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 APM  Vol.11 No.10 , October 2021
On the Uniform and Simultaneous Approximations of Functions
Abstract: We consider the relation between the simultaneous approximation of two functions and the uniform approximation to one of these functions. In particular, F1 and F2 are continuous functions on a closed interval [a,b], S is an n-dimensional Chebyshev subspace of C [a,b] and s1* & s2* are the best uniform approximations to F1 and F2 from S respectively. The characterization of the best approximation solution is used to show that, under some restrictions on the point set of alternations of F1−s1* and F2−s2*, s1* or s2* is also a best A(1) simultaneous approximation to F1 and F2 from S with F1≥F2 and n=2.

1. Introduction

The interest in the simultaneous approximation started long ago [1] [2] [3] [4]. This paper concerned with the relation between the simultaneous approximation and the uniform approximation. The setting is as follows. Let C [ a , b ] be the set of all real-valued continuous functions defined on the closed interval [ a , b ] with the uniform norm . .

For f C [ a , b ] ,

f = max { | f ( x ) | , x [ a , b ] } .

The norms F A ( p ) , 1 p , on E = C [ a , b ] × C [ a , b ] are defined as follows:

For F = ( F 1 , F 2 ) E

F A ( ) = max { F 1 , F 2 }

F A ( p ) = [ F 1 p + F 2 p ] 1 p , 1 p < .

Now if S is an n-dimensional subspace of C [ a , b ] , then U = { ( s , s ) : s S } is an n-dimensional subspace of E and there exist u * = ( s * , s * ) and v * = ( t * , t * ) where s * , t * S such that:

F u * A ( ) = inf u U F u A ( ) = inf s S max { F 1 s , F 2 s } = F k s * , k = 1 or 2.

Such s * is called a best A ( ) simultaneous approximation to F = ( F 1 , F 2 ) from S. The set of all best A ( ) simultaneous approximations to F from S will be denoted by P S ( F , ) .

For 1 p < ,

F v * A ( p ) = inf u U F u A ( p ) = inf s S { [ F 1 s p + F 2 s p ] 1 p } = [ F 1 t * p + F 2 t * p ] 1 p .

t * is called a best A ( p ) simultaneous approximation to F = ( F 1 , F 2 ) from S. The set P S ( F , p ) denotes the set of all best A ( p ) simultaneous approximation to F from S. And P S ( F k ) is the set of all best uniform approximation to F k from S, k { 1 , 2 } .

We are interested in the relation between the simultaneous approximation and the uniform approximation; in section two, we will show under certain conditions, that if s k * P S ( F k ) then s k * P S ( F , 1 ) , k { 1 , 2 } .

Definition 1 A point t [ a , b ] is called a straddle point for two functions f and g in C [ a , b ] if there exists σ = ± 1 such that

f = σ f ( t ) , g = σ g ( t ) .

Definition 2 The functions f and g C [ a , b ] are said to have d alternations on [ a , b ] if there exists d + 1 distinct points x 1 < < x d + 1 in [ a , b ] such that for some σ = ± 1 ,

f ( x i ) = σ f , if i is odd

g ( x i ) = σ g , if i is even

or

g ( x i ) = σ g , if i is odd

f ( x i ) = σ f , if i is even

We follow [5] [6] [7] [8] for the notations and the terminology of this section which will be used throughout this paper. The uniform approximation theory can be found in [9] [10]. Theorems 1 and 2 of this section and the remark thereafter which are needed for our analysis are direct consequences of theorems 1 and 3 of [6].

Theorem 1 Let S be an n-dimensional subspace of C [ a , b ] which contains a nonzero constant, F = ( F 1 , F 2 ) E then:

(a) s * P S ( F , 1 ) if and only if there exists subsets X 1 = { x i , i I 1 } , X 2 = { x i , i I 2 } of [ a , b ] and positive numbers λ i , i I 1 , μ i I 2 with

i I 1 λ i = i I 2 μ i = 1

such that

θ i ( F 1 ( x i ) s * ( x i ) ) = F 1 s * , i I 1 ,

θ i ( F 2 ( x i ) s * ( x i ) ) = F 2 s * , i I 2 ,

i I 1 θ i λ i s ( x i ) + i I 2 θ i μ i s ( x i ) = 0 s S ,

θ i = ± 1 .

(b) If s * P s ( F , 1 ) with F 1 s * = F 2 s * then s * P S ( F , p ) for all p,

1 < p .

Theorem 2 LetS be an n-dimensional Haar subspace of C [ a , b ] , if F 1 F 2 on [ a , b ] then s * P s ( F , ) if and only if F 1 s * & F 2 s * have a straddle point or n alternations on [ a , b ] with F 1 s * = F 2 s * . Furthermore, if F 1 s * & F 2 s * have n alternations on [ a , b ] then s * is unique.

Remark If t [ a , b ] is a straddle point for F 1 s * & F 2 s * , F 1 F 2 on [ a , b ] then

( F 1 F 2 ) ( t ) = ( F 1 s * ) ( t ) + ( F 2 s * ) ( t ) = F 1 s * + F 2 s * F 1 F 2 .

This implies that ( F 1 F 2 ) ( t ) = F 1 F 2 and

F 1 s * + F 2 s * = F 1 F 2 F 1 s + F 2 s s S .

Hence s * P S ( F , 1 ) .

2. The Main Result

Theorem 3 Let s k * P S ( F k ) , where F k C [ a , b ] , k { 1 , 2 } , F 1 F 2 on [ a , b ] , F = ( F 1 , F 2 ) and S is a 2-dimensional Chebyshev subspace of C [ a , b ] containing a nonzero constant function. And let X = { a = x 1 < x 2 < x 3 = b } be the alternating set for F 1 s 1 * , Y = { a = y 1 < y 2 < y 3 = b } be the alternating set for F 2 s 2 * .

(i) If ( F 1 ( x 1 ) s 1 * ( x 1 ) ) = F 1 s 1 * and ( F 2 ( y 1 ) s 2 * ( y 1 ) ) = F 2 s 2 * , then s 1 * P S ( F , 1 ) .

(ii) If ( F 1 ( x 1 ) s 1 * ( x 1 ) ) = F 1 s 1 * and ( F 2 ( y 1 ) s 2 * ( y 1 ) ) = F 2 s 2 * , then s 2 * P S ( F , 1 ) .

Proof

(i) suppose that ( F 1 ( x 1 ) s 1 * ( x 1 ) ) = F 1 s 1 * and ( F 2 ( y 1 ) s 2 * ( y 1 ) ) = F 2 s 2 * , since F 2 F 1 then

( s 1 * ( x 2 ) F 2 ( x 2 ) ) ( s 1 * ( x 2 ) F 1 ( x 2 ) ) = F 1 s 1 *

and if x [ a , b ] is such that ( F 2 s 1 * ) ( x ) 0 , then

F 1 s 1 * ( F 1 s 1 * ) ( x ) ( F 2 s 1 * ) ( x ) 0 .

Hence there exists a γ [ a , b ] such that

F 2 s 1 * = ( F 2 s 1 * ) ( γ ) .

If γ = a or γ = b then γ is a straddle point for F 1 s 1 * & F 2 s 1 * which implies that s 1 * P S ( F , 1 ) .

If a < γ < b then taking x 1 = z 1 , γ = z 2 , x 3 = z 3 we have:

( F 1 s 1 * ) ( z 1 ) = ( F 1 s 1 * ) ( z 3 ) = F 1 s 1 * ,

( F 2 s 1 * ) ( z 2 ) = F 2 s 1 * ,

a z 1 < z 2 < z 3 b .

Now, since S is a Chebyshev subspace of dimension 2, there exists μ i > 0 , i { 1 , 2 , 3 } such that

μ 1 s ( z 1 ) μ 2 s ( z 2 ) + μ 3 s ( z 3 ) = 0 s S

because 1 S , μ 2 = μ 1 + μ 3 and setting ω i = μ i μ 2 , i { 1 , 2 , 3 } we have ω 1 s ( z 1 ) ω 2 s ( z 2 ) + ω 3 s ( z 3 ) = 0 s S where ω 2 = ω 1 + ω 3 = 1 and from theorem 1 s 1 * P S ( F , 1 ) .

ii) If ( F 1 ( x 1 ) s 1 * ( x 1 ) ) = F 1 s 1 * and ( F 2 ( y 1 ) s 2 * ( y 1 ) ) = F 2 s 2 * , since F 1 F 2 then

( F 1 ( y 2 ) s 2 * ( y 2 ) ) ( F 2 ( y 2 ) s 2 * ( y 2 ) ) = F 2 s 2 *

and if x [ a , b ] is such that ( s 2 * F 1 ) ( x ) 0 , then

F 2 s 2 * ( s 2 * F 2 ) ( x ) ( s 2 * F 1 ) ( x ) 0 .

Hence there exists a γ [ a , b ] such that

F 1 s 2 * = ( F 1 s 2 * ) ( γ ) .

If γ = a or γ = b then γ is a straddle point for F 1 s 2 * & F 2 s 2 * which implies that s 2 * P S ( F , 1 ) .

If a < γ < b then taking y 1 = z 1 , γ = z 2 , y 3 = z 3 we have:

( F 2 s 2 * ) ( z 1 ) = ( F 2 s 2 * ) ( z 3 ) = F 1 s 1 * ,

( F 1 s 2 * ) ( z 2 ) = F 1 s 2 * ,

a z 1 < z 2 < z 3 b .

Now, since S is a Chebyshev subspace of dimension 2, there exists

μ i > 0 , i { 1 , 2 , 3 } such that

μ 1 s ( z 1 ) + μ 2 s ( z 2 ) μ 3 s ( z 3 ) = 0 s S

because 1 S , μ 2 = μ 1 + μ 3 and setting ω i = μ i μ 2 , i { 1 , 2 , 3 } we have ω 1 s ( z 1 ) + ω 2 s ( z 2 ) ω 3 s ( z 3 ) = 0 s S where ω 2 = ω 1 + ω 3 = 1 and from theorem 1 s 2 * P S ( F , 1 ) and the theorem is proved.

The following example shows that conditions (i) & (ii) in theorem 3 are necessary conditions.

Example 1 S = span { 1 , x } is a Chebyshev subspace of C [ 0 , 1 ] and

s 1 * = 1 8 + x is the best uniform approximation to F 1 = x , s 2 * = 1 8 + x is the

best uniform approximation to F 2 = x 2 , F 1 F 2 on [ 0 , 1 ] , s 1 * P S ( F , 1 ) and s 2 * P S ( F , 1 ) .

It is possible, under the assumptions of theorem 3 that both s 1 * and s 2 * belong to the set of best A(1) simultaneous approximation as illustrated in the following example

Example 2 S = span { 1 , x } is a Chebyshev subspace of C [ 0 , 1 ] and

s 1 * = 1 8 + x is the best uniform approximation to F 1 = x 2 , s 2 * = 1 3 3 + x is the

best uniform approximation to F 2 = x 3 , F 1 F 2 on [ 0 , 1 ] .

s 1 * , s 2 * P S ( F , 1 ) . Furthermore s 2 * = 1 3 3 + x is the unique best A ( ∞ )

simultaneous approximation to F = ( F 1 , F 2 ) from S.

Cite this paper: Alyazidi, M. (2021) On the Uniform and Simultaneous Approximations of Functions. Advances in Pure Mathematics, 11, 785-790. doi: 10.4236/apm.2021.1110052.
References

[1]   Dunham, C.B. (1967) Simultaneous Chebyshev Approximation of Two Functions on an Interval. Proceedings of the AMS, 18, 472-477.
https://doi.org/10.1090/S0002-9939-1967-0212463-6

[2]   Watson, G.A. (1993) A Characterization of Best Simultaneous Approximation. Journal of Approximation Theory, 75, 175-182.
https://doi.org/10.1006/jath.1993.1097

[3]   Huotari, R. and Shi, J. (1995) Simultaneous Approximation from Convex Sets. Computers & Mathematics with Applications, 30, 197-206.
https://doi.org/10.1016/0898-1221(95)00097-6

[4]   Pinkus, A. (1997) Uniqueness in Vector-Valued Approximation. Journal of Approximation Theory, 91, 17-92.
https://doi.org/10.1006/jath.1993.1030

[5]   Asiry, M. and Watson, G.A. (1999) On Solution of a Class of Best Simultaneous Approximation Problems. International Journal of Computer Mathematics, 75, 413-425.
https://doi.org/10.1080/00207169908804818

[6]   Asiry, M. and Watson, G.A. (2000) Simultaneous Approximation from Chebyshev and Weak Chebyshev Spaces. Communications in Applied Analysis, 4, No. 3.

[7]   Alyazidi-Asiry, M. (2016) Adjoining a Constant Function to N-Dimensional Chebyshev Space. Journal of Function Spaces, 2016, Article ID: 4813979.
https://doi.org/10.1155/2016/4813979

[8]   Alyazidi-Asiry, M. (2017) Extending a Chebyshev Subspace to a Weak Chebyshe Subspace of Higher Dimension and Related Results. Journal of Applied and Computational Mathematics, 6, Article ID: 1000347.
https://doi.org/10.4172/2168-9679.1000347

[9]   Cheney, E.W. (1966) Introduction to Approximation Theory. McGraw-Hill, New York, London.

[10]   Watson, G.A. (1980) Approximation Theory and Numerical Methods. John Wiley & Sons, Chichester.

 
 
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