A method to obtain the best uniform polynomial approximation...

A method to obtain the best uniform polynomial

approximation for the family of rational

functioncbxax 2

1

M. A. Fariborzi Araghi1 ,F. Froozanfar2

1Department of Mathematics, Islamic Azad university, Central Tehran branch,

P.O.Box 13.185.768, Tehran, Iran.

2Ms.student of Mathematics, Islamic Azad university, Kermanshah branch

, Kermanshah, Iran

*Correspondence E‐mail: M. A. Fariborzi Araghi: [email protected]

© 2015 Copyright by Islamic Azad University, Rasht Branch, Rasht, Iran

Online version is available on: www.ijo.iaurasht.ac.ir

Abstract

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion,

we obtain the best uniform polynomial approximation out of nP2 to a class of

rational functions of theform 12 cax on any non symmetric interval ed , .

Using the obtained approximation, we provide the best uniform polynomial

approximation to a class of rational functions of the form 12 cbxax for

both cases 042 acb and 042 acb .

Key words: Chebyshev’s polynomials, Chebyshev’s expansion, uniform norm,

the best uniform polynomial approximation, alternating set.

1. Introduction

.

cos21

sinsincoscoscoscos12

1

0

ptt

ppntppntpntptxTtb

pp

npnnpnpnpn

j

pj

pj

In

section 2, we characterize the best On of the important and applicable subjects in

applied mathematics is the best approximation for functions. A large number of

paper and books have considered this problem in various points of view.

Volume 7, issue 1, Winter 2015, 753-766 Research Paper

Iranian Journal of Optimization, Vol 7, Issue 1,winter 2015 754

Definition 1. [19] Suppose nP denotes the space of polynomials of degree at most

n , then for given edCf , , there exists a unique polynomial nn Pp * such that:

.,*

nn Pppfpf

We call *

np the best polynomial approximation out of nP to f on ed , .

In other words, nn Pp * is the best uniform approximation for function f on

ed , if .,maxmax,, *

nexd

nexd

n PpxpxfxpxfedfE

The main questions of this problem are existence, uniqueness and characterization

of the solution. The existence and uniqueness of the solution for the best

approximation problem have been proved in [15,19].

In resent years, some researches investigated in order to characterize the best

uniform approximation for special classes of functions. Several of these

researches were focused on classes of functions possessing a certain expansion by

Chebyshev’s polynomials. For example Jokar and Mehri in [8] studied

11

aax and 1

1

x . Also Achieser in [1,2] studied 1 ax . Lorentz in

[10] obtained the best uniform approximation for complex

function 1z , Cz , . In the sequel, Lubinesky in [11] showed that

Lagrange interpolants at the Chebyshev zeros yield the best relevant polynomial

approximation of 121

ax on 1,1 . Eslahchi and Dehghan in [6] characterized

the best uniform polynomials approximation to a class of functions 122 xa on

1,1 and cc, . They also in [5] obtained the best uniform approximation to a

class of 1 xTaT qq . Also Elliott in [9] used the generalized form of

Chebyshev’s polynomials in a specific series to obtain the best approximation.

At first some definitions and theorems that will be used throughout this article are

introduced.

Theorem 1. (Chebyshev’s alternation theorem)[15]

Let f be in ],[ edC . Let the polynomial p be in nP and xpxfxe n . Then

p is the best uniform approximation

np to f on ed , if and only if there exist at

least 2n points 221 nxxx in ed , , for which:[14]

iniexd

i xpxfxe

max , with ii xexe 1 .


Definition 2. [4,16] The Chebyshev’s polynomial in 1,1 of degree n is denoted

by nT and is defined by nxTn cos where xcos .

1

Note that nT is a polynomial of degree n with leading coefficient 12 n .

Definition 3. [12] The Chebyshev’s polynomial in ed , of degree n is denoted by *

nT and is defined by nxTn cos* where

de

edx

2cos .

2

Lemma 1. [8] For cosx , 1t and natural number p we have:

,

cos21

cos12

0

ptt

ptxTta

pp

p

j

pj

pj

uniform approximation to the class of 12 cax on ed , and in section 3, using

the results from section 2, we obtain the best uniform approximation for the class

of 12 cbxax on 1,1 .

2. Best Approximation of 12 cax on ed ,

In this section, we determine the best uniform polynomial approximation out of

nP2 to 12 cax on ed , , when 0

a

c or 0

a

c.

Now, we prove the following lemmas to verify Chebyshev’s expansion in two

mentioned cases.

Lemma 2. Suppose that edx , , 0a

c and 2

4de

a

c

. Then, we have:

3

);(

1

32

1

16112

0

2

42

2

42

2

22

xTttdea

t

tdea

t

xa

ca

caxk

k

k


where 1,)(4

2)(

1 2

tde

a

c

a

c

det

4

and nxTn cos)( wherede

x

2cos .

Proof: In the expansion of the function 1,1 2

22

x on 1,1 we have [17]:

),(1

8

1

412

0

2

4

2

4

2

22xTt

t

t

t

t

xk

k

k

5

where, cosx andt

t

2

12 and 12 t . Suppose that

a

c , then

we have

).(1

8

1

4

cos2

1

12

0

2

4

2

4

2

2

22

xTtt

t

t

t

t

tk

k

k

6

According to 2 we can write:

.

22

14

4

cos2

1

1122

22

2

2

222

ed

xt

tde

de

t

txa

c

7

Combining 6 and 7 we obtain:

).()1(

32

)1(

16

22

1

42

0

2

42

2

42

2

222

2

xTttde

t

tde

t

edx

t

tde

k

k

k

8


Suppose that

2)(

42

)(

1de

a

c

a

c

det , consequently 1t .( Note

that for )(

)(42 22

de

deaat

, the condition 1t is not true.)

Thus we have:

).(

)1(

32

)1(

16

2

42

0

2

42

2

42

2

2xTt

tde

t

tde

t

edx

a

ck

k

k

9

where with nxTn cos)( , de

x

2cos , so relation 3 is proved.

Lemma 3. Suppose that edx , and 0a

c. Then we have:

);(1

1

32

1

16112

0

2

42

2

42

2

22

xTttdea

t

tdea

t

a

cxa

caxk

k

kk

10

where

.1,2

41 2

t

a

cde

a

c

det

11

and nxTn cos)( wherede

x

2cos .

Proof: In the expansion of the function22

1

x on 1,1 we have[6]:

,1

1

8

1

41 *

2

0

2

4

2

4

2

22xTt

t

t

t

t

xk

k

kk

12


where cosx ,t

t

2

1 2 .With suppose

ac ,the rest of proof is similar to

the proof of lemma 2. Thus we omit it.

Theorem 2. The best uniform polynomial approximation out of nP2

for 12 cax where 0

a

c, on ed , and 2

4de

a

c

, is as follows:

),(1

32)(

1

32

1

162242

22

2

1

0

2

42

2

42

2*

2 xTtdea

txTt

tdea

t

tdea

txp n

n

k

n

k

k

n

13

and 242

42*

22

1

32

tdea

tpffE

n

nn ,

14 where 1,)(4

2)(

1 2

tde

a

c

a

c

det , nxTn cos)(

wherede

x

2cos .

Proof: Noting to Chebyshev’s alternation theorem we should prove that the

function )(1

)( 222 xpcax

xe nn

15

has 22 n alternating points in ed , . From 3 and 13 we have:

).(1

32)(

1

322242

22

2

2

42

2

2 xTtdea

txTt

tdea

txe n

n

k

nk

k

n

16

By replacing 2p in lemma 1 and subtracting both sides of b from a we

obtain:

.2cos21

2sin2sin2cos2cos2cos)(

24

22

2

2

tt

nntntxTt n

k

nk

k

17

By replacing 17 in 16 , we obtain:


ntt

tt

ntt

tttt

tdea

txe

n

n

2sin2cos21

2sin1

2cos2cos21

2cos2112cos1

1

32

24

42

24

2442

242

22

2

18

Noting to 4 , we have 2

2

4

1

t

tde

a

c. Then we can rewrite 18 in the

form of:

.2sin

44

2

2cos

8

1

32

22

4222

2

2

22

222

242

42

2

n

xa

cde

xdexdea

c

a

c

n

xa

cde

dea

cx

a

cde

tdea

txe

n

n

19

Now if we define:

20,

8

22

222

1

xa

cde

dea

cx

a

cde

xF

21.

44

2

22

4222

2

2

2

xa

cde

xdexdea

c

a

c

xF

Then we have:

.

282

2

22

2

2

2

1

xa

cde

dea

c

a

cx

xF

22


It is easy to conclude that if ed,0 , then xF1 is a monotonic function and if

ed,0 then xF1 is a monotonic function on each interval 0,d , e,0 . Also we

have: .,,1)()( 2

2

2

1 edxxFxF

23

Therefore, according to 22 and definition of nT for edx , , we have:

1)(1 1 xF . Hence according to mean value theorem for every edx , , there

exists a ),0( such that .,),(cos 1 edxxF

24

Therefore, from 23 we can write: ).(sin 2 xF

25

Replacing 24 and 25 in 19 we obtain:

.2cos1

32242

42

2 ntdea

txe

n

n

26

Now, if x varies from d to e , then n2cos varies from )1cos( n to

)cos( and consequently n2cos possesses at least 22 n extermal points,

where it assumes alternately the values 1 . Therefore

np2 is the best

approximation out of nP2 , and 14 will be proved with considering 26 .


for 12 cax where 0

a

c, on ed , will be:

27),(

1

132)(1

1

32

1

162242

22

2

1

0

2

42

2

42

2*

2 xTtdea

txTt

tdea

t

tdea

txp n

nn

k

n

k

kk

n

and 242

42*

22

1

32

tdea

tpffE

n

nn .

28 where

1,241 2

t

a

cde

a

c

det , nxTn cos)(

wherede

x

2cos .


Proof: The proof is similar to the proof of theorem 2.

If we obtain the best uniform polynomial approximation for 225

1

xxf

on

3,3 by using theorem 2, with 3n , xp*

6 we will see that this result is

similar to the best uniform polynomial approximation obtained in [6]. If we obtain

the best uniform polynomial approximation for 225

1

xxf

on 5,5 by

using theorem 3, with 2n , xp*

4 we will see that this result is similar to the best

uniform polynomial approximation obtained in [6].

Example 1. In figure 1, the function 192

12

x

xf has been drawn. The

dashed points show the best uniform polynomial approximation of degree 8

, xp*

8 , to 12 192

x on 2,1 .

Figure 1: The best approximation of 11922

x .

Example 2. In figure 2, both the function and its best uniform polynomial

approximation, xp*

16 , ( the dashed point ) of degree 16 to 125

x on 2,0 has

been shown.


Figure 2: The best approximation of 125

x .

3. Best Approximation of 12 cbxax

In this section, by using the previous theorems, we obtained the best polynomial

approximation for 12 cbxax on 1,1 .


for 12 cbxax on 1,1 is as follows:

29,

21

8

21

8

1

41)()( 224

22

2

1

0

2

4

2

4

2

2

a

bxT

t

t

a

bxTt

t

t

t

t

axpa n

n

k

n

k

k

n

where,

2 2

2 2

2 2

4 41, 4 4 0 , 1. 30

4 4

b ac b act b ac a t

a a

31,

21

18

21

1

8

1

41)()( 224

22

2

1

0

2

4

2

4

2

2

a

bxT

t

t

a

bxTt

t

t

t

t

axpb n

nn

k

n

k

kk

n

where,

2 2

2

2 2

4 41 , 4 0, 1. 32

4 4

b ac b act b ac t

a a

Proof: We can write the function 12 cbxax in the form of:


33

.

2

222

4

4

2

11

a

acb

a

bxa

cbxax

therefore 1,1xSince

1 ,1 , .2 2 2

b b bx d e

a a a

34

Now, by changing x to a

bx

2 in theorems 2 and 3, we have:

.

22arccos2cos

21

21

22

arccos2cos2

22

a

bxT

abxn

ab

ab

abx

na

bxT nn

Case1 : ( 042 acb ) In this case, replacing a

cby

2

2

4

4

a

acb , according to 34 ,

the defined t in theorem 2, changes to 30 where 22 44 aacb . Therefore, we

can prove a by using theorem 2.

Case2 : ( 042 acb ) In this case, replacing a

cby

2

2

4

4

a

acb , according to 34 ,

the defined t in theorem 3, changes to 32 . Therefore, we can prove b by using

theorem 3.

Example 3. In figure 3, we have drawn the function 152

12

xx

xf . Also,

the dashed points show the best uniform polynomial approximation of degree 6

, xp*

6 , to 12 152

xx on 1,1 .


Figure 3: The best approximation of 11522 xx .

Example 4. In figure 4, both the function and its best uniform polynomial

approximation , xp*

16 , ( the dashed point ) of degree 16 to 12 62

xx

on 1,1 are shown.

Figure 4: The best approximation of 1622 xx .

4. Conclusion

As seen in this article, in the sequel of previous researches, the best uniform

approximation for 12 cax was achieved. In this case, we applied the interval

ed , as general in place of 1,1 .


Also, by characterizing the best uniform approximation for 12 cbxax on

1,1 , a more general form than previous approximation in [6,8] was obtained.

References

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[2]-Achieser N.I., Theory of Approximation, Dover, New York, translated from

Russian, 1992.

[3]-Bernstein S.N., Extremal Properties of Polynomials and the Best

Approximation of Continuous Functions of Single Real Variable, State United

Scientific and Technical Publishing House, translated from Russian, 1937.

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1982.

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rational functions,Computers and Mathematics with applications, 2009.

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class of the form 122 xa , Nonlinear Anal., TMA 71 (740_750) , 2009.

[7]-Golomb M., Lectures on theory of approximation, Argonne National

Laboratory, Chicago, 1962.

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ax and

itstransforms, J. Approx. Theory 125 ( 106-115) , 2003.

[12]-Mason J. C., Handscomb D. C., Chebyshev polynomials, Chapman &

Hall/CRC, 2003.

[13]-Newman D.J.,Rivlin T. J., Approximation of monomials by lower degree

polynomials, Aeq. Math. 14 (451-455) , 1976.

[14]-Ollin H. Z., Best polynomial approximation to certain rational functions,

J.Approx. Theory 26 (389-392) , 1979.

[15]-Rivlin T. J., An introduction to the approximation of functions, Dover, New

York, 1981.


[16]-Rivlin T. J., Chebyshev Polynomials. New York: Wiley, 1990.

[17]-Rivlin T. J., Polynomials of best uniform approximation to certain rational

functions,Numer. Math. 4 (345-349) , 1962.

[18]-Timan A.F., Theory of Approximation of a Real Variable, Macmillan, New

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