A note on Whitney’s theorem Petar P. Petrov
Department of Mathematics, University of Sofia, Blvd. James Boucher 5, 1164 Sofia, BULGARIA E-mail:
[email protected] Abstract Let πm be the space of all algebraic polynomials of degree less than or equal to m. For a given function f ∈ C[a, b], Em (f ) denotes the best approximation to f in the uniform metric by polynomials from πm and ωk (f, [a, b]) is the k-th modulus of smoothness of f in C[a, b]. Whitney’s theorem states that for any f ∈ C[a, b] and k ≥ 1, Ek−1 (f ) ≤ Cωk (f, [a, b]), where C is a constant independent on f . In this note we prove that the exact constant in the above inequality for k-monotone functions is equal to 1/2, namely, sup k [a,b]\π f ∈M+ k−1
1 Ek−1 (f ) = , ωk (f, [a, b]) 2
k where M+ [a, b] ⊂ C[a, b] is the cone of all k-monotone functions on [a, b].
1
Introduction
Let πm be the space of all algebraic polynomials of degree less than or equal to m. For any function f ∈ C[a, b] we denote by ∆kh f (x) :=
k X
!
(−1)k−j
j=0
k f (x + jh), {x, x + kh} ⊂ [a, b], j
its k-th finite difference with step h > 0. The k-th modulus of smoothness of f in C[a, b] is defined by ωk (f, [a, b]) := sup |∆kh f (x)|. x,x+kh∈[a,b]
Let Em (f ) := min max |f (x) − P (x)| P ∈πm x∈[a,b]
be the best uniform approximation to f by polynomials from πm . Whitney [17] was the first who proved in 1957 that (1)
Wk :=
sup f ∈C[a,b]\πk−1
1
Ek−1 (f ) 0 such that {x, x + kh} ⊂ [a, b]. For example, f is non-negative if k = 0, monotone non-decreasing if k = 1 and convex if k = 2. An equivalent definition is: A function f ∈ C[a, b] is k-monotone if all k-th order divided differences f [x0 , x1 , . . . , xk ] :=
k X
f (xj ) k Y
j=0
,
a ≤ x0 < x1 < . . . < xk ≤ b,
(xj − xi )
i=0 i6=j
are non-negative. Denote by M+k [a, b] ⊂ C[a, b] the cone of all k-monotone functions on [a, b].
2
Main result
The main result of this note is the following Theorem 1. If f ∈ M+k [a, b], then (3)
1 Ek−1 (f ) ≤ ωk (f, [a, b]). 2
k Moreover, the constant 1/2 in (3) is exact, i.e., there exist functions {fm }∞ m=1 ⊂ M+ [a, b]\πk−1 such that Ek−1 (fm ) 1 lim = . m→∞ ωk (fm , [a, b]) 2
Proof. Let Lk−1 (f, x0 , x1 , . . . , xk−1 ; x) ∈ πk−1 be the Lagrange polynomial, interpolating f at the points a ≤ x0 < x1 < . . . < xk−1 ≤ b. Let us denote by tj := a + jh,
j = 0, 1, . . . , k, 2
h := (b − a)/k,
the equidistant points in [a, b] with step h and set P1 (x) := Lk−1 (f, t0 , t1 , . . . , tk−1 ; x), P2 (x) := Lk−1 (f, t1 , t2 , . . . , tk ; x). Let f be an arbitrary function from M+k [a, b]. By the error representation formula f (x) − Lk−1 (f, x0 , x1 , . . . , xk−1 ; x) = f [x, x0 , . . . , xk−1 ](x − x0 ) . . . (x − xk−1 ) and the fact that f is k-monotone we get (−1)j (f (x) − P1 (x)) ≥ 0,
x ∈ [tk−1−j , tk−j ]
and (−1)j+1 (f (x) − P2 (x)) ≥ 0,
x ∈ [tk−1−j , tk−j ]
for j = 0, 1, . . . , k − 1. Therefore P1 (x) ≤ f (x) ≤ P2 (x),
x ∈ [tj−1 , tj ],
j = k, k − 2, . . . ,
P2 (x) ≤ f (x) ≤ P1 (x),
x ∈ [tj−1 , tj ],
j = k − 1, k − 3, . . . ,
and f (x) − P1 (x) + P2 (x) ≤ 1 |P2 (x) − P1 (x)|. 2 2
(4)
Polynomials P1 and P2 can be represented by the Newton interpolation formula as follows: P1 (x) = f [t1 ] + f [t1 , t2 ](x − t1 ) + . . . +f [t1 , t2 , . . . , tk−1 ](x − t1 ) . . . (x − tk−2 ) +f [t0 , t1 , . . . , tk−1 ](x − t1 ) . . . (x − tk−1 ) and P2 (x) = f [t1 ] + f [t1 , t2 ](x − t1 ) + . . . +f [t1 , t2 , . . . , tk−1 ](x − t1 ) . . . (x − tk−2 ) +f [t1 , t2 , . . . , tk ](x − t1 ) . . . (x − tk−1 ). From here,
(5)
|P2 (x) − P1 (x)| = khf [t0 , t1 , . . . , tk ]|(x − t1 ) . . . (x − tk−1 )| ∆k f (t0 ) = kh h k |(x − t1 ) . . . (x − tk−1 )| k!h ≤ ∆kh f (t0 ) ≤ ωk (f, [a, b]),
since max |(x − t1 ) . . . (x − tk−1 )| = (tk − t1 ) . . . (tk − tk−1 ) = (k − 1)!hk−1 .
x∈[a,b]
Now, inequality (3) is a direct consequence by (4) and (5). Further, consider the functions
fm (x) := −Tk−1
2x b − 1/m + a − b − 1/m − a b − 1/m − a 3
+ C(k, m)(x − b + 1/m)k+ ,
m > 1/h,
where Tk−1 is the (k − 1)-st Chebyshev polynomial of the first kind and the constant k
C(k, m) = m
1 + Tk−1
b + 1/m − a b − 1/m − a
is chosen so that fm (b) = 1. Since (k) fm (x) = k!C(k, m)(x − b + 1/m)0+
is non-negative and monotone non-decreasing, then fm ∈ M+k [a, b] ∩ M+k+1 [a, b]. By a well known result for k-monotone functions (see, e.g., [1, p. 61]) it follows that (6)
ωk (fm , [a, b]) =
∆kh fm (a)
= 1 + Tk−1
b + 1/m − a b − 1/m − a
On the other hand, since fm takes the values ±1 at k + 1 points (the extremal points of 2x Tk−1 b−1/m−a ee-Pousin’s theorem − b−1/m+a b−1/m−a and the point b) with alternating signs, Vall´ ([16]) implies that Ek−1 (fm ) ≥ 1.
(7) By (3), (6) and (7) we obtain
Ek−1 (fm ) 1 1 . ≥ ≥ 2 ωk (fm , [a, b]) 1 + Tk−1 b+1/m−a b−1/m−a Therefore, lim
m→∞
1 Ek−1 (fm ) = . ωk (fm , [a, b]) 2
The proof is complete. 2
References [1] G. Anastassiou, S. Gal, Approximation theory: moduli of continuity and global smoothness preservation, Birkh¨auser Boston, 2000. [2] P. Binev, O(n) bound of Whitney constants, C.R. Acad. Bulg. Sci., 38, 10(1985), 1315-1317. [3] Y.A. Brudnyi, The approximation of functions of n variables by quasipolynomials, Izv. AN SSSR, Ser. Mat., 34(1970), 546-583 (in Russian). [4] Y.A. Brudnyi, Whitney’s constants, Modern problems of the theory of functions. Abstract All-Union school-conference, 1988 (in Russian). [5] H. Burkill, Cesaro-Perron absolute periodic functions, Proc. London Math. Soc., 3(1952), 150-174. [6] K. Ivanov, M. Takev, O(n ln(n)) bounds of constants of H. Whitney, C.R. Acad. Bulg. Sci., 36, 9(1985), 1129-1131.
4
[7] Yu.V. Kryakin, Whitney constants, Mat. Zametki, 46, 2(1989), 155-157 (in Russian). [8] Yu.V. Kryakin, Exact constants in the Whitney theorem, Mat. Zametki, 54, 1(1993), 34-51 (in Russian); Engl. transl.: Math. Notes, 54, 1-2(1993), 688-700. [9] Yu.V. Kryakin, On a theorem and constants of Whitney, Mat. Sb., 185, 3(1994), 25-40 (in Russian); Engl. transl.: Russian Acad. Sci. Sb. Math., 81, 2(1995), 281-295. [10] Yu.V. Kryakin, On functions with a bounded n-th difference, Izv. Ross. Acad. Nauk, Ser. Mat. , 61, 2(1997), 95-100 (in Russian). [11] B. Sendov, On the constants of H. Whitney, C.R. Acad. Bulg. Sci., 35(1982), 431-434. [12] B. Sendov, Modifying Steklov means, C.R. Acad. Bulg. Sci., 36(1983), 315-317. [13] B. Sendov, The constants of H. Whitney are bounded, C.R. Acad. Bulg. Sci., 38, 10(1985), 1299-1302. [14] B. Sendov, On the one theorem of Whitney, DAN SSSR, 291(1986), 1296-1300. [15] B. Sendov, V. Popov, The average moduli of smoothness, J. Wiley & Sons, 1989. ´e-Pousin, Sur les polynˆomes d’approximation et la repr´esentation [16] Ch. de la Valle approch´ee d’un angle, Bull. Acad. Belgique, (1910), 808-844. [17] H. Whitney, On the functions with bounded n-th difference, J. Math. Pures Appl., 36, 9(1957), 67-95.
5
