CARDINAL SPLINE INTERPOLATION FROM H1(Z) - American ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 9, Pages 2597–2601 S 0002-9939(00)05290-4 Article electronically published on February 21, 2000

CARDINAL SPLINE INTERPOLATION FROM H 1 (Z) TO L1 (R) FANG GENSUN (Communicated by J. Marshall Ash) Abstract. Let H 1 (Z) be the discrete Hardy space, consisting of those sequences y = {yj }j∈Z ∈ lp (Z), such that Hy = {Hyj } ∈ l1 (Z), where Hyj = P (k − j)−1 yj , j ∈ Z, is the discrete Hilbert transform of y. For a sequence k6=j

y = {yj } ∈ l1 (Z), let Lm y(x) ∈ Lp (R) be the unique cardinal spline of degree m − 1 interpolating to y at the integers. The norm of this operator, kLm k1 = sup{kLm ykL(R) kykl(Z) }, is called a Lebesgue constant from l1 (Z) to L1 (R), and it was proved that sup {kLm k1 } = ∞. m

It is proved in this paper that π π sup kLm yk1(R) (kykl(Z) + k{H(−1)j yj }kl(Z) ) 6 1 + 1+ . 2 3 m

1. Introduction Denote the classical Lebesgue space on R by Lp (R), 1 6 p 6 ∞, and let k · kp(R) denote its norm. For a natural number m, the space Sm,p (R) = {s} of cardinal splines of degree m − 1 is taken to consist of those functions satisfying: (i) s ∈ C m−2 (R), (ii) kskp(R) < ∞, 1 6 p 6 ∞, (iii) s reduces to a polynomial of degree at most m − 1 on each of the intervals [ν + m/2, ν + m/2 + 1], ν ∈ Z. For a sequence y = {yj }j∈Z ∈ lp (Z), we define the space of double infinite bounded sequences with the usual norm as follows: k{yj }klp (Z) =

X

|yj |p

1/p ,

1 < p < ∞,

j∈Z

k{yj }kl(Z) := k{yj }kl1 (Z) , k{yj }kl∞ (Z) = sup{|yj |}. j∈Z

Received by the editors January 21, 1997 and, in revised form, October 13, 1998. 2000 Mathematics Subject Classification. Primary 41A17, 42B30; Secondary 30D15, 30D55. Key words and phrases. Cardinal spline, entire function, Lebesgue constant. Project 19671012 supported by both the National Natural Science Foundation and the Doctoral Programme Foundation of Institution of Higher Education of the People’s Republic of China. c

2000 American Mathematical Society

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Schoenberg [9] proved that there is a unique element Lm y ∈ Sm,p (R) interpolating the given data at integers, i.e., Lm y(j) = yj ,

(1.1)

j ∈ Z.

The operator Lm : lp (Z) −→ Sm,p (R) is called the cardinal spline interpolation operator of order m from lp (Z) to Lp (R) and its norm kLm kp = sup{kLm ykp(R) : kyklp (Z) 6 1}

(1.2)

is referred to as the mth Lebesgue constant for cardinal spline interpolation. These numbers were investigated previously by many authors (see [6]–[8]). Theorem A ([6]). Let 1 < p < ∞. Then kLm kp 6 Cp ,

(1.3)

where the constant Cp is independent of m. Theorem B ([6]). The norms of the mth order cardinal spline interpolation operators from l1 (Z) to L1 (R) satisfy (1.4) lim kLm k1 − (4 π 2 ) log m = 2A π + 4 π 2 log(4 π) + γ], m→∞

where γ is the Euler–Mascheroni constant and Z π 2 t (1.5) dt. t−1 tan( ) − A= 2 π(π − t) 0 From Theorem B, we know that sup{kLm k1 } = ∞. m

Let H 1 (Z) be the discrete Hardy space, consisting of those double infinite bounded sequences y = {yj } ∈ l1 (Z), such that Hy = {Hyj } ∈ l1 (Z), where X yj (1.6) , j ∈ Z, Hyj = k−j k6=j

is the discrete Hilbert transform of y. Thus H 1 (Z) is the subspace of l1 (Z) consisting of those sequences y = {yj } for which the discrete Hilbert transform also belongs to l1 (Z). Clearly k{yj }kH 1 (Z) := k{yj }kl(Z) + k{Hyj }kl(Z)

(1.7) 1

is a norm of H (Z). H 1 (Z) was introduced by Coifman and Weiss [3, p. 622] as an important example of the Hardy space H 1 (X), associated with a space X of homogeneous type, in order to extend the atomic decomposition theory for the classical Hardy spaces to a more general setting. It is well known that the Hardy space H 1 (R) is a proper closed subspace of L1 (R), and many results in harmonic analysis and approximation theory are valid on H 1 (R) but are not correct on L1 (R). We have found the same situation exists with respect to the Lebesgue constant of the cardinal spline interpolation operator. Our main result is the following: Theorem 1. Let {(−1)j yj } ∈ H 1 (Z). Then for all m ∈ N

π π

{(−1)j yj } 1 . 1+ (1.8) kLm ykL(R) 6 1 + H (Z) 2 3


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2. Interpolation operator of cardinal spline Let j(x) be the unique integer satisfying j(x) − 12 6 x < j(x) + 12 , and let X0 e (2.1) yj (x − j)−1 , Hy(x) = P e is named where the sum 0 is taken over those j ∈ Z for which j 6= j(x), and Hy the mixed Hilbert transform of the sequence y = {yj }. Following some ideas of [6], we have Lemma 1. Let y ∈ H 1 (Z). Then e 1(R) 6 kHyk

(2.2)

π2 k{yj }kH 1 (Z) . 3

Proof. From the definition of j(x), we have |j(x) − x| 6 12 , and for j 6= j(x), we get |j(x) − j| 6 |j(x) − x| + |x − j| 6 therefore

1 + |x − j|; 2

j(x) − j 1 1 6 2. 61+ x−j 2 |x − j|

(2.3) For j 6= j(x),

1 (j(x) − x)(j(x) − j) 1 = + (j(x) − j)−2 . x−j j(x) − j x−j Hence

X0 X 0 y X0 y |yj | j j , 6 + x−j j(x) − j |j(x) − j|2 from which we obtain Z X X0

X0 y |yj | 0 yj

j 6 dx

+

x − j 1(R) j(x) − j |j(x) − j|2 R X Z k+ 12 X yj X |yj | = dx + 1 k−j |k − j|2 k∈Z k− 2 j6=k j6=k X X yj X X |yj | = + k−j |k − j|2 k∈Z j6=k k∈Z j6=k X 1 X X X yj = |yj | + k−j k2 (2.4)

k∈Z j6=k

k∈Z\{0}

j∈Z

1 6 k{Hyj }kl(Z) + π 2 k{yj }kl(Z) 3 1 2 (2.5) 6 π k{yj }kH 1 (Z) , 3 which completes the proof of Lemma 1. P Let Wσ y(x) = yk sinc σ(x − kπ σ), and let k∈Z

kWσ k1 = sup kWσ y(x)k1(R) : {(−1)j yj } H 1 (Z) 6 1 , where sinc x := x−1 sin x for x 6= 0 and 1 for x = 0, Wσ is the well-known Whittaker operator and Wσ y is the Whittaker cardinal series. From Lemma 1, we have


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Theorem 2. Let σ > 0. Then

1 π . kWσ k1 6 1 + π 3 σ Proof. We first consider the case σ = π: X yj sinc π(x − j) |Wπ y(x)| = (2.6)

j∈Z

sin πx X yj (−1)j 6 + |yj(x) sinc π(x − j(x))| π x−j j6=j(x) yj 1 X (−1)j 6 + |yj(x) |. π x−j j6=j(x)

Therefore, it follows from Lemma 1 that we have π kWπ y(x)kL(R) 6 k{(−1)j yj }kH 1 (Z) + k{yj }kl(Z) 3 π k{(−1)j yj }kH 1 (Z) . (2.7) 6 1+ 3 By changing scale, we obtain from (2.7) that π π . kWσ k1 6 1 + 3 σ Denote by Lm p (R), 1 6 p 6 ∞, m ∈ N, the subspace of f in Lp (R) for which the (m − 1)th derivative of f exists and is locally absolutely continuous on R, and for which kf (m) kp(R) is finite. By [4], if f ∈ Lm p (R), then k{f (j)}kp(Z) 6 kf kp(R) + kf 0 kp(R) < ∞; therefore, it follows from Schoenberg [9] that for every f ∈ Lm p (R), there is a unique Lm f ∈ Sm,p (R), such that Lm f (j) = f (j) for all j ∈ Z. Moreover, we have Lemma 2 ([5]). Let f ∈ Lm 1 (R), m ∈ N, and let Lm f be the unique cardinal spline of degree m − 1 interpolating to {f (j)}j∈Z at the integers. Then (2.8)

kf − Lm f k1(R) 6

Km (m) kf k1(R) , πm

where Km is the Favard constant, (2.9)

Km :=

∞ 4 X (−1)k(m+1) , π (2k + 1)m+1 k=0

and 4 π < · · · < K3 < K1 = . π 2 Remark 1. de Boor and Schoenberg [2] proved that equation (2.8) is also valid for m even and p = ∞. 1 = K0 < K2 < · · ·
0, be the restriction on R of entire functions of exponential type σ, and let Bσ,p = Eσ (R) ∩ Lp (R), 1 6 p 6 ∞,

Bσ := Bσ,∞ .

It is well known that Bσ,p ⊆ Bσ,q , 1 6 p < q 6 ∞.


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Lemma 3 ([1, p. 211] Inequality of Bernstein’s type). Let f ∈ Bσ,p , 1 6 p 6 ∞, σ > 0. Then kf 0 kp(R) 6 σkf kp(R) . Lemma 4 ([10]). Let y = {yj } ∈ l2 . Then there is a unique f ∈ Bπ,2 , interpolating the given data y = {yj }j∈Z at the integers, and f is represented by X yj sinc π(x − j), for all x ∈ R, f (x) = and the series

P

j∈Z

yj sinc π(x − j) converges uniformly on R.

j∈Z

Proof of Theorem 1. Let {(−1)j yj } ∈ H 1 (Z). Then {yj } ∈ l2 (Z). By Lemma 4, there exists a function f ∈ Bπ,2 such that f (j) = yj for all j ∈ Z, hence f ∈ Bπ . It follows from Theorem 2 that f ∈ L1 (R); therefore f ∈ Bπ,1 . Using Lemma 2 and Bernstein’s inequality we get Km kf − Lm f k1(R) 6 m kf (m) k1(R) π (2.10) 6 Km kf k1(R), which together with (2.9) and Theorem 2 gives kLm yk1(R) = kLm f k1(R) 6 (1 + Km )kf k1(R) π π 1+ k{(−1)j yj }kH 1 (Z) , 6 1+ 2 3 which completes the proof of Theorem 1. Acknowledgment The author would like to thank the referee for his valuable suggestions and comments. References 1. R. P. Boas, Jr., Entire Functions, Academic Press, New York, 1954. MR 16:914f 2. C. de Boor and I. J. Schoenberg, Cardinal interpolation and spline functions VIII, Lecture Notes in Mathematics, Vol. 501, Springer, Berlin, 1976, 1-79. MR 58:12097c 3. R. R. Coifman and G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83(1977), 569–645. MR 56:6264 4. Fang Gensun, Whittaker–Kotelnikov–Shannon sampling theorem and Aliasing error, J. Approx. Theory 85(1996), 115–131. MR 97b:41002 5. G. G. Magril–Il’yaev, Average dimension, widths, and optimal recovery of Sobolev classes of functions on the line, Math. USSR Sbornik 74(1993), 381–403. MR 92k:41034 6. M. J. Marsden, F. B. Richards, and S. D. Riemenschneider, Cardinal spline interpolation operators on lp data, Indiana Univ. Math. J. 24(1975), 677–689. MR 52:3807 7. F. Richards, The Lebesgue constants for Cardinal Spline Interpolation, J. Approx. Theory 14(1975), 83–92. MR 52:6254 8. F. Richards, Uniform spline interpolation in L2 , Illinois J. Math. 18 (1974), 516-521. MR 50:10620 9. I. J. Schoenberg, Cardinal Spline Interpolation, CBMS, Vol.12, SIAM, Philadelphia, 1973. MR 54:8095 10. J. M. Whittaker, Interpolatory Function Theory, Cambridge University Press, 1935. Department of Mathematics, Beijing Normal University, Beijing, 100875, People’s Republic of China E-mail address: [email protected]
