CC BY
10URAL MATHEMATICAL JOURNAL, Vol. 3, No. 1, 2017
APPROXIMATION BY LOCAL PARABOLIC SPLINES
CONSTRUCTED ON THE BASIS OF INTERPOLATION
IN THE MEAN1
Elena V. Strelkova
N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences; Ural Federal University, Ekaterinburg, Russia, [email protected]
Abstract: The paper deals with approximative and form—retaining properties of the local parabolic splines
of the form S(x) = Vj B2(x — jh), (h > 0), where B2 is a normalized parabolic spline with the uniform nodes j
and functionals Vj = Vj (f) are given for an arbitrary function f defined on R by means of the equalities
hi 2
Vj = hi / f (jh + t)dt (j e Z).
-hi 2
On the class W2 of functions under 0 < hi < 2h, the approximation error value is calculated exactly for the case of approximation by such splines in the uniform metrics.
Key words: Local parabolic splines, Approximation, Mean.
Introduction
In the function approximation theory, the local polynomial splines of the order r and of minimal defect are usually constructed as linear combinations of the corresponding B-splines Br,j (x). For a function f from the class of continuous ones, the local polynomial spline S(x) = S(f, x) is defined as follows:
S(x) = £ bj (f )Br,j (x), (0.1)
j
where bj (f) is the sequence of linear continuous functionals, whose choice determines the form of the approximation.
As the functionals bj (f), one chooses the linear combinations of the function values and its derivatives at the mesh nodes or its divided differences.
The most simple and convenient (in computation) version of this choice is bj (f) = f (xj) (here, xj are the nodes of the spline S mesh). It leads to the well known local spline (see, for example, [1-4]):
S(x) = £ f (xj)Br,j(x). (0.2)
j
In formula (0.2) instead of xj, the arithmetic mean is often used that is calculated over all nodes of the B-spline Br,j (x) ([1-3]).
xThe paper was originally published in Trudy Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 2007. Vol. 13, no. 4. P. 169-189 (in Russian).
A spline (constructed in such a way) is not an interpolation one. But in the case r = 2 (i.e., of the parabolic splines), it is a continuously differentiable function on the whole number axis R and possesses both the form-retaining and extremal features [5].
But if the function f is not continuous but only integrable, it is not natural to consider aspects of this function approximation taking into account only its values at the mesh nodes. It is so since its values at separate points are not essential for functions of such a type. In such case, one uses the interpolation on the average. Aspects of existence, uniqueness, and approximative and extremal properties of such splines were investigated in works [6-8]. (Generalizations onto the L-splines see, also, on [9, 10]).
Let two real numbers h > 0 and h1 > 0 be given. For a function integrable on the whole number axis f : R ^ R assume
( hi
2
j(f ) = y, = | hrj f (jh +t)dt■ hl > 0 (0.3)
hi 2
f (jh), hi =0.
Let B2,0(x) be a normalized parabolic B-spline (see, for example, [1]) with nodes — ^, — ^,
and B2,(x) = B2>o(x — jh).
Let, also, = (X) = {f : f e AC, ||f"||oo < 1} be a class of functions given on the set X (X = R or X = [a,b]).
Here, AC is the class of locally absolute continuous functions || • ||oo = || • ||l^(x), L0(X) is the class of the functions essentially restricted on X with the usual definition of the norm
f ||oo = ess sup\f (x)|.
xex
In the present work, we investigate in details aspects of approximation of smooth functions f by the local parabolic splines of the form (0.1) (i.e., for r = 2) with the choice of the functionals bj(f) in the form (0.3).
1. Properties of the spline S
On the axis R (on both its sides), consider the following mesh of nodes: • • • < x-2 < x-1 <
h
x0 < x1 < x2 < • • • ; and let x, = jh, h = x,+1 — x,, x,+1/2 = x, + 2 (j e Z). For xj < x < x,+1/2 from formulas (0.1) and (0.3), it follows
2 ™ ,_)2
(x — xj+1/2)'2 (x,+1 — x (x — xj+1/2)
S(x) = y,-1--—- + yj •
2h2 j h h2
(x — xj+1/2)2 x — x_
+yj+1 M -oTT2-+
2h2 h
(1.1)
and for x,+1/2 < x < xj+1
(x — xj+1/2)2 xj+1 — ^ fx — xj (x — xj+1/2)
S (x) = yj )+ yj+1 ^
+yj+2
2h2 h 2 h h2 (x — xj+1/2)2
(1.2)
2h2
2
To the function f(x) £ W^ (R), we put in correspondence the parabolic spline S(x) = S(f,x) (see (1.1)-(1.2)), where the functionals yj are defined by formula (0.3).
For h1 = 0, the form-retaining and approximation properties of such splines were investigated in [5]; so, we shall consider the case h1 > 0.
Denote by Aj the interval (xj —2; xj + "2t) , j £ Z.
Theorem 1. The local .spline S(x) defined by formulas (1.1) — (1.2), possesses the following properties:
1) locally inherits the sign of the original function f in the sense that
a) if f (x) > 0 (< 0) for x £ Aj—i U Aj If Aj+i, then S(x) > 0 (< 0) for xj < x < xj+1/2 (j £ Z);
b) if f (x) > 0 (< 0) for x £ Aj U Aj+i U Aj+2, then S(x) > 0 (< 0) for xJ+i/2<x<x1+i (j£Z);
2) locally inherits the monotonicity property of the original function f, namely,
a) if the function f (x) does not decrease (does not increase) in the interval (xj—i — -2; xj+1+ then the spline S(x) does not decrease (does not increase) in the interval (xj;xj+1/2) (j £ Z);
b) if the function f (x) does not decrease (does not increase) in the interval ^xj — ^; xj+2 + ^ then the spline S(x) does not decrease (does not increase) in the interval (xj+1/2;xj+1) (j £ Z).
Proof. For the point 1), the proof follows directly from non-negativity of the B-spline B2,0(x), formula (0.2), and non-negativity of yj for x £ Aj.
Point 2a). From condition of point 2) of Theorem 1 and definition of yj it follows yj+i > yj > yj—1. By differentiation of the right-hand side of equations (1.1), we obtain that for x £ (xj; xj+1/2) the derivative
S'(x) = ^^ + ■ (yj+i — 2yj + yj+i)
of the spline S in this interval is a linear function in the variable x. So, to prove point 2), it is sufficient to verify that for yj+i > yj > yj—i, inequalities S'(xj + 0) > 0, S'(xj+i/2 — 0) > 0 hold. Validity of these inequalities follows from the formulas
S' (xj +0) = M^ji, S'x+i/2 — 0) = ;
and point 2a) of Theorem 1 is proved.
Point 2b). From conditions of point 2b) of Theorem 1 and definition of yj, it follows yj+2 > yj+i > yj- By differentiation of the right-hand side of equality (1.2), we obtain that for x £ xj+i/2; xj , the derivative
S'(x) = ^ + ■ (yj+2 — j + yj)
of the spline S in this interval is a linear function in the variable x. So, to prove point b), it is sufficient to verify that for yj+2 > yj+i > yj, the equalities S'(xj+i/2 + 0) > 0, S'(xj+i — 0) > 0 hold. But this follows from the formulae
S'(xj+i/2 + 0)
yj+i — yj h ''
S' (xj+i — 0) = .
□
Before formulation of further statements, obtain the integral representation for the difference S(x) — f (x) in the interval [xj; Xj+\] for 0 <h1 < 2h.
Let for the beginning x £ [xj; xj+1/2]. Then, by the Taylor formula for the function f (x) £ W2 (R) we have
x
f (x) = f (xj) + f'(xj )(x — xj) + J(x — t)f"(t)dt.
(1.3)
Using (1.3) and definition yj (see (0.3)), we derive
hi
2 S+xj-i
yj-1 = f(xj) — f (xj)h + hi j ds j (xj-! + s — t)f"(t)dt,
_ hi xj
2
h1
2 S X j
yj = f (xj) + hi j ds j (xj + s — t)f"(t)dt,
hi xj " 2
hi
s+-xj+1
yj+i = f(xj) + f'(xj)h + hi j ds j (xj+i + s — t)f"(t)dt.
hi xj
' 2
Therefore, from (1.1), we obtain
S(x) =
(x — xj+i/2)2 2h2
hi
2 s+xj_i
f(xj) — f (xj)h + h- f ds j (xj-i + s — t)f''(t)dt
xj+1 — x (x — xj+1/2)2
+
/(x — xj+1/2)2 + + \ 2h2 + h
h2
hi xj " 2
hi
2 S+x j
f(xj) + -1 I ds j (xj + s — t)f"(t)dt
_ hi xj 2
hi
s+xj+i . f (xj) + f'(xj)h + h- j ds j (xj+1 + s — t)f''(t)dt
_ hi xj
2
(1.4)
Taking into account that 0 < h1 < 2h, change the integration order in the integrals entering
h
into representation (1.4). Hence, we obtain
(x — xj+i/2)2
+h1
Xj-1 + 2
S (x) = V" h1"' {f (xj) — fx )h + hi [ / 1 f"(t){t — xj—i + h^)2 dt
_h1
Xj-i 2
+
f "(t)hi(t — xj—i)dt
+ h1 Xj-1 + 2
+
xj+i — x (x — xj+i/2)2
h2
_hi
^ f x > + hi
2
1 f "(t)(t — xj + f2)2dt + / 1 f''(t)(t — xj — f2)2dt
hi
(1.5)
+
(x " Ij+i/2^+ ^^^ i f (xj)+ f'(xj)h + 1
2h2
h
_h1
Xj + 1 2
f ''(t)hi(xj+i — t)dt
xj+1+h1
1 hi 2
+ 2f (t)(t — xj+i— "r)dt r x £ [xj; xj+i/2].
_hi
Xj+1 2
By virtue of symmetry of formulas for S(x) w.r.t. the middle xj+i/2 of the interval [xj; xj+i], we obtain the following similar representation of S(x) for x £ [xj+i/2, xj+i]:
S(x) =
(x — xj+i/2)2 xj+i — x
2h2
+
h
f x) + hi
2 f ''K t—x3+hi)2dt
+ / 2f(t)(t—xj—2)2dt
+
x — xj (x — xj+i/2)2
h
h2
f (xj) + f '(xj )h
1
+hi
_h1
Xj+1 2
Xj+1 + T1
f ''(t)hi(xj+i — t)dt +
2 f ''(t)( t — xj+i — y)2dt
_h1
Xj+1 2
+
(x — xj+i/2)2 2h2
f (xj) + f '(xj )2h + hi
h1
X j+2—21
f ''(t)hi(xj+2 — t)dt
X j+2 +
+ J 2f''(t){t — xj+2 — y) dt ^ x £ [xj+i/2,xj+i].
h1
X j+2 —
□
Theorem 2. A local spline S(x) defined by formula (1.1) — (1.2), for 0 < hi < 2h, possesses the following properties:
h
j
j— 2
h
1
I : —
j— 2
h
1
X
1) inherits locally the convexity property of the original function f, namely,
a) if the function f (x) is down- (upper) convex in the interval (xj—1 —2; xj+1 + , then the spline S(x) is the down- (upper-) convex function in the interval (xj; xj+1/2) (j £ Z);
h1h1
b) if the function f (x) is down- (upper-) convex in the interval [xj —xj+2 + y ), then the
spline S(x) is the down- (upper-) convex function in the interval (xj+1/2; xj+1) (j £ Z).
h1 h11
■j-1 —xj+1 + — , the exact
| S''(x) |< 1, x £ (xj;xj+1/2) ,
h1 h1 2) a) for any function f £
w2 j—1 2 ; xj+1 + 2 J' the exact inequality holds
x2
and, moreover, for all x £ (xj; xj+1/2), the inequality sign is provided by the function f (x) = —;
h1 h1 1
xj —xj+2 + — , the exact i
S"(x) l< 1, x £ (xj+1/2;xj+1)
r h h '
b) for any function f £ W2 xj —xj+2 + — , the exact inequality holds
x2
and, moreover, for all x £ (xj+1/2; xj+1), the inequality sign is provided by the function f (x) = —.
Proof. To prove 1a), it is necessary to verify that if
f ''(x) > 0 (< 0) for x £ (xj—1 — y; xj+1 + ^,
then S''(x) > 0 (< 0) for x £ (xj; xj+1/2) .
By the twice differentiation of the function S(x), we obtain from formula (1.5)
■ + hi.
x j—i + 2 x j x j
S''(x)= j f''(t)C1(t)dt + J f''(t)C2(t)dt + J f''(t)C3(t)dt
hi + hi
xj — i — 2 xj — i+ 2 xj —
+hi _ hi + hi
xj+ 2 xj+i— 2 xj+1+ 2
+ J f''(t)C4(t)dt + J f''(t)C5(t)dt + j f''(t)C6(t)dt,
xj xj xj+i — -r
1 h1 2 1 1 h1 2
C1 (i) = 2hh I - xj—1 + i) ■ C2(t) = h(—^ C(t) = -Whi 0—xj + f
1 I. h1 \2 1. ^^ 1 / h1x2
(1.6)
where
C4(t) = — № ^ — x — Y) , C5(t) = h2 (xj+1 —1); C6(t) = 2hh7 ^ — xj+1 — T
Divide the further proof into two cases: 1) 0 < h1 < h, and 2) h <h1 < 2h. Case 1). Let 0 < h1 < h. Under this, the function S''(x) is transformed to the form
+hi _ hi
xj—i + 2 xi 2 xj
S''(x)= j f''(t)C1(t)dt + j f''(t)C2 (t)dt + J f ''(t)(C2(t) + Ca(t))dt
hi + hi _ hi xj—i 2 xj—i+~2 xj 2
+hi -hi + hi ( .
xj+ 2 xj+i 2 xj+i+ 2
+ f f''(t)(C4(t) + C5(t))dt + f f'' (t)C5 (t)dt + [ f''(t)C6(t)dt.
+hi _ hi
xj + 2 xj+i 2"
From the definitions of Cj(t) (j = 1,6), it follows that Ci(t) > 0 for t g
C2 (t) > 0 for t g hi
j
+ hi • _ hi xj-i + 2 • xj "
t
xj+i
+ hi xj+1 + TJ
2
C5 (t) > 0 for t g
+ hi • xj + o • xj+1
_hi • + hi xj-i 2 • xj-i+ 2
h- , and Ca(t) > 0 for
Now it remains to investigate the quadratic trinomials C2(t) + C3(t) for t £
C4 (t) + C5 (t) for t g
• + hi xj • xj + «
hi
xj 2 ' x
and
2
Non-negativity of the functions C2(t) + C3(t) for t G
t
• + hi1 xj • xj + «
hi 2
• xj
and C4 (t) + C5 (t) for
follows from the fact that the branches of corresponding parabolas are down-
2
directed and their values (at the ends of the intervals under investigation) are non-negative. Namely,
4h _ h
C2(x3 ) + C3 (xj ) = C4(xj ) + C5 (xj ) = > 0,
hi hi hi hi 2h hi
C2(xj - -2) + C3(xj - -2) = C4(xj + 2) + C5(xj + -1 ) =
2h2
> 0.
From the statements proved, representation (1.7), and the condition f"(t) > 0 for t g (^xj-i —2• xj+i + "?i), it follows that S"(x) > 0 for x g (xj• xJ+i/2).
Case 2). Let h < hi < 2h. Under this, the function S"(x) is transformed to the form
2
+h1
xj-i+ 2
S" (x)= y f"(t)Ci (t)dt+ J f"(t)(Ci(t)+C3(t))dt + J f"(t)(C2(t)+C3(t))dt
_h1
xj-1 2
xj 2
+h1 Xj-1+ 2
h1
Xj+1 2
xj+ 2
h1
Xj + 1+2
+
f " (t)(C4(t)+C5(t))dt+
f "(t)(C4(t)+C6(t))dt +
f"(t)C6(t)dt. (1.8)
_h1
Xj+1 2
xj + 2
From the definitions of Cj (t) (j = 1,6), it follows that Ci(t) > 0 for
tg
hi
hi
and Ca(t) > 0 for t g
22 lementary calculations show that
+ hi • + hi xj + o • xj+i +
2
Ci (xj - hi) + C3 (xj - h-) = C4 (xj + hi) + C6 fxj + hi) = > 0,
2 J 2hi
Ci (x-i + y ) + C3 (x-i + y ) = C^xj+i + y ) + C^xj+i + y ) =
hi
hi) 2h2 - (hi - 2h)2
2hih2
> 0,
C2(xj) + C3(xj) = C4(xj) + C5(xj) = 2 (h - y ) > 0
C2 (xj-i + y ) + C3 (xj-i + y ) = C4(xj+i - y^i) + C5(xj+i - y )
hi 2
hi) 2h2 - (hi - 2h)2
2h1h2
0.
Since the quadratic trinomials Ci(t) + C3(t), C2(t) + C3(t), C4(t) + C5(t), and C4(t) + C6(t) have negative leading coefficients and their values at the ends of the corresponding intervals are
h
1
h
1
h
1
h
1
non-negative, we have
C1(t) + Cs(t) > 0 for t £ C2(t) + Cs(t) > 0 for t £ C4(t) + C5(t) > 0 for t £ C4(t) + C6(t) > 0 for t £
_hk; + hi xj O ; xj — 1 + o
2
+ h1;
xj—1 + 2 ; xj
2
2
xj; xj+1
_ h.; + hi xj+1 2; xj + 2
From the statements proved, representation (1.8), and the condition f''(t) > 0 for t £ yxj—1 —
y; xj+1 + yj, it follows that S''(x) > 0 for x £ (xj; xj+1/2).
Proof of point 1b) one-to-one repeats the reasonings mentioned in the prof of point 1a) after
substitution the variable x - xj by the xj+1 - x one.
Point 2a). Estimate |S''(x)| for x £ [xj;xj+1/2]. Under 0 < h1 < h by (1.7) for any function
i ^ Tjr2 r h1 , h1
f £ W2 xj—1— y; xj+1 + yj
and this sum is equal to 1. Namely,
the value ^''(x)| is estimated from above by the sum of integrals,
+hi
xj—i+ 2
S'^K j C1 (t)dt + j C2(t)dt + j (C2(t)+ C3(t))dt
_hi
xj — i 2
+hi
xj—i+ 2
_hi
xj+i 2
hi
xj+i + 2
+ J (C4(t) + C5(t))dt + J C5(t)dt + j C6(t)dt = 1, x £ [xj;xj+1/2].
xj + i 2
Similarly, for h <h1 < 2h from (1.7), we derive that for any function
h1" x 1 for x £ [xj; xj+1/2], the following inequality holds:
f£W
xj 7T; xj+2 + ^T
2
2
x3 2
S''(x)| < f C1(t)dt +
+hi
xj—i+ 2
(C1(t)+ C3(t))dt +
(C2(t)+ C3(t))dt
_hi
xj—i 2
+hi
xj—i + 2
_hi
xj + i 2
hi
+ j (C4(t)+ C5(t))dt + j (C4(t)+ C6(t))dt + j C\(t)dt = 1.
_hi
xj+i 2
From the proofs considered above, it follows that in both cases the function that realizes exact
x2
equality in the inequalities proved is the function f (x) = —.
Point 2b). The proof follows directly from the function S(x) symmetry w.r.t. the middle point xj+1/2 of the interval [xj; xj+1].
□
h
i
x
j— 2
j
h
i
x
j— 2
h
i
x
j2
h
i
x
j2
h
i
j
h
i
x
j— 2
h
i
x
j2
h
i
x
j2
2. Estimations of approximation errors
Theorem 3. For 0 < hi < 2h, the following equality holds:
h2 h2
sup \\f — S = - + -4.
f W (R) 8 24
Xj-1 -
Proof. Consider for x £ [xj; xj+1/2] non-integral terms in the function f £ W^ h hi
—; xj+1 + — in representation (1.3) and in the function S(x) in representation (1.5). Note that they coincide since
(xjl (f (xj) - f(xj )h) + ( xl±h-x - (xj2)l) f (xj)
+ ((x - xT)2 + x-xi) (f (xj) + f'(xj)h) = f (xj) + f '(xj).
Taking this into account, we have for any x £ [xj; xj+1/2]
S(x) - f (x)=(x - xj+1/2)2 1
2h2 h1
h
3-1^-2 .
/ 2f"'(t)it - xj-1 + Yfdt + / f ''(t)hi(t - x—)dt
hi . hi __1 W ■ 1 J--1
i hi Xj-i + 2
Xj-i 2 Xj-i+ 2
X j x j + 2
+{jhrx - ixjt} ¿[/ 2f''(t)(t - xj+hr)2dt+/ 2fn - xj -
h
x.-hi xj
xj 2
2 J i - 2 J i - ■ 2
+ ((xjL + T*) t\S ™hj - t)dt + / - xj+1 - ^)2dt
xj x.,, -hi
j x3 + 1- —
x
- j f''(t)(x - t)dt. (2.1)
xj
So, for any x £ [xj; xj+1/2], the following equality holds:
i hk
xj-1+ 2 xj
S(x) - f (x) = J f''(t)K1(x,t)dt + j f''(t)K2(x,t)dt
hi + hi
xj-i T Xj-i + —
+hi _ hi
xj+~T Xj+1 2~
+ J f''(t)K3(x,t)dt + j f "(t)K4(x,t)dt + j f "(t)K5(x,t)dt (2.2)
_ hi
xj 2
+hi
xj+1+ 2 x
r/ J, / <•//
+ f''(t)K6(x, t)dt - f (t)Kj(x, t)dt,
_hi
xj+l 2
where
v { +) (x - xj+1/2)2 ( + hi \2 K1(x't) = 4h2h ^ - ^ + yJ
(x - xj+l/2)
2
K2(x,t) = —2h—(t - xj"i)'
K>(x.t)=(^) 2hl ( - *+i;.
= ( Xj+1 - x (x - xj+1/2)2 ) ( _hl
---^ /2h)Vt- x' 2/'
K,(X't) = (+ ^ |(Xj+i -1)'
(x - Xj+1/2)2 X - xA 1 / h
k»(x-'> = 1" 2^"" + ^—xj+1— tJ ■
K7(x, t) = t — x.
Further proof for x £ [xj; xj+1/2] is divided into two cases: 1) 0 < h1 < h and 2) h <h1 < 2h.
r h
Case 1). Let 0 < h1 < h. Under this, the difference S(x) — f (x) for x £ " " '
2
transformed to the form
xj ,xj + 2
is
+hx _ hx
Xj—i + — Xj 2
S(x) - f(x) = J f"(t)Ki(x,t)dt + J f"(t)K2(x't)dt _ hx + hx
xj X
+ J f"(t)(K2(x't) + K3(x,t))dt + j f"(t)(KA(x't)+ K5 (x,t) + K7(x,t))dt (2.3)
_ hx
Xj 2
+hx — hx + hx
Xj + ~2 Xj+1 2 Xj+1 + ~2
+ j f"(t)(K4(x't) + K5(x,t))dt + j f" (t)K5(x,t)dt + j f"(t)K6(x't)dt.
+hx _ hx
Xj+ 2 Xj+1 2
But for x
+ hi xj + "2' xj+1/2
its form is
+hx _ hx
Xj-1 + ~2 X3 2
S(x) - f(x) = J f"(t)K1(x,t)dt + J f"(t)K2(x,t)dt
_ hx + hx
Xj-1 2 Xj-1 + ~2
+hx X j+ 2
+ J f"(t)(K2(x,t)+ K (x,t))dt + J f"(t)(K4(x't) + K5(x,t) + K7(x,t))dt (2.4)
hx j
X ■--x j
X j 2
_ hx + hx
X X j+x 2 X j+x + ~2
+ j f"(t)(K5(x't) + K7(x,t))dt + j f" (t)K5(x,t)dt + j f"(t)K6(x't)dt.
+hx X _hx
Xj+ 2 Xj+x 2
To obtain the estimation value on the function class W^ = W^ (R), we shall prove that
t g t g
under x g [xj, xj+i/2]: Ki(x,t) > 0 for t g hi h\'
_ hi. + hi xj-i ^ 'xj-i + 2
xj-i + 2 ' xj 2 hi , hi xj+i - y. xj+i + y j
K2(x,t) + Ks(x,t) > 0 for t g
hi .
xj 2 ' x
K2 (x, t) > 0 for K6(x,t) > 0 for
under x
+hi
xj ,xj + o
for t
. . + hi
x. x j \ ^
2
: K4(x, t) + K5(x,t)+K7(x,t) > 0 for t g [xj; x], K4(x,t)+K5(x,t) > 0
, K5(x,t) > 0 for t g
under x
2
+ hi xj + y, xj+i/2
hi
hi
xj + o ; xj+i o
K7(x, t) > 0 for t G All these inequa
hi
hi
: K4(x,t) + K5(x, t) + K7(x,t) > 0 for t g [xj; xj + —^], K5(x,t) +
'x j \ ;
K5(x,t) > 0 for t g
x.xj+i
hi 2 J
ities (except only two) immediately follow from definitions of the functions
Kj(x,t) (j = 1, 7). So, it is only necessary to verify that K4(x,t) + K5(x,t) + K7(x,t) > 0 for
x
+hi
xj ,xj + ~
for x
2
+ hi xj + ~2T, xj+i/2
t g [xj ; x], for x g
t
xj \ ;
+ hi xj + y, xj+i/2
hi
t
+hi xj ; xj+ o
and K5(x,t)+ K7(x, t) > 0
The function K4(x, t) + K5(x, t) + K7(x, t) is the quadratic trinomial in the variable t with the
T hi 1
positive leading coefficient; at the ends of the intervals [xj; x], xj; xj + — this function (as one of
the variable t) takes the positive values, and abscissa of the corresponding parabola apex is placed at the left from the point xj. From this, the non-negativity of this function on the mentioned sets follows.
The function K5(x, t) + K7(x, t) is linear in the variable t and takes non-negative values at the
ends of the interval for x g
. + hi ;
x j \ ; x
2
. + hi ;
xj \ _ ;x
2
. Hence, K5(x,t) + K7(x,t) > 0 in the whole interval
L Z J
+ hi xj + y, xj+i/2
Taking into account the above proved facts, it follows from formulas (2.3) and (2.4) that to obtain the estimate from above for the value \S(x) — f (x)| (for these formulas) in the class W^(R) and, hence, in formula (2.1), the function f"(t) can be substituted by 1 in (2.3) and (2.4).
Put f "(t) = 1 and calculate for it values of integrals in the right-hand side of formula (2.1);
h,2 h2
denote this value by the symbol J. After elementary calculations, we obtain that J = — + ^4• It implies that the exact inequality
h2 h2
\f(x) — S(x)\< y + ^
holds for any function f g W
_ hi; , hi xj-i 2 ; xj+i \
2
for any x g [xj; xj+i/2]. Moreover, the sign
t2
of equality is provided by the function f (t) = — for t g
T h
Similarly, for the function f g W^ ""
the exact inequality
xj - y; xj+2 \
_ hi , hi xj-i 2; xj+i \ 2
hi 2
for any x g [xj+i/2; xj+i], we derive
n (x) - 5 (x)i < h2+hl,
and the sign of equality is provided by the function f (t) = — for t g
h1
h1
xj- y;xj+2 \ —
2
2
Case 2). Let h <h1 < 2h. In this case, the difference S(x) — f (x) for x £ transformed to the form
xj,xj+1
is
+hi
xj — i+ 2
S(x) — f (x) = J f''(t)K1(x,t)dt + j f''(t)(K1(x,t) + K3 (x,t))dt
_hi
xj — i 2
^ j x
+ J f''(t)(K2(x,t)+ K3(x,t))dt + j f''(t)(K4(x,t)+ K5(x,t) + K7(x,t))dt
+hi
xj—i+ 2
_hi
xj+i 2
+ J f''(t)(K4(x,t)+ K5(x,t))dt + J f''(t)(K4(x,t) + K6(x,t))dt
_hi
xj+i 2
1 hi
xj + i+2
+
f ''(t)K6(x,t)dt,
(2.5)
+hi
xj — i+ 2
S(x) — f (x) = J f''(t)K1(x,t)dt + j f''(t)(K1(x,t) + K3 (x,t))dt
_hi
xj—i 2
hi
xj+i 2
+
f''(t)(K2(x,t)+ K3(x,t))dt + f' (t)(K4(x,t) + K5 (x, t) + K7(x,t))dt
+hi
xj — i + 2
+
-j ' 2
f''(t)(K4(x,t) + K6(x,t)+ K7(x,t))dt + j f''(t)(K4 (x,t)+ K6(x,t))dt
_hi
xj+i 2
i hi xj+i+~2
+
f ''(t)K6(x,t)dt.
(2.6)
To obtain the error estimate in the class W^ (R), prove that
under x £ [xj,xj+1/2\, the following inequalities hold: K1(x, t) > 0 for t £
K1(x,t) + K3(x,t) > 0 for t £ K6(x,t) > 0 for t £
xj + 2 ; xj+1 + 2
under x
xj ,xj+1
h1
hi 2
_hi; + hi xj 2 ; xj—1 + 2 hi
h1 ; h1 xj—i 2; xj 2
, K2 (x, t) + K3(x, t) > 0 for t £
+hi; xj—1 + 2 ; xj
these inequalities are: K4(x,t)+ K5(x,t)+K7(x,t) > 0 for t £ [xj; x],
K4(x, t) + K5(x, t) > 0 for t £
x; xj+1
h1
K4(x, t) + K6(x, t) > 0 for t £
_ hi; + hi xj+i 2 ; xj + 2
h
1
2
h
i
x
j— 2
h
i
x
j— 2
h
i
x
j2
h
i
x
j2
h
i
x
j— 2
h
i
x
j— 2
h
i
x
h
i
x
j2
under x g hi 2 J
xj; xj+i
for t G
x; xj + 2
hi
xj+1 - y ' xj+1/2
, ^4(x,t) + K6(x,t) = K7(x,t) > 0 for t G hi
the inequalities hold: K4(x, t) + K5(x, t) + K7(x, t) > 0 for t g
hi
xj+i
2 'x
and K4(x,t) + K6(x,t) > 0
All these inequalities (except two ones) immediately follow from definitions of the function Kj(x,t) (j = 1, 7). It is only necessary to verify that K4(x,t) + K5(x,t) + K7(x,t) > 0 under
x
, t g [xj ; x] and under x g
_ hi
xj ,xj+i 2
K4(x, t) + K6(x, t) + K7(x, t) > 0 under x g
hi
y, xj+1/2 t
tg xj+1
xj; xj+1 hi 2
hi 2
and
xj+i hi
xj+1 " y, xj+1/2
The function K4(x, t) + K5(x, t) + K7(x, t) is the quadratic trinomial in the variable t with the
r h '
positive leading coefficient; at the ends of the intervals [xj; x], xj; xj+1 —— , this trinomial takes positive values, and abscissa of the corresponding parabola apex is placed at the left from the point
x j .
The function K4(x,t) + K6(x,t) + K7(x,t) for x g
xj+i
hi ;
y ; xj+1/2
t
xj+1
hi
possesses the same properties. Remind that this function is also the quadratic trinomial in the
r h
variable t with the positive leading coefficient; at the ends of the interval ~
xj+i
this
trinomial takes positive values, and abscissa of the corresponding parabola apex is placed at the
h1
left from the point xj+1 ——. It implies non-negativity of the considered functions in the mentioned sets.
Taking into account the above proved facts, it follows from formulas (2.5) and (2.6) that to
obtain the estimate from above for the value \S(x) — f (x)| (for these formulas) in the class W^(R)
and, hence, in formula (2.1), the function f"(t) can be substituted by 1.
Put f "(t) = 1 and calculate for it values of integrals in the right-hand side of formula (2.1);
h2 h2
denote this value by the symbol J. After elementary calculations, we obtain that J = — + . It implies that the exact inequality
h2 h2
\f(x) — S(x)\ < y + —4
8 24'
holds for any function f g Wr
h1
h1
xj-i o ;xj+i + o
2
2
under any x g [xj;xj+1/2]. Moreover, the
t2
equality sign is provided by the function f (t) = — for t G
r h h '
Similarly, for the function f g W^ ~ ~ 1
h
h
xj-1 o ; xj+1 + o
2
2
the exact inequality
xj- y ; xj+2 + —
2
under any x g [xj+1 /2; xj+1 ], we derive
h2 h2
\S(x) - f (x)\< y + 24,
2
t
and the equality sign is provided by the function f (t) = — for t g
h1
h1
xj- y; xj+2 + —
2
□
Acknowledgements
The Author expresses deep gratitudes to Prof. K. Etter (Germany) and to Dr. V.L. Mirosh-nichenko (Novosibirsk, Russia) for the fruitful discussions of the results obtained.
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