UDC 517.512.6

MATHEMATICS

S. A. ATAKHANOV, G. I. NATANSON

APPROXIMATION OF FUNCTIONS BY FOURIER–JACOBI SUMS

(Presented by Academician V. I. Smirnov on 3 V 1965)

Let \(P_n^{(\alpha,\beta)}(x)\) be the Jacobi polynomials, orthogonal on \([-1,1]\) with weight
\(p(x)=(1-x)^\alpha(1+x)^\beta\), normalized by the condition
\(P_n^{(\alpha,\beta)}(1)=\Gamma(n+\alpha+1)/\Gamma(n+1)\Gamma(\alpha+1)\);
\(S_n^{(\alpha,\beta)}[f;x]\) is the \(n\)-th partial sum of the Fourier series of the function
\(f(x)\) in the polynomials \(P_n^{(\alpha,\beta)}(x)\). As is known,

\[ S_n^{(\alpha,\beta)}[f;x]=\int_{-1}^{1} f(t)K_n(t,x)p(t)\,dt, \]

where

\[ K_n(t,x)=\lambda_n \frac{P_{n+1}^{(\alpha,\beta)}(t)P_n^{(\alpha,\beta)}(x) - P_n^{(\alpha,\beta)}(t)P_{n+1}^{(\alpha,\beta)}(x)} {t-x}, \]

\[ \lambda_n= \frac{2^{-\alpha-\beta}}{2n+\alpha+\beta+2} \frac{\Gamma(n+2)\Gamma(n+\alpha+\beta+2)} {\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}. \]

The aim of the present note is to establish the rate of convergence of
\(S_n^{(\alpha,\beta)}[f;x]\) to \(f(x)\) for various classes of functions.

We consider the following classes of functions, defined on the interval \([-1,1]\):
\(W^r\) is the class of functions having an absolutely continuous derivative of order
\(r-1\) and a derivative of order \(r>0\) satisfying almost everywhere the inequality
\(|f^{(r)}(x)|\le 1\);
\(W^rH_\omega\) is the class of functions having a continuous derivative of order
\(r>0\), whose modulus of continuity \(\omega(f^{(r)},\delta)\le \omega(\delta)\),
where \(\omega(\delta)\) is a given true majorant of moduli of continuity (i.e.
\(\lim_{\delta\to0}\omega(\delta)=\omega(0)=0\) and, for
\(0\le \delta_1<\delta_2\),
\(0\le \omega(\delta_2)-\omega(\delta_1)\le \omega(\delta_2-\delta_1)\));
\(H^\mu=W^0H_\mu\) is the class of functions satisfying a Lipschitz condition of order
\(\mu\) \((0\le \mu\le 1)\) with constant \(1\);
\(W^rH^\mu\) is the class \(W^rH_\omega\), where \(\omega(\delta)=\delta^\mu\).

Theorem 1. Let \(\alpha\ge -\frac12,\ \beta\ge -\frac12\),

\[ L_n(x)=\int_{-1}^{1} |K_n(t,x)|p(t)\,dt \]

be the Lebesgue function of the process under consideration. Then the estimate

\[ L_n(x)=O(1)\left\{\ln n+ \frac{n^{\alpha+1/2}}{(n\sqrt{1-x})^{\alpha+1/2}+1} + \frac{n^{\beta+1/2}}{(n\sqrt{1+x})^{\beta+1/2}+1} \right\} \]

is valid. Here \(x\in[-1,1]\), and the quantity \(O(1)\) depends only on \(\alpha\) and \(\beta\).

Remark 1. The order of the indicated estimate cannot be improved, since, as Rau showed \((^1)\),
\(L_n(1)\sim n^{\alpha+1/2}\).

Remark 2. Using the well-known inequality

\[ |S_n^{(\alpha,\beta)}[f;x]-f(x)|=[L_n(x)+1]E_n(f), \]

where \(E_n(f)\) is the best approximation of the function \(f(x)\) by algebraic polynomials of degree not exceeding \(n\), one can find estimates for the deviation of the Fourier–Jacobi sums from the approximated functions for all possible classes of functions.

* The class \(H^0\) consists of measurable functions \(f\) for which \(|f(x)-f(y)|\le 1\) for all \(x\) and \(y\).

Theorem 2. The asymptotic formula holds

\[ \sup_{f\in H^\mu}\left|S_n^{(\alpha,\beta)}[f;1]-f(1)\right|= \]

\[ =\frac{2^{2\mu}}{\pi^{1/2}}\int_0^{\pi/2}u^\mu\sin u\,du\, \frac{\Gamma(\mu/2+\alpha/2+1/4)\Gamma(\mu/2+\beta/2+3/4)} {\Gamma(\alpha+1)\Gamma(\mu+\alpha/2+\beta/2+1)} \,n^{\alpha+1/2-\mu}+\sigma_n n^{-\mu}. \]

Here \(\alpha>-1/2,\ \beta>-1/2\),

\[ \sigma_n= \begin{cases} O(n^{\alpha-1/2}\ln n), & \text{if } \alpha=1/2,\ \beta\ge -1/2 \text{ or } \alpha\ge 1/2,\ \beta=-1/2,\\ O(n^{\alpha-1/2})+O(1)+O(n^{\alpha-\beta-1}) & \text{in the remaining cases;} \end{cases} \]

the constants entering the \(O\)-terms depend only on \(\alpha\) and \(\beta\).

Remark 1. If \(-1<\alpha\le -1/2\), then

\[ \sup_{f\in H^\mu}\left|S_n^{(\alpha,\beta)}[f;1]-f(1)\right|=O(n^{-\mu}). \]

Remark 2. Since

\[ \sup_{f\in H^0}\left|S_n^{(\alpha,\beta)}[f;1]-f(1)\right|=\frac12 L_n(1), \]

from Theorem 2, for \(\mu=0\), one obtains the asymptotic formula for \(L_n(1)\) found by Lorch \({}^{(2)}\).

Theorem 3. If the function \(f(x)\in W^r\), then

\[ S_n^{(\alpha,\beta)}[f;1]-f(1) = A\{S_{n-r}^{(\alpha+r,\beta+r)}[f^{(r)};1]-f^{(r)}(1)\} + \]

\[ + B\int_{-1}^{1}g(t)P_{n-r-1}^{(\alpha+r+2,\beta+r+1)}(t)(1-t)^{\alpha+r}(1+t)^{\beta+r+1}\,dt, \tag{*} \]

where \(\alpha>-1,\ \beta>-1\),

\[ A=2^r\frac{\Gamma(\alpha+r+1)\Gamma(n+\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(n+\alpha+\beta+r+2)} \,\frac{(n-r)!}{n!}, \]

\[ B=2^{-\alpha-\beta-r-2} \frac{\Gamma(n+\alpha+\beta+2)}{\Gamma(\alpha+1)\Gamma(n+\beta+1)} \,\frac{(n-r-1)!}{n!}, \]

\[ g(t)=f^{(r)}(1)+r f^{(r)}(t)-r(r+1)\int_0^1 f^{(r)}(1-u+tu)u^{r-1}\,du. \]

Remark 1. If \(f(x)\in W^rH_\omega\), then the second term on the right-hand side of formula \((*)\) is

\[ O\bigl(\omega(n^{-1})(n^{\alpha-r-1/2}+n^{-2r})\bigr), \]

when \(\alpha+r\ne 1/2\), and

\[ O\bigl(\omega(n^{-1})n^{2r}\ln n\bigr), \]

when \(\alpha+r=1/2\). Here the constants entering the \(O\)-terms depend on \(\alpha,\beta\), and \(r\).

Remark 2. From Theorems 2 and 3 and the preceding remark there easily follows the asymptotic formula for

\[ \sup_{f\in W^rH^\mu}\left|S_n^{(\alpha,\beta)}[f;1]-f(1)\right|. \]

Theorem 4. Let \(\alpha>-1,\ \beta>-1\). Then for \(x\in(-1,1)\) and \(r\ge 0\) one has

\[ \sup_{f\in W^rH^\mu}\left|S_n^{(\alpha,\beta)}[f;x]-f(x)\right|= \]

\[ =\frac{2^{\mu+1}}{\pi^2}\int_0^{\pi/2}u^\mu\sin u\,du \left(\frac{\sqrt{1-x^2}}{n}\right)^{r+\mu} \ln n + O(n^{-r-\mu}), \]

where the remainder term depends on \(\alpha,\beta,r\), and \(x\). The estimate of the remainder term is uniform with respect to \(x\) on the interval \([-1+\varepsilon,1-\varepsilon]\).

Leningrad State
Pedagogical Institute
named after A. I. Herzen

Received
3 V 1965

REFERENCES

\({}^{1}\) H. Rau, J. reine u. angew. Math., 161 (1929). \({}^{2}\) L. Lorch, Proc. Am. Math. Soc., 10, 5 (1959).