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# is UPC in the direction of a vector v ∈ S

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(1)

n

n−1

n

r

E

n

E

n

n

n



(2)

n

n

n

n

v

n−1

n−1

n

v

n−1

v

E

r

E

n

n−1

v

v+

E

2E

2

1/2

1

n

E

1

n

E

E

|E

n

n

v+

E

ε→0+

E

(3)

n−1

n

n

v

v

n

v

v

n

v

n−1

n

n

v+

E

0≤r≤1

v

n−1

−1

12

−1

−1

1

E

E

E

(4)

1

v

0≤r≤1

v

0≤r≤1

v

0≤r≤1

v

0≤r≤1

v

0≤r≤1

v

−1

1

R→1+

−1

−1

v+

E

v+

E

ε→0+

ε→0+

−1

0≤r≤1

v

v+

E

v

(5)

n

v+

E

0≤r≤1

2

v

−1

n

v

0≤r≤1

2

v

2

1/2

v+

E

2

−α

[−1,1]

k

α

k

α

2

α

α

1/2

0

[−1,1]

0

1/2

k

1/2

k

α

k

0

(6)

1/2

1−θα

θ0

1−θ0

1−θα

n

n

n−1

v

n

m

1

n

v

2m

E

v+

E

0≤r≤1

−m

−(m−1/2)

v

2

2

−(m−1/2)

E

v

2

n

n−1

0

0

x

n

x

x

v

x

m

0

x∈E0

x

n

1

n

v

E

2m

E

M1

2m

(7)

2

p

i

n−1

i

i

1

n

2m

E

i=1,...,n

i

n−1

0

x

x

n

m

n

2m

E

1

n

√2

M

2m

n

v

m

1

n

v

−1/(2m)

v

−(1−1/(2m))

E

v

E

m−1/2

−1/2

2m

E

0

0

0

0

v

0

m

−m

2

2m

0≤r≤1

2

v

−1

−1/m

v

−(1−1/(2m))

(8)

0≤r≤1

2

m

m−1/2

−1/2

2

−(m−1/2)

v

−1/2

m−1/2

2

−(m−1/2)

E

n

n

n−1

v

E

2

E

v

v

m

0≤r≤1

2

v

−1

v

−α

n

0≤r≤1

2

v

−1

v

−α

v

−2m

−2m2

m

v

2

2

v

α

2

2

v

α

α

(9)

v

−(1+α)

−(1+α)/2

2

1+α

v

α2

v

## (tx) ≥ 2

−(1+2α+3α2+...+kαk−1+kαk)/2

2

1+α+...+αk

v

αk+1

v

−2m

−2m2

2

m

−2m

−2m2

m

v

2

−(1−|x|)−1

v

1

E

2

E

1

n

i

α

n

1

1/α1

n

1/αn

1

n

n

ei

n

j=1,j6=i

j

1/αj

αi

i

i

j6=i

j

ei

n

j=1,j6=i

j

1/αj

1/αj

αi

i

1/βi

n

j=1,j6=i

j

1/αj

αi

1/βi

n

j=1,j6=i

j

1/αj

αi

1/βi

αi

i

αi

(10)

i

j6=i

j

−αi

α

i

Eα

i

αi−1/2

−1/2i

j6=i

j

αi

i

Eα

1

n

0≤r≤1

2

ei

−1

i

i

1/2

ei

−(1−1/(2αi))

α

i

i

j6=i

j

1/2

ei

−(1−1/(2αi))

Eα

α

1

n

2

−1

1

2

e1

e2

−1

1

E

2

E

e2+

E

0≤r≤1

2

−1

2

−1/2

2

2

−1/2

2

2

−1/2

2

−1/(1+

√5)

(11)

2

2+2/(1+

√5)

E

2

2

−1/2

2

1+2/(1+

√5)

2

1/2

E

2

−1/2

E

2

E

2

E

## .

n

n

n

n

n

### [Z] M. Z e r n e r, D´eveloppement en s´erie de polynˆomes orthonormaux des fonctionsind´efiniment diff´erentiables, C. R. Acad. Sci. Paris 268 (1969), 218-220.

INSTITUTE OF MATHEMATICS INSTITUTE OF MATHEMATICS UNIVERSITY OF MINING AND METALLURGY JAGIELLONIAN UNIVERSITY

AL. MICKIEWICZA 30 REYMONTA 4

30-059 KRAK ´OW, POLAND 30-059 KRAK ´OW, POLAND E: mail: BARAN@IM.UJ.EDU.PL

### Re¸cu par la R´edaction le 27.5.1993

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