Yu. L. Bessonov

APPROXIMATION OF PERIODIC FUNCTIONS BELONGING TO THE CLASSES (W_{p,\theta}^{(r_1,r_2)}) BY FOURIER SUMS

(Presented by Academician A. I. Mal’tsev on 8 VI 1962)

Let (f(x,y)) be a periodic function and

[
f(x,y)=\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty} a_{kl} e^{i(kx+ly)}
]

its Fourier series.

We define the partial derivative of arbitrary (including fractional) order (r>0), in the Weyl sense, of (f) with respect to the variable (x) as follows:

[
f_x^{(r)}(x,y)=\sum_{k=-\infty}^{\infty}\sum_{l=-\infty}^{\infty} \gamma_k^{(r)} a_{kl} e^{i(kx+ly)},
]

where (\gamma_k^{(r)}) is a single-valued function determined by the equalities

[
\gamma_k^{(r)}=e^{r\pi i/2} k^r,\qquad
\gamma_{-k}^{(r)}=e^{-r\pi i/2} k^r
\quad (k=0,1,2,\ldots).
]

In the same way the mixed derivative is defined:

[
f_{yx}^{(r_2,r_1)}=f_{xy}^{(r_1,r_2)}
=\sum_{-\infty}^{\infty}\sum_{-\infty}^{\infty}
\gamma_k^{(r_1)}\gamma_l^{(r_2)} a_{kl} e^{i(kx+ly)},
]

which does not depend on the order of differentiation.

Definition. Let (r_1) and (r_2) be given positive, not necessarily integer, numbers. By definition, a function (f), (2\pi)-periodic in each of the variables, belongs to (W_{p,\theta}^{(r_1,r_2)}) if (f\in L_p) and has the property that the function

[
\frac{\partial^{r_1}f}{\partial x^{r_1}}
+e^{i\theta}\frac{\partial^{r_2}f}{\partial y^{r_2}}
=\varphi(x,y)
\tag{1}
]

is summable in the (p)-th power ((1\le p<\infty)), and for its norm the inequality holds

[
|\varphi|_p
=
\left(
\int_0^{2\pi}\int_0^{2\pi}
|\varphi(x,y)|^p\,dx\,dy
\right)^{1/p}
\le 1.
\tag{2}
]

In this connection one may consider only those pairs (r_1) and (r_2) for which the differential equation (1) has a unique periodic solution. It can be shown that the uniqueness property holds if and only if, for every (k),

[
\theta+\frac{(r_1+r_2)\pi}{2}\ne (2k+1)\pi,\qquad
\theta-\frac{(r_1+r_2)\pi}{2}\ne (2k+1)\pi,
]

[
\theta+\frac{(r_1-r_2)\pi}{2}\ne (2k+1)\pi,\qquad
\theta-\frac{(r_1-r_2)\pi}{2}\ne (2k+1)\pi.
\tag{3}
]

[
(k=0,\pm1,\ldots)
]

Here we shall consider only the values (\theta=0,\pi). For them the uniqueness property holds. Let

[
s_{m,n}=s_{m,n}(f,x,y)=
\sum_{-(m-1)}^{(m-1)}\sum_{-(n-1)}^{(n-1)}
\frac{b_{kl}}{(ik)^{r_1}+e^{i\theta}(il)^{r_2}}e^{i(kx+ly)}
]

be the (m,n)-th Fourier sum of the function (f). Denote by

[
\varepsilon_{m,n}\bigl(W_{\infty,\theta}^{(r_1,r_2)}\bigr)
=
\sup_{f\in W_{\infty,\theta}^{(r_1,r_2)}}|f-s_{m,n}|_{\infty};
]

[
\varepsilon_{m,n}\bigl(W_{1,\theta}^{(r_1,r_2)}\bigr)
=
\sup_{f\in W_{1,\theta}^{(r_1,r_2)}}|f-s_{m,n}|_{1}
]

the upper bounds for the deviations of functions (f) from their Fourier sums (s_{m,n}), extended to the classes (W_{p,\theta}^{(r_1,r_2)}) for (p=\infty) and (p=1). Introduce the number

[
\lambda=\frac{\left|\vartheta_{-m,n}^{(r_1,r_2)}\right|}
{\left|\vartheta_{m,n}^{(r_1,r_2)}\right|},
]

depending on (r_1,r_2,m,n), where

[
\vartheta_{m,n}^{(r_1,r_2)}
=
\frac{1}{(im)^{r_1}+e^{i\theta}(in)^{r_2}},
]

and put (\lambda_1=\dfrac1\lambda).

Theorem 1. The following asymptotic formula is valid:

[
\varepsilon_{m,n}
=
\varepsilon_{m,n}\bigl(W_{\infty,\theta}^{(r_1,r_2)}\bigr)
=
\varepsilon_{m,n}\bigl(W_{1,\theta}^{(r_1,r_2)}\bigr)
=
\frac{16E+8(\lambda^2-1)F}
{\pi^4\left|m^{r_1}+e^{i\alpha}n^{r_2}\right|}
\ln m\,\ln n
+
]

[
+
O\left{
\frac{\ln m\,[16E+8(\lambda^2-1)F]}{m^{r_1}}
+
\frac{\ln n\,[16E+8(\lambda^2-1)F]}{n^{r_2}}
\right}
+
O\left(\frac{\ln m}{m^{r_1}}+\frac{\ln n}{n^{r_2}}\right),
]

where

[
\alpha=\theta+\frac{r_2-r_1}{2}\pi,
]

[
E=E\left(\lambda,\frac{\pi}{2}\right)
=
\int_{0}^{\pi/2}\sqrt{1-\lambda^2\cos^2 t}\,dt
\quad \text{for } 0<\lambda<1,
]

[
F=F\left(\lambda,\frac{\pi}{2}\right)
=
\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-\lambda^2\cos^2 t}};
]

[
\varepsilon_{m,n}
=
\frac{16E+8(\lambda_1^2-1)F}
{\pi^4\left|m^{r_1}+e^{i\beta}n^{r_2}\right|}
\ln m\,\ln n
+
O\left{
\frac{\ln m\,[16E+8(\lambda_1^2-1)F]}{m^{r_1}}
+
\right.
]

[
\left.
+
\frac{\ln n\,[16E+8(\lambda_1^2-1)F]}{n^{r_2}}
\right}
+
O\left(\frac{\ln m}{m^{r_1}}+\frac{\ln n}{n^{r_2}}\right),
]

where

[
\beta=\theta+\frac{r_1+r_2}{2}\pi
\quad \text{for } 0<\lambda_1<1;
]

[
\varepsilon_{m,n}
=
\frac{16\ln m\,\ln n}
{\pi^4\left|m^{r_1}+e^{i\alpha}n^{r_2}\right|}
+
O\left(\frac{\ln m}{m^{r_1}}+\frac{\ln n}{n^{r_2}}\right)
\quad \text{for } \lambda=\lambda_1=1.
]

In the one-dimensional case, for integer (r), this asymptotic estimate was obtained by A. N. Kolmogorov ((^1)); in the fractional case (for all (r>0)) by V. T. Pinkevich ((^2)). In the two-dimensional case with (r_1=r_2=2), this estimate was obtained by Ya. S. Bugrov ((^3)).

Below we give the proof, which proceeds for (\lambda\ne1). In the case (\lambda=1) it is modified somewhat. We note that the case (\lambda=1) occurs when (r_1) and (r_2) are even and satisfy conditions (3).

Idea of the proof. From the periodicity of (f) it follows that the equality

[
\int_0^{2\pi}\int_0^{2\pi} \varphi(x,y)\,dx\,dy=0
\tag{4}
]

holds.

We obtain an integral representation for (f(x,y)). To this end, expand the function (\varphi(x,y)) in a Fourier series

[
\varphi(x,y)\sim \sum_{-\infty}^{\infty}\sum_{-\infty}^{\infty}{}' b_{kl} e^{i(kx+ly)} .
]

The prime on the sum means that the term (k=l=0) is absent. Then

[
f(x,y)=\frac{b_{0,0}}{4}+\frac{1}{4\pi^2}\int_D K^{(r_1,r_2)}(x-u,y-v)\,a(u,v)\,du\,dv,
\tag{5}
]

where (D={0\le x,y\le 2\pi}) and

[
K^{(r_1,r_2)}(u,v)=
\sum_{-\infty}^{\infty}\sum_{-\infty}^{\infty}{}'
\frac{e^{i(ku+lv)}}{(ik)^{r_1}+e^{i\theta}(il)^{r_2}} .
]

Putting

[
\Psi_{m,n}^{(r_1,r_2)}(u,v)
=
K^{(r_1,r_2)}(u,v)
-
\sum_{-(m-1)}^{(m-1)}{}'
\sum_{-(n-1)}^{(n-1)}{}'
\frac{e^{i(ku+lv)}}{(ik)^{r_1}+e^{i\theta}(il)^{r_2}},
]

we obtain

[
R_{m,n}(f)=f(x,y)-s_{m,n}(f)
=
\frac{1}{4\pi^2}\int_D
\Psi_{m,n}^{(r_1,r_2)}(x,-u,y-v)\,a(u,v)\,du\,dv .
]

Carrying out Abel’s transformation successively four times, we obtain the following expression for (\Psi_{m,n}^{(r_1,r_2)}(u,v)):

[
\Psi_{m,n}^{(r_1,r_2)}(u,v)
=
A_{m,n}^{(r_1,r_2)}(u,v)+Q_{m,n}(u,v),
]

where

[
A_{m,n}^{(r_1,r_2)}(u,v)
=
\frac{1}{2\sin\frac{u}{2}\sin\frac{v}{2}}
\left{
\left|\vartheta_{m,n}^{(r_1,r_2)}\right|
\cos\left(\frac{2m-1}{2}u+\frac{2n-1}{2}v+\psi\right)
+\right.
]

[
\left.
+
\left|\vartheta_{-m,n}^{(r_1,r_2)}\right|
\cos\left(\frac{2m-1}{2}u-\frac{2n-1}{2}v+\psi_1\right)
\right};
\tag{6}
]

(Q_{m,n}(u,v)) is a certain remainder containing all nonprincipal terms of the expansion, and

[
\tg\psi=
\frac{m^{r_1}\cos\frac{r_1\pi}{2}+n^{r_2}\cos\frac{r_2\pi}{2}}
{m^{r_1}\sin\frac{r_1\pi}{2}+n^{r_2}\sin\frac{r_2\pi}{2}},
\qquad
\tg\psi_1=
-
\frac{m^{r_1}\cos\frac{r_1\pi}{2}+n^{r_2}\cos\frac{r_2\pi}{2}}
{m^{r_1}\sin\frac{r_1\pi}{2}+n^{r_2}\sin\frac{r_2\pi}{2}} .
]

The equality

[
\frac{1}{4\pi^2}\int_\Delta
\left|A_{m,n}^{(r_1,r_2)}(u,v)\right|\,du\,dv
=
\frac{\ln m\ln n}{\pi^4}
\left|\vartheta_{m,n}^{(r_1,r_2)}\right|
\left[16E+8(\lambda^2-1)F\right]
+
]

[
+
O\left{
\frac{\ln m\,[16E+8(\lambda^2-1)F]}{m^{r_1}}
+
\frac{\ln n\,[16E+8(\lambda^2-1)F]}{n^{r_2}}
\right}.
\tag{7}
]

Indeed, let us partition the domain

[
\Delta_i^j=\left{\frac{4\pi}{2m-1}\leq u\leq 2\pi-\frac{4\pi}{2m-1};\ \frac{4\pi}{2n-1}\leq v\leq 2\pi-\frac{4\pi}{2n-1}\right}
]

into elementary parallelograms by two families of straight lines

[
\frac{2m-1}{2}u+\frac{2n-1}{2}v=k_1;\qquad
\frac{2m-1}{2}u-\frac{2n-1}{2}v=k_2
\quad (k_1,k_2=\mathrm{const}),
]

inside which each of the terms standing in the braces of formula (6) does not change sign. Estimating the resulting integrals with the aid of the generalized mean-value theorem and finding the order of the integrals over the incomplete parallelograms along the entire boundary strip discarded when approximating (\Delta) by parallelograms, we obtain (7).

Using equality (7), one can prove the required asymptotic formula in the norm (L_\infty).

Applying the reasoning of Ya. S. Bugrov ({}^{(3)}), one can prove the asymptotic formula also in the norm (L_1).

The formula proved is asymptotic if, for example,
(c_1\leq m^{r_1}/n^{r_2}\leq c_2), where (c_1) and (c_2) are positive constants, since in this case the first term on the right-hand side of this formula is the principal one.

Similar asymptotic estimates can be carried out for classes of equations, examples of which are the equations

[
\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\pm
\frac{\partial^4 f}{\partial x^2\,\partial y^2}
=\varphi(x,y).
]

In conclusion we give a theorem.

Theorem 2. Let (f\in L_p) ((1