MATHEMATICS

R. G. MAMEDOV

DIRECT AND INVERSE THEOREMS IN THE THEORY OF APPROXIMATION OF FUNCTIONS BY (m)-SINGULAR INTEGRALS

(Presented by Academician V. I. Smirnov on 2 I 1962)

Let (K_\lambda(t)) be an even function defined on ((-\infty,\infty)) and depending on a real parameter (\lambda). Consider the (m)-singular integral

[
T_{\lambda}^{[m]}(f;x)=(-1)^{m+1}\int_{-\infty}^{\infty}
\left[
\sum_{k=1}^{m}(-1)^{m-k}\binom{m}{k} f(x+kt)
\right]K_\lambda(t)\,dt
\tag{1}
]

for each function (f(t)) ((-\infty<t<\infty)), where (m\ge 1) is a certain fixed positive integer (see ((^5))).

The Fourier transform of a function (f(t)\in L_p(-\infty,\infty)) ((1\le p\le 2)) will be denoted by ((F)f(t)=F[f(t)]).

In addition to the Fourier transform, in what follows we shall also need the Fourier–Stieltjes transform of functions of bounded variation on ((-\infty,\infty)), i.e., functions from the class (BV(-\infty,\infty)). The Fourier–Stieltjes transform of a function (h(t)\in BV(-\infty,\infty)) will be denoted by

[
(FS)h(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-ixt}\,dh(t).
]

Let us note that if (f(t)\in L_p(-\infty,\infty)) ((p\ge 1)) and (K_\lambda(t)\in L(-\infty,\infty)), then the (m)-singular integral (1) exists almost everywhere on ((-\infty,\infty)) and

[
T_{\lambda}^{[m]}(f;x)\in L_p(-\infty,\infty)\qquad (p\ge 1).
]

In what follows, everywhere we put

[
\psi(x)=F[f(t)],\qquad \varphi_\lambda(x)=F[K_\lambda(t)].
]

Theorem 1. Let (f(t)\in L_p(-\infty,\infty)) ((1\le p\le 2)), (K_\lambda(t)\in L(-\infty,\infty)), and suppose there exists a nonnegative function (\gamma(\lambda)), monotonically decreasing to zero as (\lambda\to\infty), such that

[
\lim_{\lambda\to\infty}
\frac{
1-\sqrt{2\pi}\displaystyle\sum_{k=1}^{m}(-1)^{k-1}\binom{m}{k}\varphi_\lambda(kx)
}{
\gamma(\lambda)
}
=
r_m(x)\equiv r(x)
\tag{2}
]

for every real (x), where (r(x)) is some continuous real-valued function, and (r(x)\ne 0) for (-\infty<x<\infty).

Then from the relation

[
\left|T_{\lambda}^{[m]}(f;x)-f(x)\right|_{L_p}
=
o[\gamma(\lambda)]
\tag{3}
]

it follows that (f(x)=0) almost everywhere on ((-\infty,\infty)).

Theorem 2. Let (f(t),K_\lambda(t)\in L(-\infty,\infty)), and suppose condition (2) is satisfied.

If

[
\left|T_{\lambda}^{[m]}(f;x)-f(x)\right|_{L}
=
O[\gamma(\lambda)]
\tag{4}
]

as (\lambda \to \infty), then the function (r(x)\psi(x)) is the Fourier–Stieltjes transform of some function (h(t)\in BV(-\infty,\infty)), i.e.

[
r(x)\psi(x)=(FS)h(t)
\tag{5}
]

for every (-\infty<x<\infty).

If, in particular, (r(x)=x^k), where (k\ge 1) is an integer, then the function (f(t)) has a ((k-1))-st derivative (f^{(k-1)}(t)) and (f^{(k-1)}(t)\in BV(-\infty,\infty)).

Theorem 2 is in a certain sense reversible.

Theorem 3. Let (f(t), K_\lambda\in(t)\in L(-\infty,\infty)) and suppose that the condition

[
\frac{
1-\sqrt{2\pi}\displaystyle\sum_{k=1}^{m}(-1)^{k-1}\binom{m}{k}\varphi_\lambda(kx)
}{
\gamma(\lambda)r(x)
}
\equiv
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-ixt}\,dQ_\lambda(t)\,(FS)Q_\lambda(t),
\tag{6}
]

is satisfied, where (Q_\lambda(t)\in BV(-\infty,\infty)), ([\operatorname{Var} Q_\lambda(t)]_{-\infty}^{\infty}\le M<+\infty).

Then from condition (5), where (h(t)\in BV(-\infty,\infty)), (4) follows as (\lambda\to\infty).

Now let us additionally suppose that the condition

[
\lim_{\lambda\to\infty}(FS)Q_\lambda(t)=1
\tag{7}
]

holds.

In this case condition (2) is also satisfied. Consequently, from Theorems 2 and 3 it follows:

Theorem 4. Let (f(t), K_\lambda(t)\in L(-\infty,\infty)) and let conditions (6) and (7) be satisfied.

Then a necessary and sufficient condition for relation (4) to hold as (\lambda\to\infty) is (5).

We now give a definition (see ((^{1,2}))). Let (\gamma(\lambda)\ge 0) be a nonincreasing function and (\lim_{\lambda\to\infty}\gamma(\lambda)=0). Suppose that there exists a class (E) of functions (f(x)) such that:

(1^\circ). From (|T_\lambda^{[m]}(f;x)-f(x)|=o[\gamma(\lambda)]) it follows that (f(x)=\mathrm{const}).

(2^\circ). From (|T_\lambda^{[m]}(f;x)-f(x)|=O[\gamma(\lambda)]) it follows that (f(x)\in E).

(3^\circ). For every function (f(x)\in E) one has

[
|T_\lambda^{[m]}(f;x)-f(x)|=O[\gamma(\lambda)]
\quad\text{as }\lambda\to\infty.
]

Then the (m)-singular integral (1) is said to be saturated with order (O[\gamma(\lambda)]), and (E) is called the class of saturation.

We note that Theorems 1 and 4 determine the class and the order of saturation of the (m)-singular integrals (1) in the space (L(-\infty,\infty)). From these theorems it follows that the order of saturation of the (m)-singular integrals is (\gamma(\lambda)), and the class of saturation consists of those functions (f(x)) for which condition (5) is satisfied.

We note that the saturation classes of many known summability methods were found by A. Kh. Turetskii ((^{3,4})).

Theorem 5. Let (f(t)\in L_p(-\infty,\infty)) ((1<p\le 2)), (K_\lambda(t)\in L(-\infty,\infty)), and let condition (2) be satisfied for all real (x).

If

[
|T_\lambda^{[m]}(f;x)-f(x)|_{L_p}=O[\gamma(\lambda)]
\tag{8}
]

as (\lambda\to\infty), then the function (r(x)\psi(x)) is the Fourier transform of some function (\mu(t)\in L_p(-\infty,\infty)) ((1<p\le 2)), i.e. the relation

[
r(x)\psi(x)=F[\mu(t)]
\tag{9}
]

holds.

If, in particular, (r(x)=x^k) or (r(x)=|x|^k), then

[
f^{(k)}(t)\in L_p(-\infty,\infty).
]

Theorem 5 is converse under a small additional restriction.

Theorem 6. Let (f(t)\in L_p(-\infty,\infty)) ((1<p\le 2)), (K_\lambda(t)\in L(-\infty,\infty)), and suppose the condition

[
\frac{
1-\sqrt{2\pi}\sum_{k=1}^{m}(-1)^{k-1}\binom{m}{k}\varphi_\lambda(kx)
}{
\gamma(\lambda)r(x)
}
\equiv
\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-ixt}\,dQ_\lambda(t)
=(FS)Q_\lambda(t),
\tag{10}
]

is satisfied, where

[
Q_\lambda(t)=\int_{-\infty}^{t} b_\lambda(u)\,du,\qquad
\int_{-\infty}^{\infty}|b_\lambda(u)|\,du\le M<+\infty .
]

Then condition (9), where (\mu(t)\in L_p(-\infty,\infty)), implies (8) as (\lambda\to\infty).

From Theorems 5 and 6 it follows:

Theorem 7. Let (f(t)\in L_p(-\infty,\infty)) ((1<p\le 2)), (K_\lambda(t)\in L(-\infty,\infty)), let condition (10) be satisfied, and

[
\lim_{\lambda\to\infty}(FS)Q_\lambda(t)=1.
\tag{11}
]

Then the necessary and sufficient condition for relation (8) to hold is (9).

Theorems 1 and 7 determine the class and the order of saturation of the (m)-singular integrals (1) in the space (L_p(-\infty,\infty)) ((1<p\le 2)). More precisely, it follows from the indicated theorems that the order of saturation of the (m)-singular integrals (1) is (\gamma(\lambda)), and the saturation class consists of those functions (f(x)) for which condition (9) is satisfied.

In particular, if (m=1) and (K_\lambda(t)=\dfrac{\lambda}{\sqrt{2\pi}}K(\lambda t)), then from Theorems 1, 2, and 5 the corresponding results of Sunouchi ((^1)) follow.

In the case (m=1), (r(x)=c|x|^\alpha) ((c>0,\ 0<\alpha\le 2)), (\gamma(\lambda)=\lambda^{-\alpha}), and (K_\lambda(t)=\dfrac{\lambda}{\sqrt{2\pi}}K(\lambda t)), our theorems yield the corresponding results of Butzer ((^2)).

It is known ((^7)) that for functions (f(t)\in L_p(-\infty,\infty)) ((2