Math. Nachr. 217 (2000), 5 – 11

Rate of Convergence of Singular Integrals in Holder Norms

By Jorge Bustamante of Puebla

(Received December 16, 1997)

(Revised Version April 7, 1999)

Abstract. We present an approximation theorem in the Holder norms related with

Poisson –Cauchy and Gauss– Weiertrass singular integrals. The theorem improves a previous result

of Mohapatra and Rodriguez in [3].

1. Introduction

Let Hβ (0 < β < 1) be the linear space of all 2π – periodic real functions f definedon IR (IR is the space of real numbers) such that, for some positive constant K andevery x, y ∈ IR,

|f(x) − f(y)| ≤ K |x− y|β .(1.1)

It is known that Hβ endowed with the norm ‖f‖β := ‖f‖∞ + ‖f‖∞,β is a Banachspace, where ‖ · ‖∞ denotes the sup norm and ‖f‖∞,β is the infimun of the set of allconstant K satisfying (1.1).The interest in estimate the degree of approximation of functions in the Hβ – norm

has increased in last decades. In [8], Prossdorf studied approximation by Fejermeans of Fourier series. In [5] and [6], Mohapatra and Chandra presented resultsconcerning the degree of approximation of a function f ∈ Hβ by using operatorsrelated to the Fourier series of f . There is a special interest in to apply approximationmethods to study singular integral equations (see [9]). Kahn [4] and Mohapatraet al. [7], have considered error bounds of functions f ∈ Lp[0, 2π] by using Picard,Poisson –Cauchy and Gauss –Weierstrass singular integrals. Recall that the Poisson –Cauchy and Gauss –Weiertrass singular integrals of a function f are given by

Q(f, ζ)(x) := (ζ/π)∫ π

−π

f(x + t)/(t2 + ζ2

)dt ,(1.2)

1991 Mathematics Subject Classification. Primary: 42A10; Secondary: 42A50, 41A25.Keywords and phrases. Trigonometric approximation, singular integrals, degree of appro-

ximation.

6 Math. Nachr. 217 (2000)

and

W (f, ζ)(x) :=(1/√

πζ) ∫ π

−π

f(x + t) exp(− t2/ζ

)dt .(1.3)

Some estimates for the approximation of these singular integrals, in Hβ norms, werepresented by Mohapatra and Rodriguez as follows:

Theorem 1.1. ([7].) Let 0 < β < α ≤ 1 and f ∈ Hα. Then there exists a constantC > 0 such that, as ζ → 0+,

‖f − Q(f, ζ)‖β ≤ Cζα−β |log(1/ζ)|(1.4)

and

‖f − W (f, ζ)‖β ≤ Cζα−β−(1/2) .(1.5)

The subject of this paper is to improve the theorem above. In order to present themain result we need some notations. Let hβ (0 < β < 1) be the space of all functionsf ∈ Hβ such that limh→0+ ϕ(f, h) = 0, where

ϕ(f, h) := sup0<|x−y|≤h

| f(x) − f(y) || x − y |β .

Theorem 1.2. Let 0 < β < 1 be fixed. For each f ∈ hβ and ζ ∈ (0, 1], if Q(f, ζ)and W (f, ζ) are given by (1.2) and (1.3) then,

(i)

‖f − Q(f, ζ)‖β ≤ 2 ‖f‖β

π2ζ +

32

ϕ(f, ζ) +163

ζ

∫ π

ζ

ϕ(f, t)t2 + ζ2

dt ;(1.6)

(ii)

‖f − W (f, ζ)‖β ≤ ‖f‖β ζ + 8ϕ(f,√

ζ).(1.7)

In order to compare Theorems 1.1 and 1.2 we need the following proposition. The-orem 1.2 and Proposition 1.3 will be proved in the next section.

Proposition 1.3. If α ∈ (β, 1] and f ∈ Hα, then for every ζ ∈ (0, π)

ϕ(f, ζ) ≤ Cfζα−β and ζ

∫ π

ζ

ϕ(f, t)t2 + ζ2

dt ≤ Cfζα−β .(1.8)

It follows from Proposition 1.3 that (1.6) and (1.7) are sharper than (1.4) and (1.5)respectively. It is known that there exist functions f ∈ hβ such that, for every ε > 0,f /∈ Hβ+ε (see [1]). For every f ∈ hβ , all the expressions in (1.6) and (1.7) convergeto zero when ζ does. An analogous assertion cannot be derived from Theorem 1.1. Inparticular, for f ∈ Hα (α ∈ (β, 1]), we can infer convergence to zero in (1.5) only forα > β + (1/2), while in (1.7) we have convergence for every α > β.

Bustamante, Rate of Convergence of Singular Integrals 7

2. Proof of the results

Throughout the paper, we denote φt(x) := f(x + t) − 2f(x) + f(x − t), for x ∈ IRand t > 0. Notice that, for f ∈ hβ,

|φt(x)| ≤ 2tβϕ(f, t)(2.1)

and

|φt(x) − φt(y)| ≤ 4min{tβϕ(f, t), | x − y |β ϕ(f, | x − y |)}

≤ 4 |x− y|β ϕ(f, t) .(2.2)

For ζ ∈ (0,∞), define F (ζ) := 1ζ

(1− 2ζ

π

∫ π

0dt

t2+ζ2

). It follows that F ′(ζ) < 0, thus

for each ζ ∈ (0,∞)

0 = limx→∞F (x) < F (ζ) < lim

x→0+F (x) =

2π2

.(2.3)

P r o o f of Theorem 1.2, part (i). We begin by estimating ‖Q(f, ζ) − f‖∞. Notethat, taking into account (2.1),

ζ

2

∫ ζ

0

| φt(x) |t2 + ζ2

dt ≤ ζ

∫ ζ

0

ϕ(f, t)tβ

t2 + ζ2dt

≤ ζ1+βϕ(f, ζ)∫ ζ

0

dt

t2 + ζ2

=π

4ζβϕ(f, ζ) .

(2.4)

It follows from (2.3) and (2.4) that, for x ∈ IR

|Q(f, ζ)(x)− f(x)| ≤ |f(x)|∣∣∣∣1− 2ζ

π

∫ π

0

dt

t2 + ζ2

∣∣∣∣+

ζ

π

(∫ ζ

0

|φt(x)|t2 + ζ2

dt +∫ π

ζ

|φt(x)|t2 + ζ2

dt

)

≤ 2π2

‖f‖∞ ζ +ζβ

2ϕ(f, ζ) +

4ζπ

∫ π

ζ

ϕ(f, t)tβ

t2 + ζ2dt .

(2.5)

Now we estimate ‖Q(f, ζ) − f‖∞,β . Taking into account (2.3) and (2.2), we obtain

8 Math. Nachr. 217 (2000)

for x, y ∈ IR

|Q(f, ζ)(x)− Q(f, ζ)(y) − f(x) + f(y)|

≤ ζF (ζ) |f(x)− f(y)| +∣∣∣∣Q(f, ζ)(x)− Q(f, ζ)(y) − 2ζ

π

∫ π

0

f(x) − f(y)t2 + ζ2

dt

∣∣∣∣≤ 2ζ

π2ϕ(f, |x − y|) |x − y|β +

ζ

π

∫ π

0

|φt(x) − φt(y)|t2 + ζ2

dt

≤{2ζπ2

‖f‖∞,β +4ζπ

∫ ζ

0

ϕ(f, t)t2 + ζ2

dt+4ζπ

∫ π

ζ

ϕ(f, t)t2 + ζ2

dt

}|x− y|β

≤{2ζπ2

‖f‖∞,β + ϕ(f, ζ) +4ζπ

∫ π

ζ

ϕ(f, t)t2 + ζ2

dt

}|x− y|β .

(2.6)

Finally, it follows from (2.5) and (2.6) that

‖f − Q(f, ζ)‖β ≤ 2 ‖f‖β

π2ζ +

(1 +

ζ2

2

)ϕ(f, ζ) +

4ζπ

∫ π

ζ

(1 + tβ)ϕ(f, t)t2 + ζ2

dt . ✷

Let C1[0, 2π] be the family of all continuously differentiable 2π – periodic real func-tions. For each f ∈ hβ and t > 0, define

K(f, t) := infg∈C1 [0,2π]

‖f − g‖∞,β + t ‖g′‖∞ .

It can be proved that K(f, t) is a continuous and concave function on (0,∞) (see [2],p. 167). Thus, K(f, t)/t is a decreasing function. We remark that an inequality similarto (2.7) can be find in [1] for more general Holder – type moduli of smoothness.

Proposition 2.1. For each function f ∈ hβ and t > 0,

ϕ(f, t) ≤ K(f, t1−β

) ≤ 3ϕ(f, t) .(2.7)

Proof . Fix t > 0 and f ∈ hβ . It follows from the mean value theorem that, forg ∈ C1[0, 2π], ϕ(g, t) ≤ t1−β‖g′‖∞. Since ϕ(f, t) ≤ ϕ(f − g, t) + ϕ(g, t), the firstinequality in (2.7) is proved. For the second inequality, define for x ∈ IR

f1(x) :=1t

∫ t

0

[f(x) − f(x + u)] du and f2(x) :=1t

∫ t

0

f(x + u) du .

We have that f = f1 + f2. Moreover, if x, y ∈ IR and | x − y |≤ t, then

|f1(x) − f1(y)| ≤ 1t

∫ t

0

|f(x) − f(x + u)− f(y) + f(y + u)| du

≤ 2ϕ(f, |x − y|)|x − y|β≤ 2ϕ(f, t)|x − y|β .

On the other hand, if t < |x − y|,

|f1(x)− f1(y)| ≤ 2t

∫ t

o

ϕ(f, u)uβ du ≤ 2ϕ(f, t)|x − y|β .

Bustamante, Rate of Convergence of Singular Integrals 9

Finally, for x ∈ IR,

|f ′2(x)| =

1t|f(x + t)− f(x)| ≤ 1

t1−βϕ(f, t) .

ThusK(f, t1−β

) ≤ ‖f − f2‖∞,β + t1−β ‖f ′2‖∞ ≤ 3ϕ(f, t) . ✷

P r o o f of Theorem 1.2, part (ii). Notice that, for t ∈ [√

ζ, π] (recall that K(f, t)/tis a non increasing function),

ϕ(f, t) ≤ K(f, t1−β

) ≤ t1−β K(f, ζ(1−β)/2

)ζ(1−β)/2

≤ 3t1−β ϕ(f,√

ζ)

ζ(1−β)/2.(2.8)

On the other hand (see (2.1)),∣∣∣∣ 1√πζ

∫ π

0

(f(x + t)− 2f(x) + f(x − t))e−t2/ζ dt

∣∣∣∣≤ 2√

πζ

(∫ √ζ

0

ϕ(f, t)tβe−t2/ζ dt +∫ π

√ζ

ϕ(f, t)tβe−t2/ζ dt

)

≤ 65√

ζϕ(f,√

ζ)(

ζβ/2

∫ √ζ

0

e−t2/ζ dt +3

ζ(1−β)/2

∫ π

√ζ

te−t2/ζ dt

)

=6

5√

ζϕ(f,√

ζ)ζβ/2

(√ζ

∫ 1

0

e−u2du+

3√ζ

∫ π/√

ζ

1

√ζue−u2√

ζ du

)

≤ 65

ζβ/2ϕ(f,√

ζ)(∫ 1

0

e−u2du+ 3

∫ ∞

1

ue−u2du

)

≤ 65

ζβ/2ϕ(f,√

ζ)(20

21+

32e

)

≤ 2ζβ/2ϕ(f,√

ζ).

(2.9)

For u ∈ (0, π/2], the function h(u) := exp{−π2/u} is convex. Since h(π/2) < π/2,then h(u) < u. Therefore

∫ ∞

π

e−t2/ζ dt =ζ

2πe−π2/ζ − ζ

2

∫ ∞

π

e−t2/ζ

t2dt ≤ ζ2

4(2.10)

and ∫ π/√

ζ

1

u1−βe−u2du <

∫ π/√

ζ

1

ue−u2du =

12

(e−1 − e−π/ζ

)<

14

.(2.11)

Let us estimate the uniform norm of W (f, ζ) − f . Since

1 =2√π

∫ ∞

0

e−u2du =

2√πζ

(∫ π

0

e−t2/ζ dt +∫ ∞

π

e−t2/ζ dt

),

10 Math. Nachr. 217 (2000)

taking into account (2.9) and (2.10), we have

|W (f, ζ)(x)− f(x)| =∣∣∣∣ 1√

πζ

(∫ π

0

φt(x)e−t2/ζ dt − 2f(x)∫ ∞

π

e−t2/ζ dt

)∣∣∣∣≤ 2ζβ/2ϕ

(f,√

ζ)+ ‖f‖∞ ζ2/2 .

Moreover, taking into account (2.2), (2.8), (2.10) and (2.11),

|W (f, ζ)(x)− W (f, ζ)(y) − f(x) − f(y)| |x − y|−β

≤ 1√πζ

(∫ π

0

|φt(x)− φt(y)| e−t2/ζ dt + 2 |f(x)− f(y)|∫ ∞

π

e−t2/ζ dt

)|x− y|−β

≤ 1√πζ

(4∫ π

0

ϕ(f, t)e−t2/ζ dt +ζ2

2ϕ(f, |x − y|)

)

≤ 1√πζ

(4ϕ(f,√

ζ){∫ √

ζ

0

e−t2/ζ dt+3

ζ(1−β)/2

∫ π

√ζ

t1−βe−t2/ζ dt

}+

ζ2

2‖f‖∞,β

)

≤ 1√πζ

(4√

ζϕ(f,√

ζ)

{∫ 1

0

e−u2du+ 3

∫ π/√

ζ

1

u1−βe−u2du

}+

ζ2

2‖f‖∞,β

)

≤ 6ϕ(f,√

ζ) +ζ

2‖f‖∞,β .

Now, using the above estimates we obtain (1.7). ✷

P r o o f of Proposition 1.3. It follows from the definition of ϕ(f, t), that ϕ(f, t) ≤Cf tα−β. Moreover

ζ

∫ π

ζ

ϕ(f, t)t2 + ζ2

dt ≤ Cf ζ

∫ π

ζ

tα−β−2 dt = Cfζ

1 + β − α

[ζα−β−1 − πα−β−1

].

From these inequalities we obtain (1.8). ✷

References

[1] Bustamante, J., and Jimenez, M. A.: The Degree of Best Approximation in the Lipschitz Normby Trigonometric Polynomials, to appear

[2] Butzer, P.L., and Berens, H.: Semi – Groups of Operators and Approximation, Springer–Verlag, New York/Berlin, 1967

[3] Deeba, E., Mohapatra, R.N., and Rodriguez, R. S.: On the Degree of Approximation ofSome Singular Integrals, Rediconti di Mat. 8 (1988), 345 – 355

[4] Khan, A.: On the Degree of Approximation of K. Picard and E. Poisson–Cauchy SingularIntegrals, Rendiconti di Math. 2 (1982), 123– 128

[5] Mohapatra, R.N., and Chandra, P.: Holder Continuous Functions and Their Euler, Boreland Taylor Means, Math. Chronicle. 11 (1982), 81 – 96

[6] Mohapatra, R. N., and Chandra, P.: Degree of Approximation of Functions in the HolderMetric, Acta Math. Hung. 41 (1983), 123 – 128

[7] Mohapatra, R.N., and Rodriguez, R. S.: On the Rate of Convergence of Singular Integralsfor Holder Continuous Functions, Math. Nachr. 149 (1990), 117 – 124

Bustamante, Rate of Convergence of Singular Integrals 11

[8] Prossdorf, S.: Sur Konvergenz der Fourierreihen holderstetiger Kunktionen, Math. Machr. 69(1975), 7 – 14

[9] Prossdorf, S., and Silbermann, B.: Numerical Analysis for Integral and Related OperatorEquations, Operator Theory, Advance and Applications Vol. 52, Birkhauser, Berlin, 1991

Apartado Postal J27Colonia San Manuel 72571Puebla, PUEMexicoE–mail:jbusta@fcfm.buap.mx