Some estimations for Fourier transformsIn this paper there are given conditions, which admit a Fourier transform / of a function f elf

(1)

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVIII (1989)

Anna Musielak (Poznan)

Some estimations for Fourier transforms

In this paper there are given conditions, which admit a Fourier trans

form / of a function f e l f = l f \ — oo, oo), to belong to a generalized Orlicz class I f ( — 00, oo). There are obtained theorems which are generalizations of [2] for m = 1 and [3].

Analogous results for Fourier series in discrete case are given in [4], [6].

In this text, 1 < p ^ 2 in fixed, 1/p+l/q = 1. We define the Fourier transform as an extension to I f of the transform on L1 defined by the formula

1 00

f f{u)e ltudu for t eR = (— oo, oo),

■00

applying the Riesz-Thorin theorem (see [1], p. 208-211).

Let as write f h(t) = f ( t + h) and A (/, t, h) = f (t + h) — f (t), Then we have the following lemmas.

Lemma 1. I f f e lf, then f he lf, f , f hel3 and f h(t) = elht f(t) a.e. (see [1 ], p.

189, 212).

Lemma 2. I f f e lf, then

2 n oo

f \f(t)\qdt ^ 2~q/2\ f \A(f, t + n2~n~1, n2~n)\pdt]qlp < oo

2» - 1 — oo

and

— 2 n ~ 1 oo

f If(t)\qdt ^ 2~ql2[ J \ A ( f 1 + п 2 -п- \ n2~n)\pdt]qlp < oo.

- 2n ~00

P ro o f. It is easy to show that

(1) if h = 7i2~ 1, 2n~1 ^ ^ 2" or - 2" < t ^ - 2 n~ \ then |sin/it| ^ v/2/2.

(2)

286 Anna M u sie la k

From Lemma 1 we have

(2) \f{t)\q\sinht\q = 2~q\(fh — /_*)(0|*.

From (1) and (2) for h = n2~n~1 we get

2" 2"

j \f(t)iqdt < f 2q,2\smht\q\f{t)\q dt

2n ~ 1 2n _ 1

= 2_,/2 / 2n-i

2 ~ q12 J I ( A - Z - J W * . - 00

From inequalities ||^||, < \\g\\p (see [1], p. 211) we obtain 2f |/(f)l’ * « ' 2 - ,/2[ J M k - f - m ’ i i ] * ’

2« —1 — oo

= 2-«,2[ J |Л(/, f + A, 2h)\p d t ] qlp

—00 which gives the first inequality.

The second one we get in the same way.

Let (p be a ^-function with a parameter on ( — oo,oo)x[0, oo), i.e., (p(t, и) is measurable function of t for every и ^ 0 and a non-decreasing continuous function with respect to u,(p(t,0) = 0, ç(t, и) > 0 for и > О, (p(t, и) ->оо as и ->оо for а.е. t е( — oo, oo). Moreover, in the whole paper we shall assume additionally that ф defined as (p(f, и) = (p{t, ui/q) is concave with respect to и ^ 0 for all t g( — o o , oo).

00

In order to estimate the integral | (p(t,\f(t)\)dt the following condi-

— OO

tions for a (^-function (p will be needed:

(i) There exist a constant К > 0 and a non-negative function h e L( — 1, 1) such that

<p(f, u) ^ К \u\q + h(t) for every и ^ 0 and t e ( — 1, 1).

i

(ï) For every К > 0 there holds f (p(t, K)dt < oo.

- 1

(ii) For every natural number n there exist positive numbers m*, m~

such that

2" 1 < t ^ 2" implies (p{t, u) ^ (p(m* , u)

(3)

and

— 2” ^ t < —2" 1 implies q>(t, и) ^ (p(mn , u) for every и ^ 0.

(ii') (p(t,u) is an even function of t, non-decreasing for te (0, oo), for every и ^ 0.

Condition (i) is equivalent to the inclusion U { — 1, 1) c=Zf( — 1, 1) (see [5], p. 43, 45).

Now, we formulate:

Th e o r e m 1. Let q> satisfy condition (ii). Then for a function f e l f Î V (t,\f(t)\)d t< £ 2"~1 (p(m~, 2“ 1/22(“ "+ l)lqœp( f n2~nj)

—00 n= 1

+ £ 2- V K , 2-1'22| - " +,,'«<b(>( / > я2""))+ f \ m ) d t ,

и= 1 - 1

where

u>p(f, à) = sup [ f И (/, t, h)\pdt]llp.

\ h \ < 0 -'o o

Proof. We estimate the integral 2"

h = .f q>(t,\f{t)\)dt.

2 1

We have

2n 2n

I„= j ÿ ( t,\f( t) \9)dt ^ j (p(mt,\f(t)\q)dt.

2n— 1 2n~ 1

From Jensen’s inequality we get

2 "

2n~ 1

From Lemma 2 and the definition of cop we obtain I„ 2 " " 1 <p(m* , 2 - ” +1 2 ^ 2<op(f, n2~y).

Hence

(3) /„ = j <p(f, |/(г )|)Л < 2 " -1^ К , 2 - 1'22«-"+1»''1а)р(/, Jt2-")).

2« —1

(4)

In the same way we get the inequality _ 2n~ 1

(4) f <p(t, IM ) d t ^ 2"~1 (p(m~, 2~ ^ 2<~"-l)l«cop(f, я2~")).

-2"

From (3) and (4) we obtain the thesis, immediately.

From Theorem 1 there follow some corollaries.

Corollary 1. Let (p satisfy (i) and (ii), / e ll. Let the following condition be satisfied:

(a) the series

f 2 2-1'22(-" +1)/«coi,(/, n2~"))

n= 1

and

I 2>(m„+, ж2~’))

n= 1

are convergent. Then there holds 00

(*) .f <P(L \f{t)\)dt < oo.

- oo

1

P ro o f. From (i) we get f q>(t, \f(t)\)dt < oo.

- 1

Corollary 2. Let q> satisfy (i) and (ii'), / eLP. Let the following condition be satisfied:

(b) the series 00

I 2> (2", тс2- "))

n= 1 is convergent. Then (*) holds.

P ro o f. We get Corollary 2 from Corollary 1 with m„ —m ~ — 2".

Corollary 3. Let q> satisfy (ii) and (i'). Then, for f e L 1 n L p, if condition (a) is satisfied, then (*) holds.

Corollary 4. Let (p satisfy (i') and (ii'). Then, for f eL1 r\LP, if condition (b) is satisfied, then (*) holds.

Corollary 5. Let 0 < f ^ q, y > 0. Then, for f eU,

£ n7-/J/9jw p ^f, <oo implies f \t\y \ f (t)f dt < oo (see [2], Theorem 3).

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Proof. The function q>{t, и) = \t\y\uf satisfies the assumption of Corol

lary 2, i.e., (i) with h(t) = 1, К = 1; (ii') and the concavity of (p(t,u)

= \t\y \u\plq with respect to и for 0 < f ^ q.

In this case condition (b) is equivalent to the convergence of the series

f

2n{~m+y+1)\(ap{f, ju2-")i'.

n= 1

From equivalency of conditions

CO 00

£ 2ma2n < oo and £ ns~l a„ < oo

n= 1 n= 1

for a sequence (an) with a „ | 0 we obtain the thesis.

X

In the second part we estimate the integral f (p(t, \f(t)\)dt for

— 00

a function / of a bounded Ф-variation on ( — o o , oo).

Definition. Let Ф be a (^-function without parameter and let / be a function defined on ( — oo, oo). For an arbitrary finite sequence П

— (£i, ..., tn) with — oo < ti < ...< * „ < oo, let us write У(Ф,/, Щ = " £ Ф (|/(гк+1) - / ( У ) .

k = 1

Ф-variation of / on ( — oo, oo) is defined as Уф(Л = sup V (Ф,/, П).

n

In this part we shall assume that a non-decreasing function W on [0, oo), a ф-function Ф without parameter on [0, oo) and a constant C > 0 such that \u\p ^ СФ(и) V(u) for u ^ O are fixed. Then the following result holds.

Lemma 3. For a function f e l f with V0( f ) < oo, we have

2n

f |/(t)|«df < г-*'2 rc«-1 C«-1 ЕКрШЗ*- 1 u [4F(<»(/, Tt2-”» ]* -1

2 « ~ 1

and

- 2n ~ *

f \f(t)\qdt ^ 2~«2 nq~ 1 Cq~1 [Кф(/)]«-1 2~n(q~X) [Ф(ю(/, n 2 - n))]q- \

— 2"

where

o)(f, <5) = sup supess|d (/, t, h)\.

|ft| <ô t e R

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P roof. We shall prove the first inequality. For that we estimate the integrals

00

(5) J„ = [ J \A(f, t + k2~")\'dtY">

— GO

« J Ф ( \ Л ( / , t + я 2 -”)|)У (И (/, t + 1c2-”- ‘ ,

Ü Ог"'[У (о)(/, J Ф(М(/, г + л 2 - " - ', л 2 -”)|)Л ]4"’

“ 00

and

Nn

Л,* = f Ф (И (/,г + л2 -" -', л2'")|)Л

— Nn

J V -1 2 ” + 1 — 1 mn + (k + l ) n / 2 n + 1

= Z I J' <P(\f(t + K2-n- 1) - f ( t - K 2 - n~1)\)dt.

m = - N k = 0 m n + k n / 2 n + 1

We substitute t = u + mn + kn/2n+1 in integral

mn + ( k + l ) n / 2 n + 1

= .Ï ф( |/(н-я2- " -1) - / ( ' - * 2- " - 1)|)л тп + кп/ 2n + *

and we get я2 — *

l mk = J ф(|/(и + тя-^(к+1)тс/2"+1)-/(м + т + (/£:-1)7г/2"+1)|)^г.

о Hence

•/„,n = f Z Ф(^(и + тп + {к + \)к/2п+1)

0 k ~ 0

— f ( u + mn + {k — l)n/2n+1)\)dt.

The integrand is not greater than 2V0 (f), so we have

Taking N -*• oo, we obtain

J

Ф(И(/, г + я г-" "1, я2-" )Ц ^ n2-*V0(f).

(6)

(7)

From Lemma 2, (5) and (6) we get

J |/ ( t ) N t « : 2 - '" 2[ J \A{f, П2-”- 1, я2-")|Л]«"’

>и- 1

^ 2-«/2С9/р[ <Р(ш(/, я2-"))]ч/р

OO

x [ f Ф ^ ^ г + т и г-" -1, Tt2-”)|)d t]4"’

— 00

2 - ql2Cqlpnqlp['f'(a)(f, тг2~"))]9/р[2- пКф(/)]ч/р.

From this we have the thesis.

Theorem 2. Let q> satisfy (ii). Then for a function f e l f with V0(f) < oo we have

J‘ <p(t, \f{t)\)dt

< £ 2"_1ç) [m„+, 2" 1/2 + 1/«я1-1/«С1-1/в[Кф( /) ] 1" 1/в n= 1

x2-"[¥'(< o(/,jt2-"))]1- 1/>!

GO

+ X 2" ~ > [m~, 2 "1/2+1/«л1"1/вС1_1/«[Кф( /) ] 1" 1/в n= 1

x 2 -* [lP (« » (/,it2 -))]1- 1'«}+ f <p(t,\f{t)\)dt.

-1 P ro o f. We have

ï q> (t,\fm )d t= £ j <p(t,\fm*)dt

— oo n = 1 2 n ~ 1

+ X j <p{t, \f(t)\q)dt+ f <p(t, \f{t)\)dt.

n = 1 — 2 ” ~ 1

We estimate the sum

Si^{= f} f <p(*, l/ ( O le)<fr-

n= 12n"~ 1

From Jensen’s inequality for (p(t, •) and Lemma 3 we obtain Si ^

£ 2"~‘ ф {m„+ , 2~n~l 2~Мя~ 11-4/2 1 C4-1 [V * (/)]* '1 [Ф(а)(/, Tt2"

n= 1

(8)

Hence

X 2 " - 4 4 +, 2 -1/2it1" ^ С1- »'«

n — 1

х[Иф( / ) ] , “ ,*2‘ " [1 Р И /, Analogously, we have the inequality

■ s2^{= f} ^j‘ ^<p(f,1^{/ ( г ) И}

и =1 - 2"

< f 2 " -> |in „ -, 2-l/2nl-llqCl-llt4V0(f)T~1,q

n= 1

x2~n[^ ( w (/, л г- "))]1- 1^}.

From estimations of Sj and S2 we get the theorem.

Theorem 2 implies the following.

Corollary 6. Let (p satisfy (ii') and (i), / e U and V0(f) < oo. Let the following condition be satisfied:

(c) the series

f 2 > 2 - 1/2 + 1'вя1" 1/вС1" 1/«[^ф (/)]1" 1/<г

n= 1

х г - О И / , Л2-"))]1- 1'’ }, OO

X 2> Jm“ , 2 - 1' 2 + 1'* я1-1/«С1-1/*[Кф( /) ] 1~1/«

n= 1

x2_"[¥/ (cd(/, л2- "))]1-1/9]

are convergent.

Then (*) holds.

Corollary 7. Let q> satisfy (ii) and (i'), / e l f n L 1 and V0(f) < oo. Then, if condition (c) is satisfied then (*) holds.

Corollary 8. Let q> satisfy (ii') and (i), / e L P and V0(f) < oo. Let the following condition be satisfied:

(d) the series

f 2 > \ 2 n, 2 ~1/2+1/?л:1 “ 1, 9C1 _l l q [ y 0 ( f y f ~ 1/q2 ~ n \ 4 , ( m { f , я2“ "))]1~1/«]

n - 1

is convergent.

Then (*) holds.

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Corollary 9. Let q> satisfy (ii') and (i'). Then for f eLPnL1 with V0( f )

< oo, if condition (d) is satisfied, then (*) holds.

Corollary 10. Let 0 < f ^ q, 0 < r ^ p and y > 0. Then for f eLP with K( f ) < oo

[1] P. L. B u tz e r , R. J. N e s s e l, F o u r ie r A n a l y s is a n d A p p r o x im a tio n , Vol. 1, Birkhauser Verlag, Basel and Stuttgart 1971.

[2] P. G. M a m e d o v , G. I. O s m a n o v , S o m e p r o p e r tie s o f th e F o u r ie r tr a n s fo r m a n d a p r o p e r ty o f c o e ffic ie n ts o f t h e F o u r ie r s e r ie s (in Russian), Izv. Akad. Nauk Azerbajdzanskoj SSR, ser. fiz.-techn. i matem. nauk 2 (1966), 15-24.

[3] I. M o z e jk o , O n a b s o lu te c o n v e r g e n c e o f F o u r ie r tr a n s fo r m s , Functiones et Approximatio 2 (1976), 175-182.

[4] H. M u s ie la k , J. M u s ie la k , S o m e e s ti m a t io n s f o r s e q u e n c e s o f F o u r ie r c o e ffic i e n ts b e lo n g in g t o g e n e r a liz e d O r lic z s e q u e n c e s p a c e s , Comment. Math. 27 (1987), 159-167.

[5] J. M u s ie la k , O r lic z S p a c e s a n d M o d u l a r S p a c e s , Springer-Verlag, Berlin-Heidelberg-New York-Tokyo 1983.

[6] —, O n g e n e r a liz e d O r lic z s p a c e s o f F o u r ie r c o e ffic ie n ts f o r H a a r a n d tr ig o n o m e tr ic s y s te m ,

■ Colloquia Mathematica Societatis Janos Bolyai 49, Alfred Haar Memorial Conference, Hungary, Budapest 1985, 641-649.

implies f \t\y\f( t) fd t < oo.

Corollary 10 for p = q = 2, r = 1 implies Theorem 2 in [3].

References

Some estimations for Fourier transformsIn this paper there are given conditions, which admit a Fourier trans­form / of a function f elf

f

J

Some estimations for Fourier transformsIn this paper there are given conditions, which admit a Fourier transform / of a function f elf