# On existence theorems for some class of non-linear Tchebycheff approximations

## Full text

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ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X II (1969)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X II (1969)

t a ń k o

## On existence theorems for some class of non-linear Tchebycheff approximations

X e X

i

i = 1

P a b i ® o ): =

X 0 e X b ,

### for

X 0 e X \ X b ( х) ,

Pab*p

гаь(%)\ >

rab(x)\

rab(x )

### otherwise.

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302 S. S t a ń k o

0

= 1

0

1

2

2

2

1

0

1

### So, by 2.3 ( + + + ) it follows that тах|а^| < M c. B y the Bolzano-

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Non-linear .Tchebycheff approximation 303

1 . 1

1

2

### < 1 lj, \f(x ) ~ f ( xj)\ < 1 Ij holdS. SO \Pa0b0{CO)-f{CC)\ < \Pa0b0{ ® ) - P a 0b0{Xj)\ + + \Pa0bo(0Ci)-f(Xj)\+

\ f { X j ) — f ( x ) \

X e X \

~ f ( x )\

f

1

### (ia,b)eR

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304 S. S t a ń k o

r ab{x) m J

7 = 1 i

1

1

1 0

### ].

R e fe r e n c e s

###  N. J. A c h ie s e r ,

Theory of approximation,

###  B. B o eh m ,

Existence of best Tchebycheff approximations,

###  E. В o re l,

Leęons sur les fonctions des Variables Beetles,

###  E. W . C h en ey,

Introduction to approximation theory,

###  — and H. L. L o e b ,

Generalised rational approximation,

###  Gr. M e in a rd u s ,

Approxim ation von Funktionen und ihre numerische Behand- lung,

###  D. J. N e w m a n and H. S. S h a p iro ,

Approximation by generalised rational functions,

###  J. R. R ic e ,

On the existence o f best Tchebycheff approximations by general rational functions,

###  —

The approximation of functions,

Linear theory,

###  J. L. W a ls h ,

The existence of rational functions of best approximation,

### Trans. Amer. Math. Soc. 33 (1931), pp. 668-689.

DEPARTMENT OF MATHEMATICS I, A. MICKIEWICZ UNIVERSITY, POZNAN K ATEDRA M ATEMATYKI I, UNIW ERSYTET A. MICKIEWICZA, POZNAN

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