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MATHEMATICS
M. D. Kalashnikov, G. P. Gubanov
On polynomials of best (quadratic) approximation in a prescribed system of points
(Presented by Academician P. S. Novikov on March 8, 1963)
- Let \(\bar C\) be the space of all continuous \(2\pi\)-periodic, with respect to each of the variables \(x\) and \(y\), functions \(f(x,y)\), and let \(S_{mn}(f;x,y)\) be the Fourier sum of order \((m,n)\) of the function \(f(x,y)\in \bar C\). It is well known that, among all trigonometric polynomials \(T_{mn}(x,y)\) of order not exceeding \((m,n)\),
\[ \begin{aligned} T_{mn}(x,y)={}&\frac{\alpha_{00}}{4} +\frac{1}{2}\sum_{\mu=1}^{m}(\alpha_{\mu0}\cos\mu x+\gamma_{\mu0}\sin\mu x)\\ &+\frac{1}{2}\sum_{\nu=1}^{n}(\alpha_{0\nu}\cos\nu y+\beta_{0\nu}\sin\nu y) +\sum_{\mu=1}^{m}\sum_{\nu=1}^{n}(\alpha_{\mu\nu}\cos\mu x\cos\nu y\\ &+\beta_{\mu\nu}\cos\mu x\sin\nu y +\gamma_{\mu\nu}\sin\mu x\cos\nu y +\delta_{\mu\nu}\sin\mu x\sin\nu y) \end{aligned} \]
the least value of the integral
\[ \int_{0}^{2\pi}\int_{0}^{2\pi}|f(x,y)-T_{mn}(x,y)|\,dx\,dy \]
is delivered by \(S_{mn}(f;x,y)\).
Let a system of equidistant points \((x_i,y_j)\) be given, where \(x_i=2i\pi/M\), \(i=1,2,\ldots,M\); \(y_j=2j\pi/N\), \(j=1,2,\ldots,N\); \(M\) and \(N\) are arbitrary natural numbers greater than \(2m+1\) and \(2n+1\), respectively.
It is easy to verify that, among all trigonometric polynomials \(T_{mn}(x,y)\) of order not exceeding \((m,n)\), the least value of the sum
\[ \sum_{i=1}^{M}\sum_{j=1}^{N}[f(x_i,y_j)-T_{mn}(x_i,y_j)]^2 \]
is delivered by the polynomial with coefficients
\[ \alpha_{\mu\nu}=\frac{4}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}f(x_i,y_j)\cos\mu x_i\cos\nu y_j, \]
\[ \beta_{\mu\nu}=\frac{4}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}f(x_i,y_j)\cos\mu x_i\sin\nu y_j, \]
\[ \gamma_{\mu\nu}=\frac{4}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}f(x_i,y_j)\sin\mu x_i\cos\nu y_j, \]
\[ \delta_{\mu\nu}=\frac{4}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}f(x_i,y_j)\sin\mu x_i\sin\nu y_j \]
\[ (\mu=0,1,2,\ldots,m;\ \nu=0,1,2,\ldots,n). \]
We shall call this polynomial the polynomial of best (quadratic) approximation for the function \(f(x,y)\) in the system of points \((x_i,y_j)\), where \(x_i=2\pi i/M\), \(i=1,2,\ldots,M\); \(y_j=2\pi j/N\), \(j=1,2,\ldots,N\), and shall denote it by
\[ T_{mn}^{MN}(f;x,y). \]
Substituting the values of the coefficients \(\alpha_{\mu\nu}, \beta_{\mu\nu}, \gamma_{\mu\nu}, \delta_{\mu\nu}\), we obtain for the polynomial \(T_{mn}^{MN}(f;x,y)\) the expression
\[ T_{mn}^{MN}(f;x,y)=\frac{4}{MN}\sum_{i=1}^{M}\sum_{j=1}^{N}f(x_i,y_j)D_m(x_i-x)D_n(y_j-y), \]
where
\[ D(t)=\frac{\sin(n+\tfrac12)t}{2\sin\tfrac12 t} \]
is the Dirichlet kernel.
If \(M=2m+1,\ N=2n+1\), then \(T_{mn}^{MN}(f;x,y)\) turns into the interpolating trigonometric polynomial \(\widetilde S_{mn}(f;x,y)\) for the function \(f(x,y)\) with equidistant interpolation nodes. If \(M=r(2m+1),\ N=s(2n+1)\), then in the limit as \(r,s\to\infty\) we obtain the Fourier sum \(S_{mn}(f;x,y)\).
- Let \(H_{\omega_1,\omega_2}\) denote the class of functions \(f(x,y)\) of period \(2\pi\) with respect to \(x\) and \(y\), satisfying the condition
\[ |f(x_1,y_1)-f(x_2,y_2)|\leq \omega_1(|x_1-x_2|)+\omega_2(|y_1-y_2|), \]
where \(\omega_1(t)\) and \(\omega_2(z)\) are convex moduli of continuity. By \(\mathscr E_{mn}^{(r,s)}(x,y)\) we denote the exact least upper bound of the deviations of the functions \(f(x,y)\) from the polynomials \(T_{mn}^{r(2m+1)s(2n+1)}(f;x,y)\), extended over the entire class \(H_{\omega_1,\omega_2}\):
\[ \mathscr E_{mn}^{(r,s)}(x,y) = \sup_{f\in H_{\omega_1,\omega_2}} |f(x,y)-T_{mn}^{r(2m+1)s(2n+1)}(f;x,y)|. \]
It is not difficult to verify that the upper bound \(\mathscr E_{mn}^{(r,s)}(x,y)\) is an even function, periodic with period \(h^{(r)}=2\pi/r(2m+1)\) with respect to \(x\) and with period \(g^{(s)}=2\pi/s(2n+1)\) with respect to \(y\).
Theorem. For any integers \(r,s>1\) the following asymptotic equality holds
\[ \begin{aligned} \mathscr E_{mn}^{(r,s)}(x,y) &= \frac{8\ln m\ln n}{\pi^2rs} \left| \cos\frac{2m+1}{2}x\, \cos\frac{2n+1}{2}y \right| \\ &\quad\times \sum_{i=1}^{[r/2]}\sum_{j=1}^{[s/2]} \min\{\omega_1(2x_i),\ \omega_2(2y_j)\} \sin\frac{i\pi}{r}\sin\frac{j\pi}{s} +\rho_{mn} \end{aligned} \]
\[ \left(0\leq x\leq \tfrac12 h^{(r)},\quad 0\leq y\leq \tfrac12 g^{(s)}\right), \]
where
\[ \rho_{mn} = O\left\{ (\ln m+\ln n) \left[ \omega_1\!\left(\frac{2\pi}{2m+1}\right) + \omega_2\!\left(\frac{2\pi}{2n+1}\right) \right] \right\}, \qquad x_i=ih^{(r)},\quad y_j=jg^{(s)}. \]
Passing to the limit as \(r,s\to\infty\), from the theorem we obtain the result of P. T. Bugaits \((^1)\):
\[ \begin{aligned} \sup_{f\in H_{\omega_1,\omega_2}} |f(x,y)-S_{mn}(f;x,y)| &= \frac{2(2m+1)(2n+1)\ln m\ln n}{\pi^4} \\ &\quad\times \int_0^{\pi/(2m+1)} \int_0^{\pi/(2n+1)} \min\{\omega_1(2u),\omega_2(2v)\} \sin\frac{2m+1}{2}u\, \sin\frac{2n+1}{2}v\,du\,dv +\rho_{mn}, \end{aligned} \]
where
\[ \rho_{mn} = O\left\{ (\ln m+\ln n) \left[ \omega_1\!\left(\frac{2\pi}{2m+1}\right) + \omega_2\!\left(\frac{2\pi}{2n+1}\right) \right] \right\}. \]
- Let us also note the following. We shall regard \(T_{mn}^{MN}(f;x,y)\) as a linear operator (for fixed \(x,y\), a linear functional) in the space \(\overline C\), where the norm of \(f(x,y)\) is taken to be \(\sup_{x,y}|f(x,y)|\). Denote by \(L_{mn}^{MN}\) the norm of this operator. We have
\[ L_{mn}^{MN}=\sup_{x,y}L_{mn}^{MN}(x,y), \qquad L_{mn}^{MN}(x,y) = \frac{4}{MN} \sum_{i=1}^{M}\sum_{j=1}^{N} |D_m(x_i-x)D_n(y_i-y)|. \]
Here \(L_{mn}^{MN}(x,y)\) is a function periodic with period \(h=2\pi/M\) with respect to \(x\) and with period \(g=2\pi/N\) with respect to \(y\), and is even. We indicate an expression for the norm of the operator \(L_{mn}^{MN}(x,y)\).
Let
\[ M=2m+1+1+p,\quad 0<p\leq 2m+1; \qquad N=2n+1+q,\quad 0<q\leq 2n+1. \]
In this case
\[ \begin{aligned} L_{mn}^{MN}(x,y)=& \left( \frac{2}{\pi}\sin\frac{2m+1}{2}x\ln\frac{M}{p} + \frac{p}{\pi M\sin p\pi/2M} \sum_{\mu=1}^{p}\frac{(-1)^\mu}{\mu}\right.\\ &\left.\times \left\{ \cos\left[\left(k'_\mu+\frac12\right)\frac{\pi p}{M} +\left(m+\frac12\right)x\right] + \cos\left[\left(k_\mu+\frac12\right)\frac{\pi p}{M} -\left(m+\frac12\right)x\right] \right\}\right)\\ &\times \left( \frac{2}{\pi}\sin\frac{2n+1}{2}y\ln\frac{N}{q} + \frac{q}{\pi N\sin q\pi/2N} \sum_{\nu=1}^{q}\frac{(-1)^\nu}{\nu} \left\{ \cos\left[\left(l'_\nu+\frac12\right)\frac{\pi q}{N} +\left(n+\frac12\right)y\right]\right.\right.\\ &\left.\left.\qquad\qquad\qquad\qquad + \cos\left[\left(l_\nu+\frac12\right)\frac{\pi q}{N} -\left(n+\frac12\right)y\right] \right\}\right) +O\left(\ln\frac{M}{p}\right)+O\left(\ln\frac{N}{q}\right) \end{aligned} \]
\[ (0\le x\le \tfrac12 h,\; 0\le y\le \tfrac12 g), \]
where
\[ k'_\mu=\left[\mu\frac{M}{p}-\frac{2m+1}{p}\frac{x}{h}\right],\qquad k_\mu=\left[\mu\frac{M}{p}+\frac{2m+1}{p}\frac{x}{h}\right], \]
\[ \mu=1,2,\ldots,p,\qquad h=\frac{2\pi}{M}, \]
\[ l'_\nu=\left[\nu\frac{N}{q}-\frac{2n+1}{q}\frac{y}{g}\right],\qquad l_\nu=\left[\nu\frac{N}{q}+\frac{2n+1}{q}\frac{y}{g}\right], \]
\[ \nu=1,2,\ldots,q,\qquad g=\frac{2\pi}{N}. \]
If \(M\ge 2(2m+1)\), \(N\ge 2(2n+1)\), then
\[ \begin{aligned} L_{mn}^{MN}(x,y)=& \frac{uv}{\pi^2MN\sin(\pi v/2M)\cdot\sin(\pi u/2N)} \sum_{\mu=1}^{v}\sum_{\nu=1}^{u}\frac{(-1)^{\mu+\nu}}{\mu\nu}\\ &\times \left\{ \cos\left[\left(l_\mu+\frac12\right)\frac{\pi v}{M} -\left(m+\frac12\right)x\right] + \cos\left[\left(l'_\mu+\frac12\right)\frac{\pi v}{M} +\left(m+\frac12\right)x\right] \right\}\\ &\times \left\{ \cos\left[\left(k_\nu+\frac12\right)\frac{\pi u}{N} -\left(n+\frac12\right)y\right] + \cos\left[\left(k'_\nu+\frac12\right)\frac{\pi u}{N} +\left(n+\frac12\right)y\right] \right\}\\ &+O(\ln m)+O(\ln n) \end{aligned} \]
\[ (0\le x\le \tfrac12 h,\qquad 0\le y\le \tfrac12 g), \]
where
\[ l_\mu=\left[\mu\frac{M}{v}+\frac{x}{h}\right],\quad l'_\mu=\left[\mu\frac{M}{v}-\frac{x}{h}\right],\quad \mu=1,2,\ldots,v,\quad v=2m+1, \]
\[ h=\frac{2\pi}{M};\qquad k_\nu=\left[\nu\frac{N}{u}+\frac{y}{g}\right],\quad k'_\nu=\left[\nu\frac{N}{u}-\frac{y}{g}\right],\quad \nu=1,2,\ldots,u, \]
\[ u=2n+1,\qquad g=\frac{2\pi}{N}. \]
The corresponding equalities can be written for the cases when
\(2m+1<M\le 2(2m+1)\), \(N\ge 2(2n+1)\) and \(2n+1<N\le 2(2n+1)\),
\(M\ge 2(2m+1)\).
If \(M=r(2m+1)\), \(N=s(2n+1)\), where \(r,s\) are integers, then
\[ L_{mn}^{r(2m+1)s(2n+1)}(x,y) = \frac{4\ln m\ln n}{\pi^2} \frac{ \cos\left[\frac{\pi}{2r}-\frac{2m+1}{2}x\right] \cos\left[\frac{\pi}{2s}-\frac{2n+1}{2}y\right] }{ rs\sin(\pi/2r)\cdot\sin(\pi/2s) } +O(\ln m)+O(\ln n), \]
whence, putting \(x=0\), \(y=0\) and passing to the limit as \(r,s\to\infty\), we obtain the expression for the Lebesgue constant
\[ L_{mn}=\frac{16}{\pi^4}\ln m\ln n+O(\ln m)+O(\ln n). \]
Dnepropetrovsk State University
named after the 300th anniversary of the reunification of Ukraine with Russia
Received
26 IV 1962
CITED LITERATURE
- P. T. Bugaets, DAN, 79, 557 (1951).