On approximation of generalized almost periodic functions •

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I : PRACE MATEMATYCZNE X I I (1968)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I (1968)

Z.

(Poznań)

On approximation of generalized almost periodic functions •

1. R. A. Abbasow [1] discussed the pecularities of the basic the­

orems of the constructive theory of functions in the class of uniformly almost periodic functions. The purpose of this note is to investigate a constructive characterization of generalized almost periodic functions;

the set of these functions will be denoted by 88. (For definitions and basic properties of ^-spaces see [2].)

For readers convenience we give only the definition of a norm which we shall use in the sequel. Let /(•) denote a representant of the element oe88, then (see [2])

T

(*) ll/l!m = sup him

I

-*

21

where = (flf(-); g(-) eA; SN(\g\) ^ 1}, and N denote the set of all real-valued or complex-valued functions g(t), defined on

t <

N — summable on every finite interval ( - T , T ), such that

T

SN(\g\) = lim f N[\g(t)\]dt <

here N denoted the complementary functions (in the sense of Young) to a given functions ilf(-) which generates 88 (see [2], [3]). Elements of 88 are called 88-almost periodic functions. By the Bochner-Fejer kernel wo understand the function

—■ oo <

••• knp(Ppt)i . I щ в л / /М\2

t <

Jcn.(pit) = щ ^ sm —— j sin-^-l and p2, . .. , 0P are arbitrary real rationaly linearly independent numbers. This kernel is non-negative for every t and

T

lim - i - f KM)dt = 1.

T-*oo 2T J '

Roczniki PTM — P race M atem atyczne X I I 11

162

Z. Stypiński

2 . Let /(•) €$ł. we denote by /* and f M>v the Boehner-Fejór polyno­

mials:

T к

fl{x ) = И т - ^ г J

f M A x) lim-^—

T-t-oo %T

T

J ' /(со + ги)Кй(и)йи,

-T

by cok(fjt)m the modulus of smoothness of degree & > 1, ■ k

" * ( /, t)n = supll V + = snpP*/|fe

|Л|«и±о V7 11

and by Е и(Лт ^ e best approximation by Bochner-Fejór polynomials

where the infimum is taken over all these Bochner-Fejór polynomials of degree

T

1 . Let f ( c o , y ) ^ 0 f o r all real x , y , f ( x , - ) € & and let

||f( x , *)||-e^, where ||f( x , -)||- is the norm (*) of f ( x , y ) as a function of y.

Then

t 1

Ит — Гf(ce, y)KA x)dx

t—*oo

2

— T

Proof of this theorems is completely analogous to that given in Г2]

(Theorem 4.8, p. 61).

From the above it is obvious that the following theorems holds:

T

2. ! / / ( •) then

tf> /„ .О «Я, £ ( • ) « * ,

k

Oi) ll/Slfe < Ż (*) I O i < ( 2 * - l) ll/lt,

T

3. I /

T

f{' )e @ and lim f \u\kK A ^)du = 0(y~k),

»oo_у then

Ep(f)m < (i“ ) (/, *)m f o r h > l .

- < И т ^ - f \\f(x,-)\\-KAx)dx.

r~>oo «/

Approximation of generalized almost periodic functions

163 Proof. In account of Theorem 1 we have

В Д к < 11/2—/Urn

T к

И т ^ / 1{ ^ ) f ( X+ V' U)Z A U)du T~*°° _21 v=o ' I

T

l i m " ^ f \1АийтК Ли)йи Я -+oo J f

T

И т ^ I 0>k{f,u)mZ A U) du- T — >«oo ЛА. J

Next, in view of the properties of the modulus of smoothness we obtain

T lim

~2T

T~>°° __

T

f [l + /*M f(»k(f, f*~l)mZ A U)du

T-yoo ЛА. yf

< lim Г

^_1)^(^)йм +

Т

+ Um^F Г

2k(°k(f, yT^K^du

< Ck(y)«>k(f, P~l)m-

Т

^Нш Г = 0(1), /or ya^oo,

T-yoo ЛА. J

Йеп /or

Eft (/)m ^ (/ > *)m

w h ere ck is in d e p e n d e n t o f y .

We say t h a t / € ^ (p) if

and/(-) possesses

absolutely continuous derivatives (^ (0) = Щ.

Th e o r e m 4 . B y all a s s u m p tio n s o f T h e o re m s 3 , f o r a n a rb itra ry f u n c t i o n

/ ( • ) e ^ ( p ) г о е T m o e

164 Z. S t y p i ń s k i

Proof. In virtue of Corollary 1, E ^ f ) - < ck+p(ok+p( f , р, for/e^?.

Thus from obvious properties of the modulus of smoothness we obtain В Д ) й < 0 t+vir'tOkU™, l T \ .

C

. h is seen that Theorems 3 and 4 generalize theorems on best approximation of a 2iz-periodic function as elements of Orlicz space L*M by trigonometric polynomials Tn.

Indeed, it is sufficient to apply Theorems 3 and 4 to E n(f)n

— inf I I / — Tn\\M in place of E ^ f)^ , where ||*||M is the norm in the Orlicz space L*M of 2-n:-periodic functions ([3], § 9), and the infimum is taken over all trigonometric polynomials Tn of degree < n. Then E ^ f ) - = E n(f)M and putting у — n we obtain

E n(f)M ^ cka>k{f, n~l)M.

R eferences

[1] P. А. А б б а со в , Изв. Акад. Наук А. ССР, No 5, 1966, рр. 3 -7 .

[2] J . A lb r y c h t, The theory of Mareinkiewicz-Orlicz spaces, Rozprawy Mat.

27 (1962), pp. 1-55.

[3] M. A. K r a s n o s e l ’ skii, Y. B. R u tic k ii, Convex functions and Orlicz spaces, Groningen 1962.