On sequences of expected values of Fourier coefficients of stochastic processes from a generalized Orlicz sequence space

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1990)

W

U

(Poznan)

On sequences of expected values of Fourier coefficients of stochastic processes

from a generalized Orlicz sequence space

1. Introduction. Let X be a real-valued stochastic process on a probability space (Q, F, P) and let cp — (<p „)®=1 be a sequence of «^-functions. Denote by M the space of integrable stochastic processes.

In this paper we define on M a modular q depending on the modulus of continuity and the sequence (<pn). We denote by M Q the modular space generated by q .

Let P be the corresponding generalized Orlicz sequence space. We give sufficient conditions for a, b: M Q^ P to be continuous linear maps, where a = (a„(2Q), b = (bn(X)) are the sequences of expected values of Fourier coefficients of the stochastic process X.

These results generalize a theorem of [2] to the case of stochastic processes. As consequences, we obtain classical theorems concerning absolute convergence of Fourier series, e.g. Bernstein’s theorem, Zygmund’s theorem, etc.

2. Definitions and notation. Throughout this paper we assume that X(t, w) is a measurable 2^-periodic stochastic process in [}(Tx Q), where T = [ — n, 7 t]

and (Q, F, P) is a probability space.

Let the Fourier series of X(t, co ) e L}(T x Q) be

00

X{t,a) )~ X cn(co)eint,

n = — 00

where

c„{co) = i(a B(«)-b„(co)i),

I я I я .

an(co) = - f X(t, œ)cosntdt, bn{œ) = - f X(t, co)sinntdt,

Я

я

= ая{Х) = J an(co)dP, bn = bn(X) = J b„{œ)dP.

Si Q

Moreover, we assume that X(t, co)eLP(T) for each coeQ, where p > 1. The modulus of continuity of a stochastic process X(t, co)eLP(T), p > 1, is defined as follows:

Я

cop<q(X, (5) = sup [ j ( J \X(t + h, co) — X(t, co)\pdt)q,pdP(co)]119

|ft|<<5 a -n where 1 /p+ 1/q = 1 .

By s denote the space of all real sequences s = {x: x = (*„)*=!, xneR}.

Let (p = (cpn)™= i be a sequence of ^-functions ([2], [3]), i.e. the (pn are nondecreasing, left-continuous functions vanishing only at zero.

Now we define on s a functional q : s->R + by q ( x ) = Z®=i cpn(|x„|); q is a modular on s ([2]). The modular space lv generated by the modular q is called the Orlicz sequence space, i.e.

1< p = { x e s : @( 2 x )->-0 as Я — > 0 }.

Let a = (afc)feL0> f = (/^)*°=о be sequences of positive numbers. For XeLP(TxQ), define functionals

e f ’(J0 = а1Ч>2,(Рка м (Х, 2~ %

00

fi, X) = £ е Г Ю -

k = 0

We observe that q %,9(X), Qip,q)(X) are pseudomodulars on LP(TxQ). We denote by M p,q the modular space generated by the modular gip’q\ i.e.

M p,q(a, P) = {Х е Щ Т х й ): g(p’q)(oc, /3, XX) ^ 0 as 2-+0}.

3. Continuity of the linear maps a and b.

L emma 1. Let X(t, co)eLP(T) for all со eQ as a function of t, 1 < p ^ 2, 1 /p+ 1/q = 1. Then

Z ( K M Ie + |b„(co)|e)

n = 2 k ~ 1 + l

< 2~ql2n - qlp[ J n \X(t + n/2k + 1, c o ) - X ( t - n / 2 k + 1, co)\pdt]qlp.

— П

P ro o f. Let

00

X(t + h, œ) ~ j a 0(œ)+ Z [an(co)cosn{t + h) + bn{œ)smn(t + h)'],

n = 1

00

X(t — h, со) ~ %a0(co) + Z [fl„(m)cosn(t — h) + bn{co)smn(t — hj],

n = l

Sequences o f expected values o f Fourier coefficients 193

X(t + h, co) — X(t — h, at)

00

~ I [(-2a „ (w) sin nh) sin nt + ( — 2 bn (w) sin nh) cos nt] .

«=i Denote

Y{t, со) = X(t + h, co) — X(t — h,

The Fourier coefficients of Y(t, со) are

a'o(œ) = 0, a'„(co) = — 2a„(co)sinnh, bfn(co) = — 2bn(w)sin nh.

Since Y(t, œ)eLP(T), by the Hausdorff-Young inequality ([2], [5]) we obtain

(*)

n = l

1 ",

2n f \Y{t,co)\pdt

l Ip

Now we observe that for h = n/2k , k e N , from (*) it follows that

2 k

n = 2 k ~ 1 + i

' 1 "

2n J I Y(t, co)\pdt

1 Ip

Taking 2k 1 ^ n ^ 2 k wé have n / 4 ^ n h ^ n /2 , hence sin n h ^ 1/^/2 and a'n(œ) = 2 |a„(cü)|sinnh ^ ^/2\an(co)\, similarly \b'n{co)\ ^ y/2\bn(co)\.

Using the above inequalities we obtain

2 k

[ £ K(ft))|« + |fc'(o>)l,] 1,<'

n = 2 k ~ 1 + l

< ¹ T J I Y{t, a>r dt V ^ L 2" - .

IIP I

< ¹ "

J \Y(t, a))\pdt ¹ I p

L emma 2. Let X(t, co)eLP(TxQ), 1 < p ^ 2, 1/p+ l/g = 1. Then

I (kl’ + kl") «S

2 -" г п - ‘Ч*со1д(Х, к2~к)

n — 2 k ~ 1 + l

P ro o f. Given Ak = {n: 2k~1 + 1 ^ n < 2fe, neiV}. From the Holder inequality and Lemma 1 we have

I ([J|a„(co)|dP(«)]«+ [ j|fc„(co)|dP((o)]'>)

neAk П Q

« I Q кИ1’«/р(ш)+f |b»|W(®)]

neAfc Q О

= J I [kMl’+kMlW®)

П neAk

< \{2~ql2n - q/p)

L fi S ^4 ^ 2 k + 1,co j 2 fc + 1 , CO dt

qIp

dP(co)

13 — Commentationes Math. 30.1

I (W + W) « 2 - « 2n - 1 ’ w% (X, к2~к).

neAk

T heorem . Let ç = (ç>„)®= i be a sequence of (p-functions such that (pn{ullq) are concave for some 1 < p ^ 2 , 1/p+l/q = 1, n = 1, 2, and let q>n{u) ^ (pn + l (u) for all u ^ 0 and n = 1 ,2 ,... Let a: X-+a(X), b: X-+b(X).

Then a and b are linear, modular continuous maps from the modular space M%'sq((2k), (2~k,q)) to the generalized Orlicz sequence space lv.

P ro o f. We show the continuity of a. Since фп(и) = (pn(ullq) are concave, from Lemma 2 we have

Z ФпОФп!) < Z ^2к(Л|аи|)

neAk neAk

= 2‘ - ‘ {2-‘ + 1 £ < 2 ‘- ‘ ф2,( X 2 - ‘ +1|Aa„|«)

neAk

neAk

I ^{% (} 2 к 1 К 2 ‘ - 1 ^ [ 2 - к+ 1 2 -''' 2 Я-«"’< ,( Я Х ,я 2 - 1)]

neAk

^ 2k- 1(p2k(2-(k + 1)/q2 - 1/2K~1/pcopJ À X , 2~кк))

^ 2k' i (p2k(2-k!q2llq2~ll22 - llp22(op>q{XX, 2~k))

< 2k~1ç 2u{2~klq2ll22~ll22~ll222(Dpq(AX, 2~k))

^ 2 k~ 1 (p2k{2~klq23,2œPfq(ÀX, 2~k))

^ 2 k- 1(p2*{2-klqwpJ 3 1 X , 2 - k)).

Above we have applied the facts that 1/q ^ к < 4, — 1/p < — j and that the modulus of continuity is nondecreasing and positive homogeneous. Hence

00 2k

g(MX)) = E E <ï>„Wa„(20l)

k=

00 00

« Е Е «) 2 .( 2 K W D « i E 2 V 2.(2 -‘rtc0p.,(32X, 2 - ‘))

k

= i ^ }[(2k), (2-fc/9), ЗЯХ].

The continuity of b can be shown similarly.

R em ark s. Taking in the above theorem X(t, co) = X{t) we have

wM (X, S) = <op(X, S) = ( f iX(t + h ) - X ( t r d t ) 1"’

— n

and we obtain the theorem given in [2]. For other applications, see [2].

Hence we obtain the inequality

Sequences of expected values of Fourier coefficients 195

References

[1] J. M u s ie la k , On abssolute convergence o f Fourier series of some almost periodic functions, Ann. Polon. Math. 6 (1959), 145-156.

[2] —, On generalized Orlicz sequence spaces of Fourier coefficients for Haar and trigonometric systems, Colloq. Math. Soc. Janos Bolyai, Proceed, of the Alfred Haar Memorial Conference, August 11-17, 1985.

[3] —, Introduction to Functional Analysis, PWN, Warszawa 1976 (in Polish).

[4] W. U rb a risk a , On absolute convergence o f series o f expected values o f Fourier coefficients of random functions, I, Fasc. Math. 12 (1980) 103-112.

[5] A. Z y g m u n d , Trigonometric Series, Cambridge 1959.

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