Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants
Let f be a function we wish to approximate on the interval [x 1 ,x N ] knowing p 1 > 1,p 2 , . . . ,p N coefficients of expansion of f at the points x 1 ,x 2 , . . . ,x N . We start by computing two neighboring N -point Padé approximants (NPAs) of f, namely f 1 = [m/n] and f 2 = [m − 1/n...
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2013
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508423110524928 |
|---|---|
| author | Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. |
| author_facet | Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. |
| author_sort | Gilewicz, J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:17:19Z |
| description | Let f be a function we wish to approximate on the interval [x 1 ,x N ] knowing p 1 > 1,p 2 , . . . ,p N coefficients of expansion of f at the points x 1 ,x 2 , . . . ,x N . We start by computing two neighboring N -point Padé approximants (NPAs) of f, namely f 1 = [m/n] and f 2 = [m − 1/n] of f. The second NPA is computed with the reduced amount of information by removing the last coefficient from the expansion of f at x 1 . We assume that f is sufficiently smooth, (e.g. convex-like function), and (this is essential) that f 1 and f 2 bound f in each interval]x i ,x i+1[ on the opposite sides (we call the existence of such two-sided approximants the two-sided estimates property of f ). Whether this is the case for a given function f is not necessarily known a priori, however, as illustrated by examples below it holds for many functions of practical interest. In this case, further steps become relatively simple. We select a known function s having the two-sided estimates property with values s(x i ) as close as possible to the values f(x i ). We than compute the approximants s 1 = [m/n] and s 2 = [m − 1/n] using the values at points x i and determine for all x the weight function α from the equation s = αs 1 + (1 − α)s 2 . Applying this weight to calculate the weighted mean αf 1 + (1 − α)f 2 we obtain significantly improved approximation of f. |
| first_indexed | 2026-03-24T02:24:58Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.5
R. Jedynak (Univ. Technolog.-Humanist., Radom, Poland),
J. Gilewicz (Aix-Marseille Univ., CNRS, CPT, UMR 7332, France)
APPROXIMATION OF SMOOTH FUNCTIONS
BY WEIGHTED MEANS OF N -POINT PADÉ APPROXIMANTS
НАБЛИЖЕННЯ ГЛАДКИХ ФУНКЦIЙ ЗВАЖЕНИМИ СЕРЕДНIМИ
N -ТОЧКОВИХ АПРОКСИМАНТ ПАДЕ
Let f be a function we wish to approximate on the interval [x1, xN ] knowing p1 > 1, p2, . . . , pN coefficients of expansion
of f at the points x1, x2, . . . , xN . We start by computing two neighboring N -point Padé approximants (NPAs) of f , namely
f1 = [m/n] and f2 = [m− 1/n] of f . The second NPA is computed with the reduced amount of information by removing
the last coefficient from the expansion of f at x1. We assume that f is sufficiently smooth, (e.g. convex-like function), and
(this is essential) that f1 and f2 bound f in each interval ]xi, xi+1[ on the opposite sides (we call the existence of such
two-sided approximants the TSE property of f ). Whether this is the case for a given function f is not necessarily known a
priori, however, as illustrated by examples below it holds for many functions of practical interest. In such case further steps
become relatively simple. We select a known function s having the TSE property with values s(xi) as close as possible to
the values f(xi). We than compute the approximants s1 = [m/n] and s2 = [m− 1/n] using the values at points xi and
determine for all x the weight function α from the equation s = αs1 + (1 − α)s2. Applying this weight to calculate the
weighted mean αf1 + (1− α)f2 we obtain significantly improved approximation of f .
Розглянемо функцiю, яку ми хочемо апроксимувати на iнтервалi [x1, xN ], якщо вiдомi p1 > 1, p2, . . . , pN коефiцi-
єнтiв розкладу f у точках x1, x2, . . . , xN . Спочатку ми знаходимо двi сусiднi N -точковi апроксиманти Паде (НАП)
функцiї f , а саме f1 = [m/n] та f2 = [m− 1/n] для f . Другу НАП знаходимо за обмеженою кiлькiстю iнформацiї
шляхом видалення останнього коефiцiєнта розкладу f у точцi x1. Припустимо, що f — достатньо гладка функцiя
(наприклад, опуклого типу) та (це суттєво) f1 i f2 обмежують f у кожному iнтервалi ]xi, xi+1[ з протилежних сторiн
(умову iснування таких двостороннiх апроксимант ми називаємо TSE властивiстю f ). А priori необов’язково вiдомо,
що це припущення виконується для заданої функцiї f. Водночас, як показано на прикладах, що наведенi нижче,
воно виконується для багатьох функцiй, цiкавих з практичної точки зору. В такому випадку подальшi кроки стають
вiдносно простими. Виберемо вiдому функцiю s з TSE властивiстю та значеннями s(xi) настiльки близькими до
значень f(xi), наскiльки це можливо. Далi ми знаходимо апроксиманти s1 = [m/n] та s2 = [m−1/n] за значеннями
в точках xi i визначаємо для будь-якого x вагову функцiю α з рiвняння s = αs1 + (1−α)s2. Застосовуючи цю вагу
при знаходженнi зваженого середнього αf1 + (1− α)f2, отримуємо значно покращене наближення f .
1. Introduction. In this section we briefly summarize the essential properties of the NPAs and the
role played by the TSE property in what amounts to a magic wand in the proposed method. In
Section 2 we analyze certain technical problems related to the rescaling of the reference function and
some simplifications in calculating weights used in determining the weighted mean approximations.
In the remaining sections we illustrate application of the proposed method to some functions of
interest.
1.1. Neighboring N -point Padé approximants. Let f be an analytic function at N different
real points
−R < x1 < x2 < . . . < xN <∞ (1)
having the power expansions
pj−1∑
k=0
ck(xj)(x− xj)k +O
(
(x− xj)pj
)
, j = 1, . . . , N, (2)
c© R. JEDYNAK, J. GILEWICZ, 2013
1410 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS OFN -POINT PADÉ APPROXIMANTS 1411
then the N -point Padé approximant to f, if it exists, is a rational function Pm/Qn noted as follows:
[m/n]p1p2...pNx1x2...xN
(x) =
a0 + a1x+ . . .+ amx
m
1 + b1x+ . . .+ bnxn
, m+ n+ 1 = p = p1 + p2 + . . .+ pN , (3)
and satisfying the following relations:
f(x)− [m/n](x) = O
(
(x− xj)pj
)
, j = 1, 2, . . . , N, (4)
where each pj represents the number of coefficients ck(xj) of expansion (2) actually used for the
computation of NPA given by (3).
In the following we always label the NPA computed with all available values using index “1”, as
f1 = [m/n], and the NPA computed using the number of values reduced by one with index “2”, as
f2 = [m− 1/n]. In all cases, we remove the last coefficient from the expansion of f or s at x1 when
calculating f2 or s2.
1.2. Two-sided estimates property (TSE property). The proposed method of approximation is
based largely on the TSE property which was first proved for the Stieltjes functions by Michael
Barnsley [1]. Let us introduce (see [4, 5]) a nondecreasing step-wise function L
L(x) =
N∑
j=1
pjH(x− xj),
where H is a Heaviside function. L(x) represents the total number of coefficients in power series
expansions of f at all points xj ≤ x:
L(xk) = p1 + p2 + . . .+ pk, L(xN ) = p =
N∑
j=1
pj .
Theorem 1. Let s be a Stieltjes function, then the diagonal s1 = [k/k] and subdiagonal s2 =
= [k − 1/k] N -point Padé approximants to s satisfy the following inequality:
(−1)L(x)[m/n](x) ≤ (−1)L(x)s(x), x ∈]−R,∞[. (5)
After removing one coefficient at x1, s2 becomes exactly [k − 1/k]p1−1...pNx1...xN .
Similar theorem for particular non-Stieltjes functions was proven in [4].
Theorem 2. Let s(x) = s(x1)+ (x−x1)h(x), where h is a Stieltjes function, then the subdiag-
onal s1 = [k + 1/k] and diagonal s2 = [k/k] N -point Padé approximants to s satisfy the following
inequality:
(−1)L(x)[m/n](x) ≥ (−1)L(x)s(x), x ∈]−R,∞[. (6)
Above theorems prove the TSE property for Stieltjes and Stieltjes-like functions, however this
property also holds for a wider class of functions, as illustrated below. For x < x1 all NPAs are
smaller than s. Starting from x1, s2 bounds s on the opposite side with respect to s1 in each interval
[xi, xi+1]. The convergence of NPAs to the Stieltjes functions being very rapid, s1 is a considerably
better approximation of s in [x1, xN ] than s2 : |s−s1| � |s−s2|. In fact this is observed also for the
non-Stieltjes functions. We can further improve the approximation f1 of f by finding the weighted
means mi in each interval [xi, xi+1] such that
∀x ∈ [xi, xi+1] : |f −mi| < |f − f1|, i = 1, 2, . . . , N − 1. (7)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1412 R. JEDYNAK, J. GILEWICZ
1.3. Finding a suitable reference function. Suppose we wish to approximate some smooth
function f on the interval [x1, xN ] knowing p1 > 1, p2, . . . , pN coefficients of expansion of f at
the points x1, x2, . . . , xN . Additionally, let the values f(xi) be regularly distributed, as in the case
of monotonic interpolations of Stieltjes functions at real arguments. We proceed as follows: first we
select similar known function having the TSE property. Next we rescale this function to obtain the
reference function s which should be close to f. For instance we can require that
s(x1) = f(x1) and s(xN ) = f(xN ). (8)
Since we know the analytic form of the reference function s, we can compute for each x the exact
weight function α from the following equation:
s(x) = α(x)s1(x) + (1− α(x))s2(x) that is: α(x) =
s(x)− s2(x)
s1(x)− s2(x)
. (9)
Now we compute two NPAs f1 and f2 of f, expecting that they are located on the opposite sides of
f in each subinterval. We also compute similar NPAs s1 and s2 of s and then using (9), we calculate
the weight function α. The basic idea of our method boils down to the use of weight function α
determined for the reference function s to compute the weighted mean of approximants f1 and f2.
This is the magic wand which delivers the improved approximation.
To simplify the calculations we define in each interval the weights and the weighted means as
follows:
αi = α
(xi + xi+1
2
)
, i = 1, 2, . . . , N − 1,
mi(x) = αif1(x) + (1− αi)f2(x), x ∈ [xi, xi+1],
(10)
expecting that they will give a good approximation of f. This approximation has the following
properties due to the condition (8) of rescaling:
m1(x1) = f(x1) = f1(x1) = f2(x1) = s(x1) = s1(x1) = s2(x1),
mN−1(xN ) = f(xN ) = f1(xN ) = f2(xN ) = s(xN ) = s1(xN ) = s2(xN ).
2. General method of approximation and the problem of rescaling. The Introduction presents
the basic ideas of the presented method of approximation. Practical application of the proposed
method requires some additional considerations.
Reference function. Two properties are required for the reference function s. One is the TSE
property, i.e., the existence of the two sided estimates. This is the case for many real functions such
as Stieltjes functions, function (a+ bx)h(x), where h is the Stieltjes function, other examples are the
function s of Theorem 2, or e−x, which is not Stieltjes.
Because our goal is to improve the approximation of a smooth function f, the second property
required for the reference function s is to be characterized by the same kind of smoothness as f. All
functions presented in this paper are “convex-like”, i.e., the difference between such function and a
convex function on the considered interval is negligible.
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APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS OFN -POINT PADÉ APPROXIMANTS 1413
For instance, the interesting suggestion by Claude Brezinski to use a Hermite interpolation poly-
nomial of f as a reference function s failed (see table of Section 3): this polynomial is not sufficiently
smooth and, both s1 and s2 can be either larger or smaller than s, rather than being on the opposite
sides of s (the property TSE is not satisfied).
The accuracy of the weighted approximation is quite sensitive to the proximity of reference
function s to approximated function f. Since the process of selection of the suitable reference function
is largely “experimental” it is desirable to have a number of choices for s.
Choice of NPA. The two sided estimates may be obtained not only by using [k/k] and [k−1/k]
NPAs. Other NPAs [m/n] and [m − 1/n] can be used as well. For instance, in the following we
successfully use the NPAs [2/3] and [1/3]. The next question concerns the choice of m and n for
NPAs. For this purpose we can use the same procedures as those used for selecting the best PA
largely presented in [2, 3] for the ordinary one-point PA.
Rescaling. To preserve the Stieltjes character of s one must restrict the transformation of Stieltjes
function h to the following:
s(x) = c× h(ax+ b). (11)
Here we have a degree of flexibility: the three parameters being subject to only two conditions (9).
Occasionally the calculated parameters a, b, c can be extremely large. In such cases, we may select
just one condition listed in (9), or give up the Stieltjes character of s. We found the following
rescaling quite effective:
s(x) = (ax+ b)× h(cx+ d). (12)
To assure that reference function s is as close as possible to f, we can use some global condition
minimizing the distances between s and f. We can also use a simple condition at one point x∗ of
interval [x1, xN ]:
s(x∗) = f(x∗) (13)
in cases when we know the value f(x∗).
Calculation of the weights αi. In practical calculation we select the middle point of each
interval to obtain the inequality for αi from (7). We observed a number of times that αi/(1−αi)� 1
and f is not necessarily close to s. Consequently the result is not certain a priori and in some cases
the approximations mi can be worse than f1. Fortunately this arises quite seldom and is usually
limited to one among N − 1 subintervals.
Somewhat simplified version of the presented method of approximation can be reduced to a
calculation of the mean α of all weights:
α = (α1 + α2 + . . .+ αN−1)/(N − 1)
allowing us to define a global approximation
m = αf1 + (1− α)f2. (14)
Our examples show that this method may work equally well.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1414 R. JEDYNAK, J. GILEWICZ
3. Approximation of f(x) = e−x/x. In this example we use the following information at four
points x = 2, 4, 6, 8: f(x1), f ′(x1), f ′′(x1), f(x2), f(x3), f(x4) to compute the NPAs f1 = [2/3]31112468
and f2 = [1/3]21112468. The Stieltjes function h(x) =
1
x
log(x) is used to obtain the reference function
s rescaled by (9) and (13):
s(x) = (−.016247x+ .130118)
log(x)
x− 1
. (15)
The inequality (7) leads to the inequalities αi > Ai. The calculations were performed with:
x Ai αi
3 .93403 .967015
5 .895191 .947596
7 .8883 .944152
It is not surprising that the “bad” NPA f2 = [1/3] contributes less then 5% to the approximation. In
the next table we present the numerical results. Recall that mi in the second column correspond to
the local approximations on the intervals [2i, 2(i+ 1)], i = 1, 2, 3, and the third column corresponds
to the approximation m (14) computed with the mean weight α = .952921.
If we use the Hermite polynomials of interpolation H built using all available data for f as the
reference function s, then clearly s1 = f1 and s2 = f2, so our weighted approximation becomes
identical to s which is not convex-like, not sufficiently smooth (Fig. 1). We see that the result is quite
poor (fourth column). It is not surprising since in most cases the Hermite polynomial of interpolation
does not have the TSE property because s1 and s2 are frequently on the same side of s, as shown in
two last columns (Fig. 2).
x f − f1 f −mi f −m f −H H − f1 H − f2
2 .0 .0 .0 .0 .0 .0
2.5 .000072 −.000084 −.00015 −.00091 .00098 −.00374
3 .0001 .000001 −.000046 −.00312 .00323 −.00013
3.5 .000058 .000027 .000014 −.00337 .00343 .00250
4 .0 .0 .0 .0 .0 .0
4.5 −.000026 −.000017 −.000018 .00562 −.00565 −.00548
5 −.000025 −.000019 −.000020 .00996 −.00999 −.00987
5.5 −.000013 −.0000108 −.0000110 .00896 −.00898 −.00893
6 .0 .0 .0 .0 .0 .0
6.5 .0000087 .0000075 .0000077 −.01604 .01605 .01603
7 .000011 .0000099 .000010 −.03263 .03264 .03262
7.5 .0000079 .0000071 .0000072 −.03540 .03541 .03539
8 .0 .0 .0 .0 .0 .0
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS OFN -POINT PADÉ APPROXIMANTS 1415
Fig. 1. In this scale f and f1 seem superposed. In spite of the distance between f and s weights calculated for s result
in excellent approximation of f.
Fig. 2. Errors of approximations with reference to (15). Approximation mi is approximately 5 times better than f1.
Notice, that only at the point x = 2.5 the NPA f1 is a little better than m1, but m at this point is
rather poor.
Respecting the Stieltjes property of s and using the rescaling (11) with c = 1 we obtain the
following reference function:
s(x) =
log(50127.x− 100191)
50127.x− 100192
, (16)
Using the mean weights m = .629645f1 + .370356f2 we get (Fig. 3)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1416 R. JEDYNAK, J. GILEWICZ
Fig. 3. Errors of approximations with reference to (16). Note that here the scale is multiplied by 10 with respect to Fig. 2.
This illustrates why the choice of the reference function and its rescaling must be very careful.
x f − f1 f −mi f −m
2 .0 .0 .0
2.5 .0000718 −.001723 −.00167
3 .00011 −.00117 −.0011
3.5 .000058 −.000296 −.00029
4 .0 .0 .0
4.5 −.000026 .000035 .000034
5 −.0000251 .000018 .000017
5.5 −.000013 .0000041 .0000038
6 .0 .0 .0
6.5 .0000087 .0000012 .00000091
7 .000011 .0000028 .0000024
7.5 .0000079 .0000026 .0000023
8 .0 .0 .0
4. Approximation of f(x) = e−x. The enormous coefficients in the last rescaled function s
suggest the need to restrict the adjustment of s and f to one point x = 2 and to manipulate three
coefficients a, b, c in (11). As in the last example we consider four points 2, 4, 6, 8. The reference
function becomes simpler, namely:
.303
log(2x)
2x− 1
. (17)
Using the same notations as previously f1 = [2/3], f2 = [1/3] and m = .875895f1 + .1241045f2
we obtain (Fig. 4)
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APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS OFN -POINT PADÉ APPROXIMANTS 1417
Fig. 4. Errors of approximations with reference to (17). Observe the quality of mi and m with respect to f1 for x > 3.5.
The NPA f1 is a little better than m1 only at point x = 3.
x f − f1 f −mi f −m
2 .0 .0 .0
2.5 .000047 −.00017 −.00033
3 .000102 −.000112 .00027
3.5 .000069 −.000015 −.000077
4 .0 .0 .0
4.5 −.000046 .0000013 −.0000028
5 −.000052 −.000013 −.000016
5.5 −.0000305 −.000013 −.000014
6 .0 .0 .0
6.5 .000024 .0000126 .0000155
7 .000025 .0000197 .000023
7.5 .000025 .000016 .000018
8 .0 .0 .0
5. Approximation of gaussian distribution from the tribology problem. This practical problem
prompted us to take a look at different methods of approximation. The following integral of the
Gaussian distribution of asperity heights x in the tribology model of contact of two surfaces appears
in many calculus problems [6] :
g(x) =
1√
2π
e
x2
2
∞∫
x
(s− x)
5
2 e−
s2
2 ds. (18)
A number of analytical formulas for g based on the tabulated numerical values of g have been
published, however, as we demonstrated in [7] the simple Padé approximation gives more accurate
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1418 R. JEDYNAK, J. GILEWICZ
Fig. 5. Errors of approximations with reference to (19). A little superiority of mi and m with over g1 is hardly detectable
on this graph.
Fig. 6. Characteristic gap between approximated function g and reference function s.
results than all previously proposed formulas. As demonstrated below, the accuracy can be further
improved by using the method presented here. Following Greenwood we consider five values of g in
the interval [0, 2]. g(0) = .62, g(2) = .04. For x > 2. g is very close to 0: g(x) < .001. The Stieltjes
function
1
x
log(1 + x) rescaled following (9) at x = 0 and x = 2, and (12) leads to the reference
function
s(x) =
log
(x
2
+ 1
)
x
2
(−.279408x+ .616634). (19)
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APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS OFN -POINT PADÉ APPROXIMANTS 1419
The considered four points are .5; 1; 1.5; 2 (Figs. 5 and 6). The NPAs, as previously, are g1 = [2/3]
and g2 = [1/3]. The weights computed at .75; 1.25; 1.75 are α1 = .992, α2 = .982 and α3 = .974.
The weighted mean is m = .982725g1 + .017275g2.
Computed values of g and s used to obtain the approximation formulas
and errors of approximations
x g s g − g1 g −mi g −m
.0 .616634 .616634 −.00039 −.00043 −.00048
.25 .0404421 .515213 −.0000126 −.000015 −.000018
.5 .272411 .425695 .0 .0 .0
.75 .188069 .345695 .00000046 .00000038 .00000027
1 .132825 .273467 .0 .0 .0
1.25 .0957906 .207699 −.00000076 −.00000065 −.00000066
1.5 .0704232 .147382 .0 .0 .0
1.75 .0526961 .0917191 .0000012 .0000011 .0000011
2 .0400761 .0400761 .0 .0 .0
6. Conclusion. The presented method is based on a novel idea of constructing the weighted mean
approximation of a given function of interest by first determining a similar weighted approximation
of a known function. The accuracy of the approximation is sensitive to the proximity of the two func-
tions. Efficient use of the method requires that a rather large collection of reference functions having
the TSE property is available, this will happen with a wider use. A few examples presented here
clearly demonstrate that the idea is very promising, opening an exciting area for further investigation
in the numerical approximation of functions.
Acknowledgments. The authors wish to thank Andrzej Szechter, an old friend of JG from the
days at Warsaw University (circa 1954), for valuable suggestions.
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Received 17.09.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
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| id | umjimathkievua-article-2518 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:58Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dd/6ffb558c7fd968aeb85781f024a4f1dd.pdf |
| spelling | umjimathkievua-article-25182020-03-18T19:17:19Z Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants Наближення гладких функцій зваженими середніми N-точкових апроксимант паде Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. Let f be a function we wish to approximate on the interval [x 1 ,x N ] knowing p 1 > 1,p 2 , . . . ,p N coefficients of expansion of f at the points x 1 ,x 2 , . . . ,x N . We start by computing two neighboring N -point Padé approximants (NPAs) of f, namely f 1 = [m/n] and f 2 = [m − 1/n] of f. The second NPA is computed with the reduced amount of information by removing the last coefficient from the expansion of f at x 1 . We assume that f is sufficiently smooth, (e.g. convex-like function), and (this is essential) that f 1 and f 2 bound f in each interval]x i ,x i+1[ on the opposite sides (we call the existence of such two-sided approximants the two-sided estimates property of f ). Whether this is the case for a given function f is not necessarily known a priori, however, as illustrated by examples below it holds for many functions of practical interest. In this case, further steps become relatively simple. We select a known function s having the two-sided estimates property with values s(x i ) as close as possible to the values f(x i ). We than compute the approximants s 1 = [m/n] and s 2 = [m − 1/n] using the values at points x i and determine for all x the weight function α from the equation s = αs 1 + (1 − α)s 2 . Applying this weight to calculate the weighted mean αf 1 + (1 − α)f 2 we obtain significantly improved approximation of f. Розглянемо функцію, яку ми хочемо апроксимувати на iнтервалi $[x_1,x_N]$, якщо відомі $p_1 > 1, p_2,... , p_N$ коефіцієнтів розкладу $f$ у точках $x_1, x_2,... , x_N$. Спочатку ми знаходимо дві сусідні $N$-точкові апроксиманти Паде (НАП) функції $f$, а саме $f_1 = [m/n]$ та $f_2 = [m — 1/n]$ для $f$. Другу НАП знаходимо за обмеженою кількістю інформації шляхом видалення останнього коефіцієнта розкладу $f$ у точці $x_1$. Припустимо, що $f$ — достатньо гладка функція (наприклад, опуклого типу) та (це суттєво) $f_1$ i $f_2$ обмежують $f$ у кожному інтервалі $]x_i, x_{i+1}\[$ з протилежних сторін (умову існування таких двосторонніх апроксимант ми називаємо TSE властивістю $f$). А priori необов'язково відомо, що це припущення виконується для заданої функції $f$. Водночас, як показано на прикладах, що наведені нижче, воно виконується для багатьох функцій, цікавих з практичної точки зору. В такому випадку подальші кроки стають відносно простими. Виберемо відому функцію s з TSE властивістю та значеннями $s(x_i)$ настільки близькими до значень $f(x_i)$, наскільки це можливо. Далі ми знаходимо апроксиманти $s_1 = [m/n]$ та $s_2 = [m — 1/n]$ зазначеннями в точках $x_i$ і визначаємо для будь-якого $x$ вагову функцію a з рівняння $s = \alpha s_1 + (1 — \alpha)s_2$. Застосовуючи цю вагу при знаходженні зваженого середнього $\alpha f_1 + (1 — \alpha)f_2$, отримуємо значно покращене наближення $f$. Institute of Mathematics, NAS of Ukraine 2013-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2518 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 10 (2013); 1410–1419 Український математичний журнал; Том 65 № 10 (2013); 1410–1419 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2518/1800 https://umj.imath.kiev.ua/index.php/umj/article/view/2518/1801 Copyright (c) 2013 Gilewicz J.; Jedynak R. |
| spellingShingle | Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants |
| title | Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants |
| title_alt | Наближення гладких функцій зваженими середніми N-точкових апроксимант паде |
| title_full | Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants |
| title_fullStr | Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants |
| title_full_unstemmed | Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants |
| title_short | Approximation of Smooth Functions by Weighted Means of N-Point Padé Approximants |
| title_sort | approximation of smooth functions by weighted means of n-point padé approximants |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2518 |
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