UDC 517.512.6

MATHEMATICS

A. V. ZORSHCHIKOV

ON THE UNIFORM CONVERGENCE OF FOURIER SERIES IN JACOBI POLYNOMIALS

(Presented by Academician M. A. Lavrent’ev, 14 XI 1966)

Let the function \(f(x)\) be continuous and have bounded variation on the interval \([-1,1]\), and let the series

\[ \sum_{k=0}^{\infty} c_k \hat P_k^{(\alpha,\beta)}(x) \tag{1} \]

be its Fourier series in Jacobi polynomials, orthonormal on the interval \([-1,1]\) with weight
\(p(x)=(1-x)^\alpha(1+x)^\beta\). From the equiconvergence theorem of G. Szegő \(\left({}^{5}\right.\), p. 254) it follows that for \(\alpha>-1,\ \beta>-1\) the series (1) converges uniformly on the interval \([-1+\mu,\,1-\mu]\), where \(0<\mu<1\). In the case of Legendre polynomials \((\alpha=\beta=0)\), E. V. Hobson \(\left({}^{4}\right.\), p. 325) also established that the series (1) converges at the points \(x=\pm1\); if, moreover, \(-1<\alpha<-1/2\) and \(-1<\beta<-1/2\), then the convergence of the series (1) at these points follows from the results of G. Rau \(\left({}^{1}\right)\).

In the present paper, the uniform convergence of the series (1) is established on the whole interval \([-1,1]\) under the conditions
\(-1<\alpha<1/2,\ -1<\beta<1/2\), and \(|\alpha-\beta|<1\) (as is known, the case \(\alpha=\beta=-1/2\) has been well studied).

Introduce the notation

\[ K_n^{(\alpha,\beta)}(t,x) = \sum_{k=0}^{n} \hat P_k^{(\alpha,\beta)}(t)\hat P_k^{(\alpha,\beta)}(x). \tag{2} \]

Lemma 1. For the kernel (2) the following formula holds

\[ K_n^{(\alpha,\beta)}(t,x) = (1-x^2)K_{n-1}^{(\alpha+1,\beta+1)}(t,x) + \tag{3} \]

\[ + \theta_n^{(\alpha,\beta)} \left[ (n+\alpha+\beta+2)P_n^{(\alpha+1,\beta+1)}(t)P_n^{(\alpha,\beta)}(x) + (n+1)P_{n-1}^{(\alpha+1,\beta+1)}(t)P_{n+1}^{(\alpha,\beta)}(x) \right], \]

where

\[ P_n^{(\alpha,\beta)}(x) = \sqrt{ \frac{ 2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1) }{ (2n+\alpha+\beta+1)\Gamma(n+\alpha+\beta+1)\Gamma(n+1) } } \,\hat P_n^{(\alpha,\beta)}(x), \tag{4} \]

\[ \theta_n^{(\alpha,\beta)}=O(1), \qquad K_{-1}^{(\alpha+1,\beta+1)}(t,x)\equiv 0, \qquad P_{-n}^{(\alpha+1,\beta+1)}(t)\equiv 0. \]

Formula (3) is a consequence of known relations for Jacobi polynomials \(\left({}^{4}\right)\), contained in \(\left({}^{5}\right)\), Ch. IV.

Put

\[ \Phi_n^{(\alpha,\beta)}(a,b;x) = \int_a^b p(t)K_n^{(\alpha,\beta)}(t,x)\,dt, \qquad -1\le a<b\le 1,\quad x\in[-1,1], \tag{5} \]

and by \(\Phi_n^{(\alpha,\beta)}(E;x)\) we shall denote the same integral, taken over a set \(E\subset[-1,1]\). In view of the validity of the formula \(\left({}^{5}\right.\), p. 71)

\[ P_n^{(\alpha,\beta)}(x)=(-1)^nP_n^{(\beta,\alpha)}(-x) \tag{6} \]

it is sufficient to consider the case when \(x\in[0,1]\).

Lemma 2. Let \(-\frac12 \leq \alpha \leq \frac12\), \(\beta>-1\), and \(x\in[0,1]\). Then, provided the condition \(\alpha-\beta \leq 1\) is satisfied, there exists a constant \(C_1\), depending only on \(\alpha\) and \(\beta\), such that for all \(n\) the inequality

\[ \left|\Phi_n^{(\alpha,\beta)}(a,b;x)\right|\leq C_1,\qquad -1\leq a<b\leq 1. \tag{7} \]

holds.

Proof. Let \(E_1=\{t;\ |t-x|<\delta\}\), where \(0<\delta<1\). We first consider the case when \([a,b]\cap E_1=\varnothing\). Using formula ((5), p. 83)

\[ K_n^{(\alpha,\beta)}(t,x) = l_n^{(\alpha,\beta)} \frac{ P_{n+1}^{(\alpha,\beta)}(t)P_n^{(\alpha,\beta)}(x) - P_n^{(\alpha,\beta)}(t)P_{n+1}^{(\alpha,\beta)}(x) }{t-x}, \tag{8} \]

where \(l_n^{(\alpha,\beta)}=O(n)\), and the second mean-value theorem, from (5) we obtain

\[ \left|\Phi_n^{(\alpha,\beta)}(a,b;x)\right| \leq C_2\delta^{-1}n \int_{a_1}^{b_1} p(t)\left[ P_{n+1}^{(\alpha,\beta)}(t)P_n^{(\alpha,\beta)}(x) - P_n^{(\alpha,\beta)}(t)P_{n+1}^{(\alpha,\beta)}(x) \right]\,dt. \tag{9} \]

where \([a_1,b_1]\subset [a,b]\). For the polynomials (4) the following relation holds ((5), p. 175)

\[ \max_{x\in[0,1]}\left|P_n^{(\alpha,\beta)}(x)\right| = \left|P_n^{(\alpha,\beta)}(1)\right| = \frac{\Gamma(n+\alpha+1)}{\Gamma(\alpha+1)\Gamma(n+1)}, \qquad \alpha\geq -\frac12, \tag{10} \]

and the inequality

\[ (1-x)^{\alpha/2+1/4}(1+x)^{\beta/2+1/4} \left|P_n^{(\alpha,\beta)}(x)\right| \leq \frac{C_3}{\sqrt n}, \]

\[ \min(\alpha,\beta)=\rho\geq -\frac12,\qquad x\in[-1,1], \tag{11} \]

which is a consequence of formula 7.32.5 ((5), p. 176). If \(\beta\geq -\frac12\), then by means of (10), (11), and formula ((5), p. 107)

\[ (1-x)^\alpha(1+x)^\beta P_n^{(\alpha,\beta)}(x) = -\frac{1}{2n}\frac{d}{dx} \left\{ (1-x)^{\alpha+1}(1+x)^{\beta+1} P_{n-1}^{(\alpha+1,\beta+1)}(x) \right\} \tag{12} \]

from (9) we find

\[ \left|\Phi_n^{(\alpha,\beta)}(a,b;x)\right| \leq C_4\delta^{-1}n^{\alpha-1/2}. \tag{13} \]

If \(-1<\beta<-\frac12\), then instead of (11) we use the inequality of G. Szegő \((^2)\)

\[ (1+x)^\lambda \left|P_n^{(\alpha,\beta)}(x)\right| \leq C_5 n^{\beta-2\lambda}, \tag{14} \]

where \(\beta>-\frac12\), \(0\leq\lambda<\beta/2+\frac14\), \(x\in[-1,1-\mu]\), and obtain

\[ \left|\Phi_n^{(\alpha,\beta)}(a,b;x)\right| \leq C_6\delta^{-1}n^{\alpha-\beta-1}. \tag{15} \]

Now let \([a,b]\cap E_1=E_2\) and \(E_2\neq\varnothing\), while \(E_3=[a,b]\setminus E_2\). For \(\Phi_n^{(\alpha,\beta)}(E_3;x)\) estimates analogous to (13) or (15) hold. To obtain an estimate for \(\Phi_n^{(\alpha,\beta)}(E_2;x)\), we make the substitution \(t=\cos\theta\). To the set \(E_2\) there corresponds a set \(D\subset[0,\pi]\), which we split into parts:
\(D_1=\{\theta;\ |\theta-\theta_0|\leq 1/n\}\) and \(D_2=D\setminus D_1\); here \(\theta_0=\arccos x\).

a) Let \(D_1\neq\varnothing\). From (10) and (11) we obtain

\[ \left|P_n^{(\alpha,\beta)}(\cos\theta)\right| \leq C_7\theta^{-\alpha-1/2}[1-\varphi_n(\theta)]n^{-1/2} + C_8 n^\alpha\varphi_n(\theta), \tag{16} \]

\[ \alpha\geq -\frac12,\qquad 0\leq\theta\leq\pi/2,\qquad \varphi_n(\theta)= \begin{cases} 1, & 0\leq\theta\leq n^{-1},\\ 0, & n^{-1}<\theta\leq\pi/2. \end{cases} \]

Taking into account (4), (5), and (16), we arrive at the inequality

\[ \left|\Phi_n^{(\alpha,\beta)}(D_1;\cos\theta_0)\right| \leq C_9 \left[ \int_{D_1}\theta^{2\alpha+1}\,d\theta \right]^{1/2} \left[ \sum_{k=0}^{n} \left[ \widehat P_k^{(\alpha,\beta)}(\cos\theta_0) \right]^2 \right]^{1/2} \leq \]

\[ \leq C_{10} \left[ \int_{D_1} \sum_{k=0}^{n} \left\{ [1-\varphi_k(\theta_0)] \left(\frac{\theta}{\theta_0}\right)^{\alpha+1/2} + (k\theta)^{\alpha+1/2}\varphi_k(\theta_0) \right\}^2 d\theta \right]^{1/2} \leq C_{11}. \tag{17} \]

b) Let, next, \(D_2^{(1)}=D_2\cap(\theta_0+n^{-1},\arccos(x-\delta))\). Then, using formula (3) and introducing for the integrals obtained the notation \(I_1,I_2,I_3\), we have

\[ \begin{aligned} \left|\Phi_n^{(\alpha,\beta)}(D_2^{(1)};\cos\theta_0)\right| &\leq C_{12}\Biggl\{ \left|\int_{D_2^{(1)}}\left(\sin\frac{\theta}{2}\right)^{2\alpha+1} \left(\cos\frac{\theta}{2}\right)^{2\beta+1} \sin^2\theta_0\,K_{n-1}^{(\alpha+1,\beta+1)}(\cos\theta,\cos\theta_0)\,d\theta\right| \\ &\quad+\left|\int_{D_2^{(1)}}\left(\sin\frac{\theta}{2}\right)^{2\alpha+1} \left(\cos\frac{\theta}{2}\right)^{2\beta+1} nP_n^{(\alpha,\beta)}(\cos\theta_0)P_n^{(\alpha+1,\beta+1)}(\cos\theta)\,d\theta\right| \\ &\quad+\left|\int_{D_2^{(1)}}\left(\sin\frac{\theta}{2}\right)^{2\alpha+1} \left(\cos\frac{\theta}{2}\right)^{2\beta+1} nP_{n+1}^{(\alpha,\beta)}(\cos\theta_0)P_{n-1}^{(\alpha+1,\beta+1)}(\cos\theta)\,d\theta\right| \Biggr\} \\ &=C_{12}\{I_1+I_2+I_3\}. \end{aligned} \tag{18} \]

To estimate \(I_1\) we use inequality (11), the asymptotic formula ((5), p. 205)

\[ P_n^{(\alpha,\beta)}(\cos\theta) =n^{-1/2}k(\theta)\{\cos(N\theta+\gamma)+(n\sin\theta)^{-1}O(1)\}, \tag{19} \]

where

\[ k(\theta)=\pi^{-1/2}\left(\sin\frac{\theta}{2}\right)^{-\alpha-1/2} \left(\cos\frac{\theta}{2}\right)^{-\beta-1/2},\quad N=n+\frac{\alpha+\beta+1}{2}, \]

\[ \gamma=-(\alpha+1/2)\pi/2,\quad \alpha>-1,\ \beta>-1,\quad cn^{-1}\leq\theta\leq\pi-cn^{-1}, \]

\(c>0\) is a fixed number, and the identity ((5), p. 83)

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

\[ =(n+\alpha+1)P_n^{(\alpha,\beta)}(x)-(n+1)P_{n+1}^{(\alpha,\beta)}(x). \tag{20} \]

After transformations from (18) we obtain

\[ \begin{aligned} |I_1|\leq C_{13}\Biggl\{ &\left|\int_{D_2^{(1)}}\left(\frac{\theta_0}{\sin\frac{\theta}{2}}\right)^{1/2-\alpha} \frac{\theta_0[\cos(N_1\theta+\gamma_1)+(n\sin\theta)^{-1}]} {\left(\cos\frac{\theta}{2}\right)^{1/2-\beta} \sin\frac{\theta-\theta_0}{2}\,\sin\frac{\theta+\theta_0}{2}}\,d\theta\right| \\ &+\int_{D_2^{(1)}}\left(\frac{\theta_0}{\sin\frac{\theta}{2}}\right)^{1/2-\alpha} \frac{\sin\frac{\theta}{2}[\cos(N_2\theta+\gamma_2)+(n\sin\theta)^{-1}]} {\left(\cos\frac{\theta}{2}\right)^{1/2-\beta} \sin\frac{\theta-\theta_0}{2}\,\sin\frac{\theta+\theta_0}{2}}\,d\theta \Biggr\}, \end{aligned} \]

whence, after applying the second mean-value theorem, the estimate \(|I_1|\leq C_{14}\) follows. With the help of (10) and (19) it is also not difficult to establish the boundedness of the integrals \(I_2\) and \(I_3\) in (18), which leads to the inequality

\[ \left|\Phi_n^{(\alpha,\beta)}(D_2^{(1)};\cos\theta_0)\right|\leq C_{15}. \]

c) Let, finally, \(D_2^{(2)}=D_2\setminus D_2^{(1)}\) and \(D_2^{(2)}\ne\varnothing\). If \(1/n\in D_2^{(2)}\), then, arguing as in case a), we find the inequality \(\left|\Phi_n^{(\alpha,\beta)}(\cos 1/n,b;x)\right|\leq C_{16}\). The further arguments are carried out with the aid of relations (8), (11), (12), (19), and (20), after which we obtain the estimate

\[ \left|\Phi_n^{(\alpha,\beta)}(D_2^{(2)};\cos\theta_0)\right|\leq C_{17}, \]

and Lemma 2 is proved.

Let us note that inequality (7) in the case when \(\alpha=\beta=1/2\) (Chebyshev polynomials of the second kind) is not difficult to verify directly.

Lemma 3. If \(-1<\alpha<-1/2\) and \(\beta>-1\), then there exists a constant \(C_{18}\), depending on \(\alpha\) and \(\beta\), such that for all \(x\in[0,1]\) and all \(n\) the inequality holds

\[ \left|\Phi_n^{(\alpha,\beta)}(a,b;x)\right|\leq C_{18},\qquad -1\leq a<b\leq 1. \tag{21} \]

The proof is essentially the same as the preceding one. To estimate \(\Phi_n^{(\alpha,\beta)}(E_2;x)\), the relation

\[ \begin{aligned} \Phi_n^{(\alpha,\beta)}(E_2;x) &=\Phi_{n-1}^{(\alpha+1,\beta+1)}(E_2;x)+ \\ &\quad+\theta_n^{(\alpha,\beta)}\int_{E_2} p(t)\bigl[(n+\alpha+\beta+2) P_n^{(\alpha+1,\beta+1)}(x)P_n^{(\alpha,\beta)}(t)+ \\ &\qquad\qquad\qquad +(n+1)P_{n-1}^{(\alpha+1,\beta+1)}(x)P_{n+1}^{(\alpha,\beta)}(t)\bigr]\,dt, \end{aligned} \]

which is a consequence of (3), is additionally used.

Thus, under the conditions \(-1<\alpha\leq 1/2\), \(-1<\beta\leq 1/2\), and \(|\alpha-\beta|\leq 1\), Lemmas 2, 3 and relation (7) imply the inequality

\[ \left|\Phi_n^{(\alpha,\beta)}(a,b;x)\right|\leq C_{19}, \qquad -1\leq a<b\leq 1,\qquad -1\leq x\leq 1. \tag{22} \]

Theorem. If \(-1<\alpha<1/2\), \(-1<\beta<1/2\), and \(|\alpha-\beta|<1\), then every continuous function \(f(x)\) of bounded variation on the interval \([-1,1]\) is expanded in a Fourier–Jacobi series converging uniformly on this interval.

Proof. It is enough to consider the case where \(f(x)\) is continuous and nondecreasing on the interval \([-1,1]\). As is known (3), the remainder of the series (1) may be represented in the form

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

and the validity of the assertion of the theorem follows from relations (8), (22), (11) or (14).

In conclusion we give two monotone continuous functions with divergent Fourier–Jacobi series under some conditions on \(\alpha\) and \(\beta\). Let \(f_1(x)=\sqrt{1+x}\) for \(x\in[-1,0]\), and \(f_1(x)\equiv 1\) for \(x\in(0,1]\). Then for \(\alpha=\beta=1/2\) we obtain
\[ R_n^{(1/2,1/2)}(f;1)=-(-1)^{[(n+1)/2]}/\pi . \]
If, however, we put
\[ f_2(x)=(1+x)^{(\alpha-\beta-1)/2}, \]
where \(\alpha-\beta>1\), then, using a result of G. Szegő ((5), p. 264), we find that the Fourier–Jacobi series of this function also diverges at the point \(x=1\).

The author expresses sincere gratitude to P. K. Suetin, under whose supervision this work was carried out.

Ural State University
named after A. M. Gorky

Received
20 IX 1966

REFERENCES

¹ H. Rau, J. reine u. angew. Math., 161 (1929).
² G. Szegő, Schriften der Königsberger Gelehrten Gesellschaft, Naturwiss. Klasse, 10 (1933).
³ I. P. Natanson, Constructive Function Theory, Moscow–Leningrad, 1949.
⁴ E. V. Gobson, Theory of Spherical and Ellipsoidal Functions, IL, 1952.
⁵ G. Szegő, Orthogonal Polynomials, Moscow, 1962.