UDC 517.512.6

MATHEMATICS

V. M. BADKOV

ON THE APPROXIMATION OF FUNCTIONS BY FOURIER–JACOBI SUMS

(Presented by Academician M. A. Lavrent’ev on 30 VI 1965)

Let \(\{\hat P_k^{(\alpha,\beta)}(x)\}_0^\infty\) be a system of Jacobi polynomials orthonormal on the interval \(-1 \le x \le 1\) with weight \(n(x)=(1-x)^\alpha(1+x)^\beta\) \((\alpha,\beta>-1)\), and let \(R_n^{(\alpha,\beta)}(f,x)=f(x)-S_n^{(\alpha,\beta)}(f,x)\) be the \(n\)-th remainder of the Fourier–Jacobi series of a function \(f(x)\) belonging to the class \(W^{(r)}H^{(\gamma)}\) of all functions whose \(r\)-th derivative \((r\ge 0)\) on the interval \(-1\le x\le 1\) satisfies a Lipschitz condition of order \(\gamma\) \((0<\gamma\le 1)\) with constant equal to one.

In the present note, for \(\alpha\) and \(\beta \ge 0\), an estimate is obtained for the quantity

\[ \sup_{f\in W^{(r)}H^{(\gamma)}} \|R_n^{(\alpha,\beta)}(f,x)\|_C, \]

where \(\|u(x)\|_C=\max_{-1\le x\le 1}|u(x)|\). In addition, for the case of ultraspherical polynomials \((\alpha=\beta=\lambda-\tfrac12)\) it is proved that the estimate obtained is exact in the sense of order.

In what follows \(q=\max(\alpha,\beta)\), and all constants \(C_1,\ldots\) depend neither on \(x\), nor on \(n\), nor on the function \(f\in W^{(r)}H^{(\gamma)}\).

Theorem 1. Let \(r\ge 0\), \(0<\gamma\le 1\), \(r+\gamma>\tfrac12\). If \(\alpha,\beta>0\), \(q\ge \tfrac12\), then the estimate

\[ \sup_{f\in W^{(r)}H^{(\gamma)}} \|R_n^{(\alpha,\beta)}(f,x)\|_C \le \frac{C_1}{(n+1)^{r+\gamma-q-\tfrac12}}, \tag{1} \]

is valid, while if \(\alpha,\beta>0\), \(q<\tfrac12\), or if \(\alpha\beta=0\) \((\alpha\ge 0,\ \beta\ge 0)\), then

\[ \sup_{f\in W^{(r)}H^{(\gamma)}} \|R_n^{(\alpha,\beta)}(f,x)\|_C \le \frac{C_2\ln(n+2)}{(n+1)^{r+\gamma-q-\tfrac12}}. \tag{2} \]

Theorem 2. Let \(a>-\tfrac12\), \(r\ge 0\), \(0<\gamma\le 1\). Then there exists a constant \(C_3>0\) such that

\[ \sup_{f\in W^{(r)}H^{(\gamma)}} |R_n^{(\alpha,\alpha)}(f,1)| \ge \frac{C_3}{(n+1)^{r+\gamma-\alpha-\tfrac12}}. \tag{3} \]

The proof of Theorem 1 is carried out with the aid of the strengthened Jackson theorem (see \((^1)\), p. 276), by virtue of which, for any \(n>r\), for every function \(f(x)\in W^{(r)}H^{(\gamma)}\) one can construct an algebraic polynomial \(Q_n(x)\) of degree \(\le n\) such that

\[ |f(x)-Q_n(x)| \le \frac{C_4}{n^{r+\gamma}} \left[ (\sqrt{1-x^2})^{r+\gamma} + \frac{1}{n^{r+\gamma}} \right] \qquad (-1\le x\le 1). \]

It follows from this that

\[ |R_n^{(\alpha,\beta)}(f,x)| \le |f(x)-Q_n(x)| + \int_{-1}^{1} |f(t)-Q_n(t)|\,|K_n^{(\alpha,\beta)}(x,t)|\,n(t)\,dt \le \]

\[ \le \frac{C_5}{(n+1)^{r+\gamma}} + \frac{C_6}{(n+1)^{r+\gamma}} \int_{-1}^{1} (1-t)^{r/2+\gamma/2+\alpha} (1+t)^{r/2+\gamma/2+\beta} |K_n^{(\alpha,\beta)}(x,t)|\,dt + \]

\[ +\frac{C_7}{(n+1)^{2(r+\gamma)}}\int_{-1}^{1}\left|K_n^{(\alpha,\beta)}(x,t)\right|\,n(t)\,dt= \]
\[ =\frac{C_5}{(n+1)^{r+\gamma}}+\frac{C_6}{(n+1)^{r+\gamma}}I_1+ \frac{C_7}{(n+1)^{2(r+\gamma)}}I_2, \tag{4} \]

where

\[ K_n^{(\alpha,\beta)}(x,t)=\sum_{k=0}^{n}\hat P_k^{(\alpha,\beta)}(x)\hat P_k^{(\alpha,\beta)}(t). \]

For the integral \(I_2\) we have

\[ I_2\leq \left\{\int_{-1}^{1}\left|K_n^{(\alpha,\beta)}(x,t)\right|^2 n(t)\,dt\right\}^{1/2} \left\{\int_{-1}^{1} n(t)\,dt\right\}^{1/2} = \tag{5} \]

\[ = \left\{\sum_{k=0}^{n}\left|\hat P_k^{(\alpha,\beta)}(x)\right|^2\right\}^{1/2} \left\{\int_{-1}^{1} n(t)\,dt\right\}^{1/2} \leq C_8\left\{\sum_{k=0}^{n}(k+1)^{2q+1}\right\}^{1/2} \leq C_9(n+1)^{q+1}. \]

From estimate (5) it follows that the last term in (4) is
\(O\bigl((n+1)^{q+1-2r-2\gamma}\bigr)\) uniformly in \(x\) and \(f\).

The estimate of the integral \(I_1\) in the case \(\alpha,\beta>0,\ q\geq 1/2\) is carried out with the help of the formula

\[ P_{n-1}^{(\alpha,\beta)}(x)= \frac{1}{(1-x)^{\alpha/2+1}(1+x)^{\beta/2+1}}\times \]

\[ \times \left\{ \int_{-1}^{x}(1-t)^{\alpha/2}(1+t)^{\beta/2} \left[ \frac{\alpha}{2}-\frac{\beta}{2} +\left(\frac{\alpha}{2}+\frac{\beta}{2}-2\right)t \right] P_{n-1}^{(\alpha,\beta)}(t)\,dt \right. \]

\[ \left. -2n\int_{-1}^{x}(1-t)^{\alpha/2}(1+t)^{\beta/2} P_n^{(\alpha-1,\beta-1)}(t)\,dt \right\} \quad(\alpha,\beta>0), \tag{6} \]

which follows from the differential equation for Jacobi polynomials. From this formula it follows that for arbitrary \(\alpha\) and \(\beta>0\) for which \(q\geq 1/2\), and for \(-1<t<1\), the inequality

\[ \left| \frac{\hat P_n^{(\alpha,\beta)}(x)-\hat P_n^{(\alpha,\beta)}(t)}{x-t} \right| \leq \frac{C_{10}(n+1)^{q+1/2}} {(1-t)^{\alpha/2+1}(1+t)^{\beta/2+1}} \quad(-1\leq x\leq 1), \tag{7} \]

holds, where the constant \(C_{10}\) depends neither on \(x\), nor on \(t\), nor on \(n\). Using the Christoffel–Darboux formula ((2), p. 83) and inequality (7), for \(\alpha\) and \(\beta>0,\ q\geq 1/2\) we obtain

\[ I_1\leq C_{11}(n+1)^{q+1/2}\times \]

\[ \times \int_{-1}^{1}(1-t)^{(r+\gamma+\alpha-2)/2}(1+t)^{(r+\gamma+\beta-2)/2} \left[ \left|\hat P_n^{(\alpha,\beta)}(t)\right| + \left|\hat P_{n+1}^{(\alpha,\beta)}(t)\right| \right]\,dt. \tag{8} \]

But the integral in inequality (8) is bounded as \(n\to\infty\) (see (2), p. 180). From inequalities (4), (5), and (8) the validity of estimate (1) follows.

In the cases where \(\alpha,\beta>0,\ q<1/2\), or where \(\alpha\beta=0\) \((\alpha\geq0,\ \beta\geq0)\), the integral \(I_1\) is estimated with the help of a device applied by P. K. Suetin (5) to Legendre polynomials \((\alpha=\beta=0)\), and of an inequality proved by S. N. Bernstein (3):

\[ \left|\hat P_n^{(\alpha,\beta)}(x)\right| (1-x)^{\alpha/2+1/4}(1+x)^{\beta/2+1/4} \leq C_{12} \quad(-1\leq x\leq1). \]

In proving Theorem 2 it is enough to restrict oneself to the consideration of functions
\(f(x)\in W^{(r)}H^{(\gamma)}\), for which
\(f(1)=f'(1)=\ldots=f^{(r)}(1)=0\).

We have, for any \(\lambda>0\) and \(0<\theta<\pi\),

\[ (\sin \theta)^{2\lambda-1}P_n^{(\lambda)}(\cos\theta) = \frac{2^{2-2\lambda}}{\Gamma(\lambda)} \frac{\Gamma(n+2\lambda)}{\Gamma(n+\lambda+1)} \sum_{\nu=0}^{\infty} f_{\nu n}^{(\lambda)} \sin (n+2\nu+1)\theta, \tag{9} \]

\[ f_{0n}^{(\lambda)}=1;\qquad f_{\nu n}^{(\lambda)} = \frac{(1-\lambda)(2-\lambda)\cdots(\nu-\lambda)}{\nu!} \times \]

\[ \times \frac{(n+1)(n+2)\cdots(n+\nu)} {(n+\lambda+1)(n+\lambda+2)\cdots(n+\lambda+\nu)}, \qquad \nu=1,2,\ldots \]

In the case of nonintegral \(\lambda\), this formula is given by G. Szegő \((^2)\). Using the uniform convergence in \(\lambda\) of the series in (9), one can prove formula (9) for natural \(\lambda=k\), with \(f_{\nu n}^{(k)}=0\) if \(\nu\ge k\). With the aid of representation (9), the expression for the remainder \(R_n^{(\lambda-1/2,\lambda-1/2)}(f,1)=R_n^{(\lambda)}\), for arbitrary \(\lambda>0\), is brought to the form

\[ R_n^{(\lambda)} = a_n^{(\lambda)} \left\{ \int_0^{2\pi} f(\cos\psi) \sum_{\nu=0}^{\infty} f_{\nu n}^{(\lambda)} \left[ \cos(n+2\nu+1)\psi + \right.\right. \]

\[ \left.\left. + \left(1-\frac{\lambda}{n+\nu+1+\lambda}\right) \cos(n+2\nu+2)\psi \right]\,d\psi - \right. \]

\[ \left. - \int_0^{2\pi} f(\cos\psi) \sum_{\nu=0}^{\infty} f_{\nu n}^{(\lambda)} \frac{2\lambda}{n+\nu+1+\lambda} \frac{\sin (n+2\nu+\tfrac32)\psi}{2\sin \psi/2}\,d\psi \right\}, \tag{10} \]

where

\[ a_n^{(\lambda)} = \Gamma(n+2\lambda+1) \left[ 2^{2\lambda}\sqrt{\pi}\,\Gamma(\lambda+\tfrac12)\Gamma(n+\lambda+1) \right]^{-1}. \]

The application of \(r\)-fold integration by parts and the Abel transformation in the first integral in (10), as well as the application of Jackson’s estimate \((^4)\) for the \(n\)-th remainder of the Fourier series of a \(2\pi\)-periodic differentiable function in estimating the second integral in (10), makes it possible to bring expression (10) to the form

\[ R_n^{(\lambda)} = a_n^{(\lambda)} \left\{ \frac{(-1)^r 2^{\lambda+1}}{(n+1)^r} \int_0^{\pi} \cos\left[ \left(n+\frac12+\lambda\right)\psi + \frac{(r+1-\lambda)\pi}{2} \right] \times \right. \]

\[ \left. \times \cos\frac{\psi}{2}\, \sin^{r+\lambda-1}\psi\, f^{(r)}(\cos\psi)\,d\psi + I_3 \right\}, \tag{11} \]

where

\[ I_3 = O\left( \frac{\ln(n+2)}{(n+1)^{r+\gamma+1}} \right), \qquad \text{if } \lambda\ge 1; \]

\[ I_3 = O\left( \frac{\ln(n+2)}{(n+1)^{r+\gamma+\varepsilon}} \right), \qquad \text{if } 0<\varepsilon<\lambda<1. \]

From (11) and from the fact that

\[ \sup_{\varphi(x)\in H^{(\gamma)}} \left| \int_0^{\pi} \cos\left[ \left(n+\frac12+\lambda\right)\psi + \frac{(r+1-\lambda)\pi}{2} \right] \cos\frac{\psi}{2}\, \sin^{r+\lambda-1}\psi\, \varphi(\cos\psi)\,d\psi \right| \]

has order \(n^{-\gamma}\), the validity of Theorem 2 follows. The indicated order is attained on the function \(v_n(x)\in H^{(\gamma)}\) \((-1\le x\le 1)\), defined by the formula

\[ v_n(x) = \frac12 u_n(x)\operatorname{sign} \left\{ \cos\left[ \left(n+\frac12+\lambda\right)\arccos x + \frac{(r+1-\lambda)\pi}{2} \right] \right\}, \]

where \(u_n(x)=(x-a)^\gamma\) for \(x\in [a,(a+b)/2]\) and \(u_n(x)=(b-x)^\gamma\) for \(x\in [(a+b)/2,b]\), while \([a,b]\) is any one of the parts into which the segment

\([-1,1]\) is divided by the zeros of the function

\[ \cos\left[\left(n+\frac{1}{2}+\lambda\right)\arccos x+\frac{(r+1-\lambda)\pi}{2}\right]. \]

Let us note that in the case when \(\alpha=\beta\) is an integer, inequality (1) follows from the estimate of the Lebesgue function of the Fourier–Jacobi series obtained by I. K. Daugavet \({}^{6}\).

Sverdlovsk Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
21 VI 1965

REFERENCES

\({}^{1}\) A. F. Timan, Theory of Approximation of Functions of a Real Variable, Moscow, 1960.
\({}^{2}\) G. Szegő, Orthogonal Polynomials, Moscow, 1962.
\({}^{3}\) S. N. Bernstein, On polynomials orthogonal on a finite interval, Works, 2, Publishing House of the Academy of Sciences of the USSR, 1954, p. 7.
\({}^{4}\) D. Jackson, The Theory of Approximation, N. Y., 1930.
\({}^{5}\) P. K. Suetin, DAN, 158, No. 6, 1275 (1964).
\({}^{6}\) I. K. Daugavet, Siberian Mathematical Journal, 6, No. 1, 70 (1965).