C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 89–98 (2018)
D O I: 10.1501/C om mua1_ 0000000864 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
A THEOREM ON WEIGHTED APPROXIMATION BY SINGULAR INTEGRAL OPERATORS
OZGE OZALP GULLER AND ERTAN IBIKLI
Abstract. In this paper, pointwise approximation of functions f 2 L1;'(R) by the convolution type singular integral operators given in the following form:
L (f ; x) = Z
R
f (t) K (t x) dt; x 2 R; 2 R+0;
is studied. Here, L1;'(R) denotes the space of all measurable functions f for which 'f is integrable on R and ' : R ! R+ is a corresponding weight function.
1. Introduction
The purpose of approximation theory is the approximation of functions by simply calculated functions. This theory is one of the most fundamental and important arm of mathematical analysis. The Weierstrass approximation theorem says that every continuous function de…ned on a closed and bounded interval of real numbers can be uniformly approximated by polynomials. Also, this well-known theorem plays signi…cant role in the development of analysis. Then, Bernstein also proved Weierstrass’s theorem by describing speci…c approximate polynomials known as Bernstein polynomials in the literature. Bernstein polynomials were changed by Kantorovich in order to approximate to the integrable functions. These polynomials and the generalizations were studied in [2], [8] and [11].
Taberski [21] studied the pointwise approximation of integrable functions and the approximation properties of derivatives of integrable functions in L1h ; i, where h ; i is an arbitrary closed, semi-closed or open interval, by a two parameter
Received by the editors: September. 09, 2016; Accepted: June 16, 2017.
2010 Mathematics Subject Classi…cation. Primary 41A35; Secondary 41A25,45P05.
Key words and phrases. -generalized Lebesgue point, pointwise convergence, rate of convergence.
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family of convolution type singular integral operators of the form:
T (f ; x) = Z
f (t) K (t x) dt; x 2 h ; i ; 2 R+0; (1)
where K (t) is the kernel satisfying appropriate assumptions for all 2 and is a given set of non-negative indices with accumulation point 0.
Then, based on Taberski’s indicated analysis, Gadjiev [10] and Rydzewska [16] proved some theorems concerning the pointwise convergence and the order of point-wise convergence of the operators of type (1) at a generalized Lebesgue point and
generalized Lebesgue point of f 2 L1( ; ), respectively.
Further, the results of Taberski [21], Gadjiev [10] and Rydzewska [16] were ex-tended by Karsli and Ibikli [12]. They proved some theorems for the more general integral operators de…ned by
T (f ; x) = b Z a
f (t) K (t x) dt; x 2 ha; bi ; 2 R+0:
Here, f 2 L1ha; bi ; where ha; bi is an arbitrary interval in R such as [a; b], (a; b), [a; b) or (a; b]. As concerns the study of integral operators in several settings, the reader may see also, e.g., [13], [18], [23], [24], [25], [26] and [27].
The main aim of this paper is to investigate the pointwise convergence of convo-lution type singular integral operators in the following form:
L (f ; x) = Z
R
f (t) K (t x) dt; x 2 R; 2 R+0; (2)
where L1;'(R) is the space of all measurable functions f for which f' is inte-grable on R and ' : R ! R+ is a corresponding weight function, at a common -generalized Lebesgue point of f' and '. In this paper, we studied a theorem of the Faddeev type similar to that of Taberski [19].
The paper is organized as follows: First, we introduce the fundamental de…ni-tions in the sequel of Introduction part. In Section 2, we prove the existence of the operators of type (2). Later, we present a theorem concerning the pointwise con-vergence of L (f ; x) to f (x0) whenever x0 is a common generalized Lebesgue point of f' and '.
Consequently, given that linear integral operators have become important tools in many areas, including the theory of Fourier series and Fourier integrals, ap-proximation theory and summability theory, it is possible to use this article in the mathematical theorem.
De…nition 1. A point x0 2 ha; bi is called generalized Lebesgue point of the function f 2 L1ha; bi, if
lim h!0 0 @ 1 (h) h Z 0 jf(t + x0) f (x0)j dt 1 A = 0;
where the function : R ! R is increasing and absolutely continuous on [0; b a] and (0) = 0. Here, also holds when the integral is taken from h to 0 [12] and [16].
De…nition 2. (Class A') Let R+0 be an index set and 0 2 be an accu-mulation point of it. Let the weight function ' : R ! R+ be bounded on arbitrary bounded subsets of R and satis…es the following inequality:
'(t + x) '(t)'(x), x; t 2 R.
Suppose that there exists a function K : R ! R+such that the following conditions hold there: a) k'K kL1(R) M < 1, for all 2 . b) For every > 0, lim ! 0 sup jtj ['(t)K (t)] = 0. c) For every > 0, lim ! 0 Z jtj '(t)K (t)dt = 0. d) lim (x; )!(x0; 0) 1 '(x0) Z R '(t)K (t x)dt 1 = 0.
e) For any 2 ; K (t) satis…es the following inequality: jK (t)j K (t)
and there exists 0 > 0 such that K (t) is non-decreasing on ( 0; 0] and non-increasing on [0; 0) for any 2 .
If the above conditions are satis…ed, then the function K : R ! R belongs to class A'.
2. Main Theorem
De…nition 3. Let L1;'(R) is the space of all measurable functions for which f (t)'(t) is integrable on R. Here ' : R ! R+ be a weight function and the norm in this space is given by the equality:
kfkL1;'(R)=
Z R
f (t) '(t) dt:
Throughout this paper we suppose that the weight function ' : R ! R+ [13]. The following lemma gives the existence of the operators de…ned by (2). Lemma 1. Let ' : R ! R+ be a weight function. If f 2 L
1;'(R); then L (f; x) de…nes a continuous transformation from L1;'(R) to L1;'(R).
Proof. By the linearity of the operator L (f ; x), it is su¢ cient to show that the expression
kL k1= sup f6=0
kL (f; x)kL1;'(R)
kfkL1;'(R)
remains bounded. Now, using Fubini’s Theorem (see, e.g., [7]), we can write kL (f; x)kL1;'(R) = Z R 1 '(x) Z R f (t)'(t) '(t)K (t x)dt dx Z R 1 '(x) 0 @ Z R f (t + x)'(t + x) '(t + x)K (t) dt 1 A dx Z R jK (t)j 0 @Z R f (t + x) '(t + x) '(t)'(x) '(x) dx 1 A dt Z R '(t)K (t)dt Z R f (t + x) '(t + x) dx M kfkL1;'(R).
Thus, the proof is completed.
The following theorem gives a pointwise convergence of the integral operators of type (2) at a common generalized Lebesgue point of f 2 L1;'(R) and the weight function ' : R ! R+.
Theorem 1. If x0 is a common generalized Lebesque point of functions f 2 L1;'(R) and ' : R ! R+; then
lim (x; )!(x0; 0)
on any set Z on which the function sup t2N (x0) '(t) 8 > < > :2K (0) (jx0 xj) + Z N (x0) K (t x) f (jx0 tj)g 0 t dt 9 > = > ; is bounded as (x; ) tends to (x0; 0); where N (x0) = (x0 ; x0+ ).
Proof. Suppose that x0is a generalized Lebesque point of function f 2 L1;'(R). Set E = jL (f; x) f (x0)j : According to condition (d), we shall write
E = jL (f; x) f (x0)j = Z R f (t)K (t x)dt f (x0) Z R f (t) '(t) f (x0) '(x0) '(t) jK (t x)j dt + f (x0) '(x0) Z R '(t)K (t x)dt '(x0) = I1+ I2.
By condition (d) of class A', I2! 0 as (x; ) ! (x0; 0). Now, we investigate the integral I1 i.e: I1 = 8 > < > : Z RnN (x0) + Z N (x0) 9 > = > ; f (t) '(t) f (x0) '(x0) '(t) jK (t x)j dt = I11+ I12:
The following inequality holds for the integral I11i.e:
I11 = Z RnN(x0) f (t) '(t) f (x0) '(x0) '(t) jK (t x)j dt Z RnN(x0) f (t + x) '(t + x) f (x0) '(x0) '(t + x) jK (t)j dt sup jtj['(t)K (t)] '(x) kfkL1;'(R) + f (x0) '(x0) '(x) Z jtj '(t)K (t)dt.
According to conditions (c) and (d) of class A', I11! 0 as ! 0. Next, we can show that I12 tends to zero as (x; ) ! (x0; 0) on N (x0).
I12 = Z N (x0) f (t) '(t) f (x0) '(x0) '(t) jK (t x)j dt = 8 < : x0 Z x0 + xZ0+ x0 9 = ; f (t) '(t) f (x0) '(x0) '(t) jK (t x)j dt sup t2N (x0) '(t) 8 < : x0 Z x0 + xZ0+ x0 9 = ; f (t) '(t) f (x0) '(x0) jK (t x)j dt = sup t2N (x0) '(t) fI121+ I122g .
Let us consider …rst the integral I121. By de…nition of generalized lebesgue point for every " > 0 there exists a > 0 such that
x0 Z x0 h f (t) '(t) f (x0) '(x0) dt < " (h) for all 0 < h < 0. De…ne the new function as
F (t) = x0 Z t f (u) '(u) f (x0) '(x0) du: (2.1)
Then, for every t satisfying 0 < x0 t we have
jF (t)j " (x0 t). (2.2)
Hence, by (2.1) we can write jI121j = x0 Z x0 f (t) '(t) f (x0) '(x0) jK (t x)j dt = (LS) x0 Z x0 jK (t x)j d [ F (t)] ;
where (LS) denotes Lebesgue-Stieltjes integral. Applying integration by parts method to the Lebesgue-Stieltjes integral, we have
jI121j jF (x0 )j jK (x0 x)j + x0
Z x0
According to (2.2) and condition (e) of class A'; we obtain jI121j " ( )K (x0 x) + " x0 Z x0 (x0 t) j(dtK (t x))j .
Now, we de…ne the variations:
A(t) = 8 > < > : t _ x0 x K (s) ; x0 x < t x0 x 0 ; t = x0 x : (2.3)
Taking above variations and applying integration by parts method to last inequality, we get jI121j " ( )K (x0 x) + " xZ0 x x0 x f (x0 x t)g p tA(t)dt = "(i1+ i2):
Let us consider the integral i2. Write
i2 = xZ0 x x0 x f (x0 x t)g p tA(t)dt = 8 < : 0 Z x0 x + xZ0 x 0 9 = ;f (x0 x t)g p tA(t)dt = i21+ i22.
From (2.3), we shall write
i21 = 0 Z x0 x " t _ x0 x K (s) # f (x0 x t)g p tdt = 0 Z x0 x [K (t) K (x0 x )] f (x0 x t)g p tdt (2.4)
and i22 = xZ0 x 0 " t _ x0 x K (s) # f (x0 x t)g p tdt = xZ0 x 0 " 0 _ x0 x K (s) + t _ 0 K (s) # f (x0 x t)g p tdt = xZ0 x 0 (2K (0) K (x0 x ) K (t)) f (x0 x t)g p tdt. (2.5)
Combining (2.4) and (2.5), we obtain i2 = i21+ i22 2K (0) (x0 x) K (x0 x ) ( ) + xZ0 x x0 x K (t) f (x0 x t)g p tdt. Thus jI121j " (i1+ i2) 2"K (0) (x0 x) + " xZ0 x x0 x K (t) f (x0 x t)g p tdt 2"K (0) (x0 x) + " x0 Z x0 K (t x) f (x0 t)g p tdt. (2.6)
We can use a similar method for estimating I122. Then we …nd the inequality
jI122j " xZ0+ x0 K (t x) f (t x0)g p tdt. (2.7)
Consequently, from (2.6) and (2.7), we can write the following inequality:
I12 sup t2N (x0) '(t) fI121+ I122g " sup t2N (x0) '(t) 2 42K (0) (x0 x) + xZ0+ x0 K (t x) f (jx0 tj)g p t dt 3 5 .
Note that in the above inequality we used the hypothesis of the theorem, i.e., boundedness of the following function:
sup t2N (x0) '(t) 2 42K (0) (x0 x) + xZ0+ x0 K (t x) f (jx0 tj)g p t dt 3 5 : Since the remaining expression is bounded by the hypothesis, I12! 0 as (x; ) ! (x0; 0). Thus, we obtain
lim (x; )!(x0; 0)
L (f ; x) = f (x0) and the proof is completed.
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Current address, O.Guller: Ankara University, Faculty of Science, Department of Mathemat-ics, Ankara, Turkey.
E-mail address : ozgeguller2604@gmail.com
ORCID Address: http://orcid.org/0000-0002-3775-3757
Current address, E.Ibikli: Ankara University, Faculty of Science, Department of Mathematics, Ankara, Turkey.
E-mail address : Ertan.Ibikli@ankara.edu.tr
