UDC 517.512.2

MATHEMATICS

V. A. ABILOV, S. A. AGAKHANOV

APPROXIMATION OF DIFFERENTIABLE FUNCTIONS BY FOURIER–HERMITE SUMS

(Presented by Academician I. N. Vekua on 29 II 1968)

Let (H_n(x)) be the Hermite polynomials, orthogonal on ((-\infty,\infty)) with weight (e^{-x^2}) and normalized by the condition
[
\int_{-\infty}^{\infty} e^{-x^2} H_n^2(x)\,dx = 1
]
(the leading coefficient of (H_n(x)) is assumed positive); let (W^r H_\omega^0) be the class of functions having on the whole axis a continuous derivative of order (r=0,1,\ldots), whose modulus of continuity satisfies (\omega(f^{(r)},\delta)\le \omega(\delta)), where (\omega(\delta)) is a prescribed true majorant of moduli of continuity and
[
f^{(r)}(0)=0;\quad
s_{n-1}(f;x)=\sum_{k=0}^{n-1} c_k(f)H_k(x),\quad
c_k(f)=\int_{-\infty}^{\infty} e^{-x^2} f(x)H_k(x)\,dx;
]
[
\Delta_n(f;x)=f(x)-s_{n-1}(f;x);\quad
c_\omega^{(m)}=\sup_{f\in H_\omega(2\pi)}\int_{-\pi}^{\pi} f(t)\cos nt\,dt,
]
where (H_\omega(2\pi)) is the class of continuous (2\pi)-periodic functions (f(x)) for which (\omega(f,\delta)\le \omega(\delta)); (p=\sqrt{2n}), (m=[p]), (q=\sqrt{2k}).

In ((^1)) an asymptotic formula was obtained for
[
\sup_{f\in W^0H_\omega^0}|\Delta_n(f;x)|
]
with a remainder term uniformly bounded for (x) in a finite interval. The purpose of our work is to extend this result to the class (W^rH_\omega^0) ((r=1,2,\ldots)).

Theorem. For (r=1,2,\ldots) and any (x\in(-\infty,\infty)),
[
\sup |\Delta_n(f;x)|=\frac{\ln m}{\pi^2 m^r}\,c_\omega^{(m)}+\rho_m(x),
]
where
[
\rho_m(x)=e^{x^2/2}(1+|x|^{11/2})\,O!\left(\frac1{m^r}\,\omega!\left(\frac1m\right)\right),
]
and the constant contained in (O) depends only on (r).

Remark. In the proof of the theorem, the usual method, based on the summation of sums of the form (\sum_{k=n}^{N}\sin kx), which works in the theory of Fourier series ((^2)), Fourier–Jacobi series ((^3)), does not apply to Fourier–Hermite series, since here sums of the form (\sum_{k=n}^{N}\sin\sqrt{k}\,x) arise. We overcome this difficulty by means of Lemma 4 (see below).

The proof of the theorem rests on the following lemmas.

Lemma 1. Let
[
x_k=\frac{n-2[n/2]+2k}{\sqrt{2(n+1)}}\,\frac{\pi}{2}\quad (k=1,2,\ldots,n).
]
Then
[
\sum_{k=1}^{n} e^{-\theta x_k^2}=O(p).
]
Here (0<\theta<1).

Lemma 2. If (f(t)\in H_{\omega}^{0}), then for (|h|\leq \delta \leq 1), for (F(t)=e^{-t^{2}/2}f(t)), one has

[
F(t+h)-F(t)=e^{-\varepsilon t^{2}}O(\omega(\delta)),
]

[
F(t+h)\sin a(t+h)-F(t)\sin at
=
e^{-\varepsilon t^{2}}O(a\omega(\delta)),
]

where (a) is an arbitrary number and (\varepsilon>0).

Lemma 3. Let

[
F(t)=\frac{1}{t-x}\left(e^{(x^{2}-t^{2})/2}-1\right).
]

Then for (|t-x|\leq 1) one has

[
F(t)=O(1+|x|^{3})e^{|x|}
]

and, moreover,

[
\int_{c/p\leq |t-x|\leq 1}
F(t)[f(t)-f(x)]\sin p(t-x)\,dt
=
p^{r}\rho_{n}(x),
]

if (f(t)\in H_{\omega}^{c}).

Lemma 4. The equality

[
\sum_{k=n}^{\infty}\frac{\cos(qt+r\pi/2)}{q^{r+1}}
=
\int_{p}^{\infty}\frac{\cos(tz+r\pi/2)}{z^{r}}\,dz
+
O(1+t^{2})p^{-r-1}
]

is valid.

Lemma 5. If (f(t)\in H_{\omega}^{0}), then

[
c_{n}(f)
=
\sqrt{\frac{2}{\pi p}}
\int_{-\infty}^{\infty}
e^{-x^{2}/2}f(x)
\cos\left(px-\frac{n\pi}{2}\right)\,dx
+
O\left(p^{-3/2}\omega(p^{-1})\right).
]

Lemma 6. If (f(t)\in H_{\omega}^{0}), then

[
\int_{c/p}^{1} f(t)\frac{\cos pt}{t^{2}}\,dt
=
O(p\omega(p^{-1})),
\qquad
\int_{p}^{\infty}\frac{dz}{z^{r+2}}
\left(
\int_{c/p}^{1} f(t)\frac{\cos zt}{t^{2}}\,dt
\right)
=
O(p^{-r}\omega(p^{-1})).
]

These formulas remain valid if (\cos t) is replaced in them by (\sin t).

The authors express their gratitude to G. I. Natanson for remarks that contributed to simplifying some of the arguments.

Dagestan State University
named after V. I. Lenin

Received
5 II 1968

CITED LITERATURE

1. S. A. Agakhanov, G. I. Natanson, Uch. zap. Kazansk. gos. univ., 124, book 4, 20 (1964).
2. V. T. Pinkevich, Izv. AN SSSR, ser. matem., 4, 521 (1940).
3. S. A. Agakhanov, G. I. Natanson, DAN, 116, No. 1, 9 (1966).