Approximation of several dimensional functions by trigonometric polynomials

Abstract

Let f(x, y) be a periodic function defined on the region D$0 \leqslant x \leqslant 2\pi ,0 \leqslant y \leqslant 2\pi $ with period 2π for each variable. If f(x, y) ∈ C^p (D), i.e., f(x, y) has continuous partial derivatives of order p on D, then we denote by ω_α,β(ρ) the modulus of continuity of the function $\frac{{\partial ^p f(x,y)}}{{\partial x^\alpha \partial y^\beta }}(\alpha ,\beta \geqslant 0,\alpha + \beta = p)$ and write $\omega _p (\rho ) = \mathop {\max }\limits_{\alpha ,\beta \geqslant 0,\alpha + \beta = p} \omega _{\alpha ,\beta } (\rho ).$ For p = 0, we write simply C(D) and ω(ρ) instead of C⁰(D) and ω₀(ρ).

Let T(x,y) be a trigonometrical polynomial written in the complex form $T(x,y) = \sum {C_{m,n} } e^{i(mx + ny)} .$ We consider R = max(m² + n²)^1/2 as the degree of T(x, y), and write T_R(x, y) for the trigonometrical polynomial of degree ⩾ R.

Our main purpose is to find the trigonometrical polynomial T_R(x, y) for a given f(x, y) of a certain class of functions such that $\mathop {\max }\limits_{x,y} \left| {f(x,y) - T_R (x,y)} \right|$ attains the same order of accuracy as the best approximation of f(x, y).

Let the Fourier series of f(x, y) ∈ C(D) be $f(x,y) \sim \sum\limits_{ - \infty }^\infty {C_{m,n} e^{i(mx + ny)} } ,$ and let $A_\nu (x,y) = \sum\limits_{m^2 + n^2 = \nu } {C_{m,n} e^{i(mx + ny)} } .$ Our results are as follows

Theorem 1 Let f(x, y) ∈ C^p(D (p = 0, 1) and $S_R^\delta (x,y;f) = \sum\limits_{\nu = R^2 } {\left( {1 - \frac{\nu }{{R^2 }}} \right)^\delta A_\nu (x,y)(\delta > {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})} .$Then $S_R^\delta (x,y;f) - f(x,y) = O\left[ {\frac{1}{{R^p }}\omega _p \left( {\frac{1}{R}} \right)} \right](p = 0,1)$holds uniformly on D.

If we consider the circular mean of the Riesz sum S_R ^δ(x, y) ≡ S_R ^δ(x, y; f): $\mu _t \left[ {S_R^\delta (x,y)} \right] = \frac{1}{{2\pi }}\int_0^{2\pi } {S_R^\delta (x + t\cos \theta ,y + t\sin \theta )d\theta ,} $ then we have the following

Theorem 2 If f(x, y) ∈ C^p (D) and ω_p(ρ) = O(ρ^α (0 < α ⩾ 1; p = 0, 1), then $\mu _{\frac{{\lambda _0 }}{R}} [S_R^\delta (x,y)] - f(x,y) = O\left( {\frac{1}{{R^{p + \alpha } }}} \right)(p = 0,1;\delta \geqslant 0)$holds uniformly on D, where λ₀is a positive root of the Bessel function J₀(x)

It should be noted that either $S_R^\delta (x,y;f) - f(x,y) = o({1 \mathord{\left/ {\vphantom {1 {R^2 }}} \right. \kern-\nulldelimiterspace} {R^2 }})$ or $\mu _{\frac{{\lambda _0 }}{R}} [S_R^\delta (x,y)] - f(x,y) = o({1 \mathord{\left/ {\vphantom {1 {R^2 }}} \right. \kern-\nulldelimiterspace} {R^2 }})$ implies that f(x, y) ≡ const.

Now we consider the following trigonometrical polynomial $S_R^\delta (x,y;f) = \sum\limits_{\nu \leqslant R^2 } {\left( {1 - \frac{{\nu ^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{R^k }}} \right)^\delta A_\nu (x,y)(k \in \mathbb{Z}^ + )} .$ Then we have

Theorem 3 If f(x, y) ∈ C^p(D), then uniformly on D, $S_R^{(k)} (x,y;f) - f(x,y) = \left\{ \begin{gathered} o[\tfrac{1}{{R^p }}\omega _p (\tfrac{1}{R})],p = 0,1,...,k - 1forkeven, \hfill \\ o[\tfrac{1}{{R^p }}\omega _p (\tfrac{1}{R})lnR],p = k - 1forkodd. \hfill \\ \end{gathered} \right.$

Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem of Zygmund, which can be extended to the multiple case as follows

Theorem 3′ Let f(x₁, ..., x_n) ≡ f(P) ∈ C^pand let $S_R^{(k)} (P;f) = \sum\limits_{\nu \leqslant R^2 } {\left( {1 - \frac{{\nu ^{\tfrac{k}{2}} }}{{R^k }}} \right)^{\sigma _m } A_\nu (P)} ,$where $\sigma _m = \left[ {\frac{{m - 1}}{2}} \right] + 1$and $A_\nu (P) = \sum\limits_{n_1^2 + ... + n_m^2 = \nu } {C_{n_1 ,...,n_m } e^{i(n_1 x_1 + ... + n_m x_m )} } ,$$C_{n_1 , \cdots ,n_m } $$being the Fourier coefficients of f(P). Then $S_R^{(k)} (p;f) - f(P) = \left\{ \begin{gathered} o\left[ {\tfrac{1}{{R^p }}\omega _p \left( {\tfrac{1}{R}} \right)} \right](p \leqslant k - 2;p = k - 1forkeven), \hfill \\ o\left[ {\tfrac{1}{{R^p }}\omega _p \left( {\tfrac{1}{R}} \right)lnR} \right](p = k - 1forkodd) \hfill \\ \end{gathered} \right.$holds uniformly.

Keywords

trigonometrical polynomial of several variables / approximation of function / Riesz mean / modulus of continuity / 42B05 / 33C10 / 41A10 / 42A10

References

[1] Chandrasekharan K., Minakshisundaram S. Some results on double Fourier series. Duke Math J, 1947, 14: 731-753.

[2] Bernstein C. Coobs Hark Math Ob-va, 1912 (in Russian)

[3] Bernstein C. Comptes Reudus hebdomadaires des Sciences del’ Acadeémie des Sciences, 1931, 191: 976-979.

[4] Zygmund A. The approximation of functions by typical means of their Fourier series. Duke Math J, 1945, 12: 695-704.

[5] Inogemchev O. I. IEEE Dokp AN SSSR, 1953, 91: 15-18.

[6] Bochner S. Summation of multiple Fourier series by spherical means. Trans Amer Math Soc, 1936, 40: 175-207.