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Georgian Mathematical Journal Volume 15 (2008), Number 1, 77–86

ON THE APPROXIMATION PROPERTIES OF A CLASS OF CONVOLUTION TYPE NONLINEAR SINGULAR INTEGRAL OPERATORS HARUN KARSLI

Abstract. In the present paper we obtain both the pointwise convergence and the rate of pointwise convergence theorems of a class of operators defined by Zb T (f ; x, λ) = K(t − x, λ, f (t)) dt, x ∈ ha, bi, a

as (x, λ) → (x0 , λ0 ) in L1 ha, bi, where ha, bi is an arbitrary interval in R. Here λ ∈ Λ and Λ is a nonempty set of indices. 2000 Mathematics Subject Classification: Primary 41A35, 41A25; Secondary 41-41. Key words and phrases: Pointwise convergence, rate of convergence, nonlinear singular integral.

1. Introduction Let λ be a positive real parameter, varying on a certain set of indices Λ having a (finite or infinite) point of accumulation λ0 . We take a family K of functions K : R × Λ × R → R, where K(t, λ, 0) = 0 for all t ∈ R and λ ∈ Λ such that K(t, λ, u) are integrable over R in the sense of Lebesgue measure with respect to the first variable for all values of the third variable, for every λ ∈ Λ. The family K will be called the kernel. In [10] Taberski showed the pointwise convergence of transformations f ∈ L1 (−π, π) at the Lebesgue points of f , by the family of convolution type singular integral operators depending on two parameters of the form Zπ (1) U (f ; x, λ) = f (t) K(t − x, λ) dt, x ∈ (−π, π), −π

where K(t, λ) is the kernel satisfying suitable assumptions. Moreover, Taberski also showed in [10] the approximation properties of the derivatives of these operators. After this fundamental study of Taberski [10], these operators were studied by some other authors. Gadjiev [4] and Rydzewska [8] obtained the pointwise convergence of these operators at generalized Lebesgue points and µ-generalized Lebesgue points of integrable functions in L1 (−π, π), respectively. In 1984, Bardaro and Gori Cocchieri [1] estimated the degree of pointwise approximation c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

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of F´ejer-type singular integrals, which are the special case of (1), in the class of functions f ∈ L1 (R) at the for which the points are generalized Lebesgue point. ˙ Very recently Karsli and Ibikli [5], [6] studied both the pointwise convergence and the pointwise convergence of the rth derivatives of the operators T (f ; x, λ), which are defined by Zb T (f ; x, λ) =

f (t) K(t − x, λ) dt,

x ∈ ha, bi.

a

In approximation theory, however, applications are limited to linear integral operators because the notion of singularity of integral operators is closely connected with their linearity. In 1981, Musielak [7] considered nonlinear integral operators defined by Z (2) Tw f (s) = Kw (t − s, f (t)) dt, s ∈ R, R

replacing the assumption of operator linearity by the Lipschitz condition for Kw with respect to the second variable. This study allows us, using the classical method for linear integral operators [3] to obtain the convergence of the nonlinear integral operators, although the notion of singularity of integral operators is closely connected with their linearity [2]. Recently Swiderski and Wachnicki [9] estimated the pointwise convergence of operators (2) when f belongs to Lp (−π, π) and Lp (R) at the points, where x0 is both a continuous point and a Lebesgue point. The purpose of this note is to study both the pointwise approximation and the degree of pointwise approximation of operators Zb T (f ; x, λ) =

K(t − x, λ, f (t)) dt,

x ∈ ha, bi,

(3)

a

at a generalized Lebesgue point of f ∈ L1 (ha, bi) as (x, λ) → (x0 , λ0 ), where ha, bi is an arbitrary interval in R. Although integrals are natural generalization of integrals of the forms (1) and (2), the restrictions on the function K are heavy but at present the author cannot manage to avoid them. Note that, all of the operators which we have mentioned above are singular integrals. It is not surprising. Indeed, when investigating convergence problems of orthogonal series one encounters the so-called singular integrals. The fact is that for values of λ close to λ0 , the values of the singular kernel which correspond to the values of t, which are at an arbitrary nonzero distance from x0 , are very small, so that the values of singular integrals depend on the value of a function near x0 . This means, for values of λ close to λ0 , f (t) is near f (x0 ) and therefore the singular integrals is roughly equal to f (x0 ).

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2. Preliminaries Let L1 ha, bi be the class of functions Lebesgue integrable in the interval ha, bi, where ha, bi is an arbitrary interval in R. ∼

Denote the auxiliary function f ∈ L1 (R) as ( ∼ f (t), t ∈ ha, bi, f (t) := 0, t∈ / ha, bi.

(4)

We assume that the function K : R × Λ × R → R satisfies the following conditions: a) L(t, λ) is an integrable function with respect to t such that |K(t, λ, u) − K(t, λ, v)| ≤ L(t, λ) |u − v| , for every t,Ru, v ∈ R and for any λ ∈ Λ. b) lim L(t, λ)dt = 0 for every U ∈ U(0) (by U(0) we denote the family λ→λ0

R\U

of all neighbourhoods of zero in R). c) lim [ sup L(t, λ)] = 0 for every δ > 0. λ→λ0 |t|≥δ R d) lim K(t, λ, u) dt = u for every u ∈ R. λ→λ0 R

e) There exists a δ0 > 0 such that L(t, λ) is non-decreasing on (−δ0 , 0] and non-increasing on [0, δ0 ) as a function of t, for each λ ∈ Λ. The condition a) implies that the class of all functions K(t, λ, u) is a kernel. Theorem 1 ([2]). Let 1 ≤ p < ∞ and assume that a function K(t, λ, u) satisfies the condition a). If f ∈ Lp ha, bi, then T (f ) ∈ Lp ha, bi and kT (f )kLp ha,bi ≤ H(λ) kf kLp ha,bi for each λ ∈ Λ. 3. Pointwise Convergence Theorem 2. Suppose that a kernel function K(t, λ, u) satisfies the conditions a), b), c), d) and e). Then at each point x0 for which lim

Zh

1

|f (x0 + t) − f (x0 )| dt = 0,

h→0+ hα+1

0 ≤ α < 1,

(5)

0

holds we have lim

(x,λ)→(x0 ,λ0 )

|T (f ; x, λ) − f (x0 )| = 0,

on any planar set Z on which the function xZ0 +δ L(t − x, λ) |t − x0 |α dt x0 −δ

is bounded, where 0 < δ < δ0.

(6)

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Proof. We assume that δ x0 − δ > a and 0 < x0 − x < , 2 for any 0 < δ < δ0 . From (4) and d) we can write x0 + δ < b,

(7)

Zb T (f ; x, λ) − f (x0 ) =

K(t − x, λ, f (t))dt − f (x0 ) Za

=

Z

∼

K(t − x, λ, f (t))dt − R

∼

K(t − x, λ, f (x0 ))dt R

Z

∼

+

K(t − x, λ, f (x0 ))dt − f (x0 ). R

Taking here the absolute Z |I(x, λ)| ≤

value of both sides, we get ¯ ¯ ∼ ∼ ¯ ¯ K(t − x, λ, f (t)) − K(t − x, λ, f (x )) ¯ 0 ¯ dt

R

¯ ¯ ¯Z ¯ ∼ ¯ ¯ + ¯¯ K(t − x, λ, f (x0 ))dt − f (x0 )¯¯ . ¯ ¯

(8)

R

According to a), ¯ ¯ ¯∼ ¯ ∼ ∼ ∼ ¯ ¯ ¯ ¯ ¯K(t − x, λ, f (t)) − K(t − x, λ, f (x0 ))¯ ≤ L(t − x, λ) ¯f (t) − f (x0 )¯

(9)

holds for any λ ∈ Λ. According to (9) we can rewrite (8) as follows: Z ¯∼ ¯ ∼ ¯ ¯ |I(x, λ)| ≤ L(t − x, λ) ¯f (t) − f (x0 )¯ dt R

¯ ¯ ¯Z ¯ ∼ ¯ ¯ ¯ + ¯ K(t − x, λ, f (x0 ))dt − f (x0 )¯¯ . ¯ ¯ R

Using the property of the point, we split the above inequality into six terms Z ¯∼ ¯ ∼ ¯ ¯ |T (f ; x, λ) − f (x0 )| ≤ ¯f (t) − f (x0 )¯ L(t − x, λ)dt t∈ha,bi /

  x −δ xZ0 +δ Zb  ¯∼ Zx0 ¯  Z0 ∼ ¯ ¯ + f (t) − f (x ) + + + ¯ 0 ¯ L(t − x, λ)dt   x0 a x0 +δ x0 −δ ¯ ¯ ¯ ¯Z ∼ ¯ ¯ + ¯¯ K(t − x, λ, f (x0 ))dt − f (x0 )¯¯ ¯ ¯ R

= I0 (x, λ) + I1 (x, λ) + I2 (x, λ) + I3 (x, λ) + I4 (x, λ) + I5 (x, λ). (10)

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Thus, it is sufficient to show that all terms on the right hand side of (10) tend to zero as (x, λ) → (x0 , λ0 ). First we consider I5 (x, λ). From the condition d) it is easy to see that ¯ ¯ ¯Z ¯ ∼ ¯ ¯ ¯ I5 (x, λ) = ¯ K(t − x, λ, f (x0 ))dt − f (x0 )¯¯ → 0 (11) ¯ ¯ R

as (x, λ) → (x0 , λ0 ). Next we consider I1 (x, λ).

∼

From (4), it is seen that for t ∈ ha, bi, f (t) = f (t). Hence we can write xZ0 −δ

I1 (x, λ) =

|f (t) − f (x0 )| L(t − x, λ)dt. a

According to (7), we obtain xZ0 −δ

I1 (x, λ) ≤

sup L(t − x, λ) x0 −δ 2δ

Thus we get

³

´ I1 (x, λ) + I4 (x, λ) ≤ 2 sup L(u, λ) kf kL1 ha,bi + |f (x0 )| (b − a) ,

(12)

|u|> 2δ

which in view of c) tends to zero as (x, λ) → (x0 , λ0 ). Next we consider I2 (x, λ). Denote Zx0 F (t) := |f (y) − f (x0 )| dy.

(13)

t

According to (5), for each ε > 0 there exists δ > 0 such that F (t) ≤ ε (x0 − t)α+1 . We now fix this δ. Let us deal with I2 (x, λ) in the following way: Zx0 I2 (x, λ) =

|f (t) − f (x0 )| L(t − x, λ)dt x0 −δ

(14)

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By (13) we can write

¯ ¯ ¯ Zx0 ¯ ¯ ¯ ¯ |I2 (x, λ)| = ¯− L(t − x, λ)dF (t)¯¯ . ¯ ¯ x0 −δ

Integration by parts gives Zx0 |I2 (x, λ)| ≤ |F (x0 − δ)| L(x0 − δ − x, λ) +

|F (t)| dt L(t − x, λ). x0 −δ

According to (14) we obtain Zx0 |I2 (x, λ)| ≤ ε δ

α+1

(x0 − t)α+1 dt L(t − x, λ).

L(x0 − δ − x, λ) + ε x0 −δ

Integrating by parts again, we get Zx0 L(t − x, λ)(x0 − t)α dt.

|I2 (x, λ)| ≤ ε (α + 1)

(15)

x0 −δ

We can use a similar method for estimating I3 (x, λ). Then we find the inequality xZ0 +δ

L(t − x, λ)(t − x0 )α dt.

|I3 (x, λ)| ≤ ε (α + 1)

(16)

x0

Combining (15) and (16), we obtain xZ0 +δ

L(t − x, λ) |t − x0 |α dt.

|I2 (x, λ)| + |I3 (x, λ)| ≤ ε(α + 1) x0 −δ

In view of (6) the left-hand side Finally, we consider I0 (x, λ). Z I0 (x, λ) =

of (17) tends to zero as (x, λ) → (x0 , λ0 ). ¯∼ ¯ ∼ ¯ ¯ ¯f (t) − f (x0 )¯ L(t − x, λ)dt

t∈ha,bi / ∼

From (4), f (t) = 0 for t ∈ / ha, bi. Hence

Z

I0 (x, λ) = |f (x0 )|

L(t − x, λ)dt.

t∈ha,bi /

According to (5), for t ∈ / ha, bi we have t < a or t > b. This implies that δ t − x < a − x < x0 − x − δ < − < 0 2 and t − x > b − x > x0 − x + δ > δ > 0.

(17)

NONLINEAR SINGULAR INTEGRALS. APPROXIMATION PROPERTIES

83

Thus for any δ > 0 there exists a neighborhood U of zero such that Z Z Z L(t − x, λ)dt ≤ L(t − x, λ)dt = L(u, λ)du. |t−x|> 2δ

t∈ha,bi /

R\U

Hence we get

Z I0 (x, λ) ≤ |f (x0 )|

L(u, λ)du,

(18)

R\U

which in view of b) tends to zero as (x, λ) → (x0 , λ0 ). Since f and L(t−x, λ) satisfy conditions (5) and (6), respectively, by combining (11), (12), (17) and (18) we get the required result. It is obvious that the proof remains in force even when 0 < x0 − x < 2δ . The relation − 2δ < x0 − x < 0 is obtained in a similar way. This completes the proof of the theorem. ¤ Remark 1. Now we consider the case ha, bi = (−π, π), K(t, λ, u) = Lλ (t) u, f (t) and Lλ (t) being 2π-periodic functions with respect to t. This case was widely used in approximation theory (see [3]). 4. Rate of Pointwise Convergence In this section we shall estimate the rate of the pointwise convergence, which we have obtained in the previous section. Theorem 3. Suppose that a kernel function K(t, λ, u) satisfies the conditions a), b), c), d) and e). Let xZ0 +δ

∆(x, λ, δ) =

L(t − x, λ) |t − x0 | dt, x0 −δ

such that for every |x − t| > 0 and for arbitrary 0 < α < 1, L(x − t, λ) = o(∆α (x, λ, δ)),

(19)

Z L(x − t, λ)du = o(∆α (x, λ, δ))

(20)

R\U

and

¯ ¯ ¯ ¯Z ∼ ¯ ¯ ¯ K(t − x, λ, f (x0 ))dt − f (x0 )¯ = o(∆α (x, λ, δ)) ¯ ¯ ¯ ¯ R

as (x, λ) → (x0 , λ0 ). Then we get |T (f ; x, λ) − f (x0 )| = o(∆α (x, λ, δ)).

(21)

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H. KARSLI

Proof. In Theorem 2 we have obtained the following inequality; Z |T (f ; x, λ) − f (x0 )| ≤ |f (x0 )| L(t − x, λ)dt R\U

h i + sup L(u, λ) kf kL1 ha,bi + |f (x0 )| (b − a) . |u|> 2δ

xZ0 +δ

+ ε(α + 1)

|t − x0 |α L(t − x, λ) dt

x −δ

0 ¯ ¯ ¯ ¯Z ∼ ¯ ¯ ¯ + ¯ K(t − x, λ, f (x0 ))dt − f (x0 )¯¯ . ¯ ¯

R

Here, we take p = α1 , and using the H¨older inequality, we get xZ0 +δ

|t − x0 |α L(t − x, λ) dt x0 −δ xZ0 +δ

[|t − x0 | L(t − x, λ)]α L(t − x, λ)1−α dt

= x0 −δ

 x +δ 1−α  x +δ α Z0 Z0 1 £ ¤ 1  ≤ L(t − x, λ)1−α 1−α dt ([|t − x0 | L(t − x, λ)]α ) α dt x0 −δ



Zb

≤

x0 −δ

1−α

L(t − x, λ)dt

∆α (x, λ, δ)

a α

= ∆ (x, λ, δ). From conditions (19)–(21) we have |T (f ; x, λ) − f (x0 )| = o(∆α (x, λ, δ)). Thus the proof of the theorem is complete. Example 1. Setting K(t, λ, u) =

(

λ u + sin λ2u , t ∈ [0, λ1 ], 0, t∈ / [0, λ1 ],

¤

(22)

where λ ≥ 1 is a real parameter having a point of accumulation λ0 = ∞. For (22), it is seen that for u, v ∈ R and t ∈ [0, λ1 ], |K(t, λ, u) − K(t, λ, v)| ≤ λ |u − v|

(23)

NONLINEAR SINGULAR INTEGRALS. APPROXIMATION PROPERTIES

85

and for u, v ∈ R and t ∈ / [0, λ1 ], |K(t, λ, u) − K(t, λ, v)| = 0.

(24)

Combining (23) and (24), we get

( λ, t ∈ [0, λ1 ], L(t, λ) = 0, t ∈ / [0, λ1 ].

Moreover

Z

Z L(t, λ)dt =

R

and

λdt = 1 1 [0, λ ]

Z ·

Z K(t − x, λ, u) dt = R

¸ · ¸ λu λu 1 λ u + sin dt = λu + sin 2 2 λ

1 [0, λ ]

=u+ Also,

1 λu sin → u as (x, λ) → (x0 , ∞). λ 2

( K(t − x, λ, u) =

λ u + sin λ2u , t − x ∈ [0, λ1 ], 0, t−x∈ / [0, λ1 ].

Hence the kernel function K satisfies the conditions a), b), c), d) and e). If xZ0 +δ L(t − x, λ) |t − x0 |α dt ≤ M < ∞

(25)

x0 −δ

on any planer set, then we obtain lim

(x,λ)→(x0 ,∞)

|T (f ; x, λ) − f (x0 )| = 0.

(26)

Acknowledgments The author is grateful to the referee for his/her valuable comments and suggestions leading to a better presentation of this paper. References 1. C. Bardaro and C. Gori Cocchieri, On the degree of approximation for a class of singular integrals. (Italian) Rend. Mat. (7) 4(1984), No. 4, 481–490. 2. C. Bardaro, J. Musielak, and G. Vinti, Nonlinear integral operators and applications. de Gruyter Series in Nonlinear Analysis and Applications, 9. Walter de Gruyter & Co., Berlin, 2003. 3. P. L. Butzer and R. J. Nessel, Fourier analysis and approximation. Volume 1: Onedimensional theory. Pure and Applied Mathematics, Vol. 40. Academic Press, New York– London, 1971.

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4. A. D. Gadjiev, On the order of convergence of singular integrals depending on two parameters. Special questions of functional analysis and its applications to the theory of differential equations and functions theory, Baku, 1968, 40–44. 5. H. Karsli and E. Ibikli, On the convergence of singular integral operators depending on two parameters. Abstracts of International Conference on Mathematics and Mechanics devoted to the 50-th anniversary from birthday of the correspondent member of National Acad. Sci. Azerbaijan, Professor I. T. Mamedov, Baku, 2005, 107. 6. H. Karsli and E. Ibikli, Approximation properties of convolution type singular integral operators depending on two parameters and of their derivatives in L1 (a, b). Proceedings of the 16th International Conference of the Jangjeon Mathematical Society, 66–76, Jangjeon Math. Soc., Hapcheon, 2005. 7. J. Musielak, On some approximation problems in modular spaces. Constructive function theory ’81 (Varna, 1981), 455–461, Publ. House Bulgar. Acad. Sci., Sofia, 1983. 8. B. Rydzewska, Approximation des fonctions par des integrales singulieres ordinaires. (French) Fasc. Math. No. 7 (1973), 71–81 (1974). ´ 9. T. Swiderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters. Comment. Math. Prace Mat. 40(2000), 181–189. 10. R. Taberski, Singular integrals depending on two parameters. Prace Mat. 7(1962), 173– 179.

(Received 1.05.2006; revised 13.08.2006) Author’s address: Abant Izzet Baysal University Faculty of Science and Arts Department of Mathematics 14280, Golkoy-Bolu Turkey E-mail: karsli [email protected]