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48
Математика. Физика
УДК 517.518
EXPONENTIAL METHODS OF SUMMATION OF THE FOURIER SERIES
A. D. Nakhman1, B. P. Osilenker2
Departments: "Applied Mathematics and Mechanics", TSTU (1); "Higher Mathematics", Moscow State University of Civil Engineering, Moscow (2);
Key words and phrases: convergence almost everywhere; convex; estimates of Lp-norms; piecewise-convex sequences.
Abstract: We consider the semi-continuous methods Л = (Xk(h), k = 0,1,...;h > 0} of summation of Fourier series and conjugated Fourier
series, generated by exponential functions X(x, h) = exp(-hua (х)), a> 0. The
estimates of Lp-norms of the corresponding maximal operators are obtained. As consequence, we get some results about exponential method of summation of the Fourier series almost everywhere and in Lp-metric.
Introduction
Consider f = f (x) e L([—n, n]), and let
Uh (f) = U (f, x; X, h) = XX|k|(h)ck (f) exp(ikx), (1)
к=—то
CO
Uh(f) = U(f,x;X,h) = — i £(sgnk)X|k|(h)ck(f)exp(ikx) (2)
к=—да
be the set of a linear means of Fourier series and conjugate Fourier series respectively. In various questions of the analysis there is a problem of behaviour of (1) and (2)
1 n
when h —^ +0. Here ck(f) = —Jf(t)exp(-ikt)dt, к = 0,± 1,±2,... are complex
—П
Fourier coefficients,
Л = (Xk (h), k = 0,1,...} (3)
is infinite sequence defined by the values of parameter h > 0. This sequence defines so-called semi-continuous method of summation. The regularity conditions of such methods will be the following [1, p. 79]:
X0(h) = 1, lim Xk(h) = 1, k = 0,1,...; (4)
h—0
sup £|ДХ k (h)i<®. (5)
h>0k=0
The similar problems for (1) have been studied by L. I. Bausov [2] in case of discrete h.
We consider the semi-continuous methods of summation corresponding, basically, to the case of
X0(h) = 1, Xk(h) = X(x,h) |x=k, k = 1,2,...,
where
X( x, h) = exp(-h<( x)), (6)
2
and function <(x) e C (0, + ro). Note that if Xk (h) = exp(-hk) we get Poisson-Abel means [3, vol. 1, p. 160 - 165].
rn p
Let
|\f ( x)\ pdx
L = L1; \\ f \\=\\ f 11 ) and
\-n
be a norm in Lebesgue space Lp (p > 0;
~ 1 f t f (x) = -- lim I f(x +1) ctg—dt
2 s—^+0 2
E<|t|<n
be a conjugate function; this function exists almost everywhere for each f e L [3, vol. 1, p. 402]. Define
U* (f) = U* (f, x; X) = sup| U (f, x; X, h)|; U* (f) = U* (f, x; X) = sup | U( f, x; X, h)|. h>0 h>0
Estimates of L^-norms
The sequence (3) is called a convex (concave), if A2 = A(AXk (h)) = Xk (h)- 2Xk+1 (h) + Xk+2 (h) > 0 (a^ < 0), k = 0,1,... The sequence
(3) is piecewise-convex, if Ak changes sign a finite number of times,k = 0,1,... Theorem 1. If the sequence (3) is a convex (a concave) and
Xk (h)ln k = 0(1), k — <», (7)
for each h > 0 then the estimates
||U*(f )||p +1| U*(f )||p< Cp,A ||(f )||p, p > 1; (8)
||U*(f)|| +||U*(f)||< Ca (1+1| f (ln+ | f |) ||); (9)
||U*(f )||p +HU.( f)||p < C p,A || (f )||, 0 < p < 1 (10)
hold.
Here C will represent a constant, though not necessarily one such constant. The estimates (8) - (10) remain valid, if a piecewise-convex sequence (3) satisfies to the condition (7) and there is constant C = CA, such, that
| X k (h)|+k|AX k (h)|< Ca (11)
for all h >0, k = 1,2,...
Proofs of both statements will be based on the Abel transform of sums (1), (2) and on the estimates of Fejer means [3, vol. 1, p. 148] by maximal operators
x+h
^ % 1 j*
f = f (x) = sup - I \ f (t) \ dt and f = f (x) = sup
h>0 hx-h h>0
I Kx+Ü dt
h<\t\<n 2tg 2
Thus the inequalities (8) - (10) occur when ||U*(f)\\p +1|U*(f)\\p is replaced by llf% + llf%, [1, vol. 1, p. 58-59, 404].
Theorem 2. Let the sequence (3) be a convex (a concave) and at every h > 0 satisfies the conditions (7) and (4). Then relations:
lim Uh (f) = f; (12)
h^0
lim Uh (f) = f (13)
h^0
hold almost everywhere (a.e.) for every f e L and in the metrics Lp for every p > 1.
The statements remain valid for every piecewise-convex sequence (3), satisfying to conditions (4), (7), (11).
Besides, under the formulated conditions the relation (12) holds in each point of a continuity of function f The relation (12) holds uniformly over x for everyone continuousf. Last statement does not extend, generally speaking, on a case (13).
Auxiliary statements
Lemma. If a piecewise-convex sequence (3) satisfies to conditions (7) then the following relations hold:
U* ( f, x; X) < С f * (x) Z (k +1)|Д2Хk (h)
k=0
(14)
U* ( f, x; X) < С ( * ( x) + f * ( x)) (k +1)| Д2Х k (h)|. (15)
k=0
Proof. We shall prove (15); the relation (14) one can deduce exactly in a similar way. According to the integrated form of Fourier coefficients and Abel transform we obtain
Uh ( f ) = U( f, x; X, h) = lim 1 f f (/)<! V Xk (h) sin k(x -1) U =
N ük=1 J
1 Г n ~ n ~
= - lim ]XN (h) f f (x + t)DN (t) dt + ЫДХN-1(h) + f f (x + t)FN-1(t) dt +
L -n -n
N - 2 n
+ V (k + 1^2Xk (h) f f (x + t)Fk (t) dt к (16)
Here
k л cosl k +1 It k
1 I 9 ] ~ 1 ~ 1 ~ Dk (t) = £ sin v t =-1---; Fk (t) = -— X D(t) =-p - Fk (t)
v=1 2tg-1 2sin-1 k +1 v=0 2tg—t
2 2 2
are conjugate Dirichlet and Fejer kernels, respectively, and
cos(k + 1)t
Fk (t ) =
2 1 '
2(k +1) sin2 21
Further, we shall establish the following estimates for the integrals containing in the right part of (16):
I f ( x+1 )Dk (t) dt
< Cf (x)lnk, k = 2,3,...;
J f (x + t)Fk (t) dt
< С(*(x) + f *(x)), k = 0,1,...
(17)
(18)
(19)
(20)
(21)
2S
and choose a natural numberS = S(k), k = 0,1,..., such that -<n<-.
k +1 k +1
According to (19), (20), we have
For this purpose we shall use the obvious inequalities:
|Dk(t)| + |Fk(t)| < C(k +1), 0 < t <n, k = 0,1,.; |D5k(t)| < C—, 0 <111<n, k = 0,1,...;
Fk (t)
< C-
\t\ 1
(k + 1)t2
0 <\t \<n, k = 0,1,...
-.S-1
I f (x+1 )Dk (t) dt
< С
(k +1) J \ f ( x + t)\dt + £ j J \ f ( x + t)\dt J=12 2J-1 21
\t\<— k+1
-<t <—
k+1 k+1
< C(1 + 2S)/ (x) < Cf (x)ln k, k = 2,3,... Further, we shall prove (18). In view of relations (19) and (21) we obtain
J f ( x +1 )Fk (t ) dt
< С
(k +1) J \ f (x +1)\ dt +
\f\< A
V k+1
J f ( x + t)
-L<и<п 2tg2 k2
dt
S k +1 <•
+ J \ f ( x +1)\dt
J=1(2 ) 2J-1 2 J
2-<t <—
k+1 k+1
< C(f (x) + f * (x)).
The statement (15) is now a direct consequence of equality (16), estimates (17), (18), relations (7) and ДХk(h) = k ^ да. The last relation is valid [3, vol. 1, p. 156] for any convex or piecewise-convex sequence.
Proof of the theorems 1, 2
2
We consider a case of a piecewise-convex sequence Л. Then Д Xк (h) keeps the sign for m < к < n, where m and n are some natural numbers. By Abel summation formula we obtain
£ (к + 1)Д2Хк (h) = Xm+1 (h) - X„+1 (h) + (т + 1)ДХт (h) - (п + 1)ДХ„+1 (h). (22)
к=m
+
Hence, £ (k +1)|A2Xk (h)| is equal to finite number of sums, each of which looks
k=0
like (22); if n then Xn+j(h) + (n + 1)AXn+j(h) ^ 0 [3, vol. 1, p. 155-156]. Now,
using relations (11) (14), (15), and (22) we obtain the second statement of the theorem 1; the first statement can be received by similar arguments.
The statements of the theorem 2 (convergence a.e. and in metrics Lp) follow from (22) and (11) by standard arguments [3, vol. 2, p. 464-465]. It is necessary to notice, that the conditions of regularity of A -method are valid; the validity of (5) follows [4, p. 748] from (22) and (11).
The convergence (12) in points of continuity and uniformly over x follows from (14) and Banach-Shteinhaus theorem. To use this theorem it is enough obtain the boundedness of Lebesgue constants of summation method. In turn, it will be follow from (14), (22), (11) for f = 1 if to notice that Fk (t) > 0.
The last statement cannot be extended to a case (13) because the conjugate function f can lose a continuity in points of continuity f [5, p. 554].
Convex and piecewise-convex exponential summarising sequences
We shall address to consideration of a case (6). It be required to us
X'x (x, h) = -h exp(-h<(x))<'(x), X'^ (x, h) = h exp(-h<(x))(h(<'(x)) - <''(x)). (23) Let restrict oneself, basically, to consideration of functions
<(x) = u a (x), a > 0. Theorem 3. Let u e C2(0,+a>), u > 0, u''<0(xe (0,+»)), 0 <a< 1,
X( x, h) = exp(-h ua (x)), (24)
and
exp (-hua (x)) ln x = 0(1), x ^ (25)
for every h > 0. Then the estimates (8) - (10) are valid and the relations (12), (13) hold a.e. for every f e L and in the metrics Lp, p > 1. These assertions remain valid if
V = V (x) = ahua (u' )2 - (a- 1)(u')2 - uu", a> 0 (26)
has on (0, + finite number of zeros, the conditions (25) is satisfied and there is constant C = Cu a, such that for all h > 0, x e (1,+ro)
xh exp(-hua(x))ua-1(x)|u'(x)| < Cu a. (27)
Proof. We shall apply the results of theorems 1, 2. The condition (7) is satisfied by (25). It is remain to prove that (24) is convex for 0 < a < 1. According to (24), (23), (26) we have
X'xx (x,h) = ahexp(-hua(x))ua-2(x)V(x). (28)
Then X 'xx (x, h) < 0 for u'' (x) < 0 and 0 < a < 1 as it was required to obtain.
Further we shall notice that the formulated condition on function V (x) in (26) provides a piecewise-convexity of sequence (6), defined by (24). Really, let, for example, V(x) is a function of constant sign for m < x < n + 2 (m and n are some
non-negative integers). We shall apply to X(x, h) (as functions of x) the Lagrange theorem twice (on [k, k+1] and on [k + 0j,k + 2] respectively):
AX k (h) = -X'x (k + 91, h);
(h) = (1 -0i)X"x (k + 0i +02, h), (29)
where 01 =01(k), 02 =02(k), 01, 02 e (0,1). Let 0 =01 +02. If m < k < n, then
m < k + 0< n + 2, such that A X k (h) is sequence of constant sign by (29).
Since a number of intervals (with the integer ends) on which V (x) is a function of 2
constant sign, is finite, then A Xk (h) has finite number of changes of a sign. It remain
to note the validity of (11), if condition (28) holds. Theorem 3 is proved.
Examples
1. Let u(x) = ln x, then
X0(h) = 1, X(x, h) = exp(- h lna x) x > 0, a > 0. (30)
It is evidently that (25) holds. For 0 < a < 1 the sequence (6), defined by (30), is convex.
If a > 1, then function (26) vanishes once; hence, the summarising sequence is
piece-convex. It is remain to note that h exp(- h lna x)lna-1 x = —^
ln x
(h lna
x)exp(- h lna x)< Ca at all a > 1 and x > 2.
So, the statements of theorem 3 hold for a case (30) at all a > 0. In particular (a = 1)
1 M 1 ~
c0(f) + X ~ck(f)exp(ikx) ^ f (x) and -i X (sgnk)—Ok(f)exp(ikx) ^ f (x)
1<|k|<»k k=-<» k a.e. for everyone f e L and in metrics Lp, p > 1.
2. Let u( x) = x, then
X0(h) = 1, X(x, h) = exp(- hxa), x > 0, a> 0. (31)
It is evidently that (25) holds. For 0 <a< 1 [6] the sequence (6), defined by (31), is onvex. If a > 1, then function (26) vanishes once; hence, the summarising sequence is piece-convex. It is remain to note that
h exp(- h xa)xa< Ca at all a > 1 and x > 0.
So, the statements of theorem 3 hold for a case (31) at all a > 0. In particular
ia = 1, h = ln—■, 0 < r < 1I we obtain the convergence of Poisson-Abel means
Ur (f, x) = Xr|k|Ck (f) exp(ikx) and Ur (f, x) = -i X (sgnk)r|k|Ck (f)exp(ikx).
r|k|
k=-» k=-<»
3. Consider a method of summation defined by the function
X(x, h) = exp(-h Pn (x)), x > 0, (32)
where Pn(x) = anxn + an-1xn-1 +... + ao, a = an >0 is any polynomial, n = 1,2,... By relation (23) X'Xx (h, x) = exp(-hPn (x))hT(h, x), where
m x) = h(P'n (x))2 - P'n (x) . (33)
The right hand part of (33) is a polynomial of degree of (2n - 2), so it has no more (2n - 2) changes of signs. Hence, the condition of piecewise-convexity of sequence (6) is satisfied.
Verify a condition (11). The production k|AXk (h)| is a value of function
n( x) = -
xhPn ( x) = hPn ( x) Qn ( x)
exp(hP„ (x)) exp(hP„ (x)) Pn (x) where
0n (x) = xPn (x). (34)
Then | n(x) | is bounded, since (x) a n (x a +®), and the first fraction
Pn (x)
hP (x) t
nK ' in (34) looks like -—, t > 0.
exp(hPn ( x)) exp t
So, the statements of theorem 3 hold for a case (32) at all n = 1,2,...
Operator of the translation type
1. Let f e L and
Th (f ) : f (x) a f (x + h) ~ £ ck (f ) exp(zkh) exp(ikx) ( h > 0)
k=-œ
is a translation operator. Using the integral form of Fourier coefficients Ck ( f ) we have
f (x + h)-
(J n (i <» ] ^ « i® 1 ^
f(t)]"" + Scoskh cosk(x-t)\dt —| f(t)j£sinkh sink(x-t)\dt I k=1 J 1k=1 J J
~ (35)
By analogy with (35) we will consider two summarising sequences A, A and operator of translation type
Th(f) = Th((; A, A;x): f (x) a U(f,x;X,h) - ¿7(f,x;X,h); (36)
denote t*( f) = T*(f; A, A; x )= supl Th (/; A, A; x).
h>0
Applying to (36) theorems 1, 2, we obtain the following statements. Theorem 4. If the elements of each sequence A and A satisfy to condition (7) and both sequences have certain character of convexity, then the estimates (8) - (10) hold
with replacement || U*(f) ||p +1| U*(f) ||p on || t*(f) ||p . The statement remains valid
for any piecewise-convex sequences A and A which elements satisfy to conditions of a kind (7), (11).
If the condition (4) is besides, satisfied, then the relation
lim Th (f) = f — f
h^0
holds almost everywhere for everyone f e L and in metrics Lp at any p > 1. 2. The result of theorem 4 can be applied to the operator Th (f)
f (x) a Th (f;u, a;ra, P; x) = ^exp(—hua (| k \))ck (f )exp(ikx) +
k=—»
+ i X (sgnk)exp('- 7®ß (I k I) \k (/) exp(/'kr).
k v h
k=—a)
Let u(x) be one of the following functions: u(x) = lnx, or u(x) = x, and ra(х) = lnx
or ra(х) = x. Then for every a > 0, P > 0 the relation
lim Th (f;u, a; ra, P;х) = f (x) h^0
holds a.e. for every f e L and in the metrics Lp (p > 1).
References
1. Kuk R. Beskonechnye matritsy i prostranstva posledovatel'nostei (Infinite matrices and sequence spaces), Moscow: GIFML, 1960, 471 p.
2. Bausov L.I. Matematicheskii sbornik, 1965, vol. 68(110), no. 3, pp. 313-327.
3. Sigmund A. Trigonometricheskie ryady (Trigonometric series), 2 vol., Moscow: Mir, 1965.
4. Efimov A.V. Mathematics of the USSR - Isvestiya, 1960, no. 24, pp. 743-756.
5. Bari N.K. Trigonometricheskie ryady (Trigonometric series), Moscow: Fizmatlit, 1961, 936 p.
6. Nakhman A.D. Transactions of the Tambov State Technical University, 2009, vol. 15, no. 3, pp. 653-660.
Экспоненциальные методы суммирования рядов Фурье
А. Д. Нахман1, Б. П. Осиленкер2
Кафедры: «Прикладная математика и механика», ФГБОУ ВПО «ТГТУ» (1); «Высшая математика», ФГБОУ ВПО «Московский государственный строительный университет» (2); [email protected]
Ключевые слова и фразы: выпуклые, кусочно-выпуклые экспоненциальные суммирующие последовательности; оценки £р-норм; сходимость почти всюду.
Аннотация: Рассматриваются полунепрерывные методы
Л = (И), к = 0,1,...;к > 0} суммирования рядов Фурье и сопряженных рядов
Фурье, порожденные экспоненциальными функциями х, к) = ехр(- ки а (х)),
а > 0. Получены оценки ¿р-норм соответствующих максимальных операторов. В качестве следствия приводятся некоторые результаты об экспоненциальных методах суммирования рядов Фурье почти всюду и в метрике Ьр.
Список литературы
1. Кук, P. Бесконечные матрицы и пространства последовательностей : монография / Р. Кук. - М. : ГИФМЛ, 1960. - 471 с.
2. Баусов, Л. И. О линейных методах суммирования рядов Фурье / Л. И. Баусов // Мат. сб. - 1965. - Т. 68(110), № 3. - С. 313 - 327.
3. Зигмунд, А. Тригонометрические ряды : пер. с англ. В 2 т. / А. Зигмунд. -М. : Мир, 1965. - 2 т.
4. Ефимов, А. В. О линейных методах суммирования рядов Фурье / А. В. Ефимов // Изв. Акад. наук СССР. Отд-ние мат. и естеств. наук. Сер. мат. -1960. - № 24. - С. 743 - 756.
5. Бари, Н. К. Тригонометрические ряды : монография / Н. К. Бари. - М. : Физматлит, 1961. - 936 с.
6. Nakhman, A. D. Weigted Norm Inequalities for the Convolution Operators / A. D. Nakhman // Вестн. Тамб. гос. техн. ун-та. - 2009. -Т. 15, № 3. - С. 653 - 660.
Exsponentialmethoden der Summierung der Reihen von Fourier
Zusammenfassung: Es werden die halbununterbrochenen Methoden A = {Xk(h), k = 0,1,...;h > 0} der Reihen von Fourier und der verknüpften Reihen von
Fourier, die von den Expontionalfunktionen X( x, h) = exp(-hwa (х)), a> 0. erzeugt
wurden. Es sind die Einschätzungen von Lp-Norm der entsprechenden maximalen Operatoren erhaltet. Als Untersuchung werden einige Ergebnisse über die Expontionalmethoden der Summierung der Reihen von Fourier fast überall und in der
Metrik Lp gebracht.
Méthodes exponentielles de la summation des séries Fourier
Résumé: Sont examinées les méthodes semi-continues A = {Xk(h), k = 0,1,...;h > 0} de la summation des séries de Fourier et des séries
conjuguées générées par les fonctions exponentielles X( x, h) = exp(-hwa ( х)), a> 0.
Sont reçues les estimations Lp-normes des opérateurs correspodants. En qualité de conséquence sont cités les résultats sur les méthodes exponentielles de la summation
des séries Fourier presque partout et dans la métrique Lp.
Авторы: Нахман Александр Давидович - кандидат физико-математических наук, доцент кафедры «Прикладная математика и механика», ФГБОУ ВПО «ТГТУ»; Осиленкер Борис Петрович - доктор физико-математических наук, профессор кафедры высшей математики, ФГБОУ ВПО «Московский государственный строительный университет», г. Москва.
Рецензент: Куликов Геннадий Михайлович - доктор физико-математических наук, профессор, заведующий кафедрой «Прикладная математика и механика», ФГБОУ ВПО «ТГТУ».
