M o t o r n y˘ı, Approximation of periodic functions by interpolation polynomials in L1, Ukrain. Math. J. 42 (1990), 690–693.

[3] K. I. O s k o l k o v, Inequalities of the “large sieve” type and applications to problems of trigonometric approximation, Analysis Math. 12 (1986), 143–166.

[4] A. F. T i m a n, Theory of Approximation of Functions of a Real Variable, Macmillan, New York 1963.

[5] T. F. X i e and S. P. Z h o u, On approximation by trigonometric Lagrange interpolat- ing polynomials, Bull. Austral. Math. Soc. 40 (1989), 425–428.

DEPARTMENT OF MATHEMATICS DEPARTMENT OF MATHEMATICS, HANGZHOU UNIVERSITY STATISTICS AND COMPUTING SCIENCE

HANGZHOU, ZHEJIANG, CHINA DALHOUSIE UNIVERSITY

HALIFAX, NOVA SCOTIA CANADA B3H 3J5

Re¸cu par la R´edaction le 13.9.1991 ; en version modifi´ee le 15.2.1992