Trigonometric analogs of the Szegő equiconvergence theorem for Fourier–Jacobi series

V. M. Badkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 68-79

Résumé

Let $\{\Phi^{\alpha,\beta}_k(\tau)\}_{k=0}^\infty$ be an orthonormal system of trigonometric Jacobi polynomials obtained by orthogonalizing the sequence $1,\sin\tau,\cos\tau,\sin2\tau,\cos2\tau,\dots$ by Schmidt method on $[0,2\pi]$ with a weight $\varphi^{\alpha,\beta}(\tau):=(1-\cos\tau)^{\alpha+1/2}(1+\cos\tau)^{\beta+1/2}$; $s_n^{\alpha,\beta}(F;\theta):=\sum_{k=0}^nc_k(\varphi^{\alpha,\beta};F)\Phi^{\alpha,\beta}_k(\theta)$ is $n$-th Fourier sum of function $F$ in system $\Phi^{\alpha,\beta}_k(\tau)\}_{k=0}^\infty$; $s_n(F;\theta)=s_{2n}^{-1/2,-1/2}(F;\theta)$ is usual Fourier sum. It is proved that if $\alpha,\beta>-1$, $A:=\min\{\alpha+1/2,\alpha/2+1/4\}$, $B:=\min\{\beta+1/2,\beta/2+1/4\}$, $\varepsilon\in(0,\pi/2)$, $F$ is measurable, $F(\tau)(1-\cos\tau)^A(1+\cos\tau)^B\in L^1$ and $\varepsilon\in(0,\pi/2)$ $F\varphi^{\alpha,\beta}\in L^1$ and the sum $s_{2n}^{\alpha,\beta}(F;\theta)$ equiconverges with each of sequences $s_n(F\sqrt{\varphi^{\alpha,\beta}};\theta)/\sqrt{\varphi^{\alpha,\beta}(\theta)}$ and $s_n(F\varphi^{\alpha,\beta};\theta)/\varphi^{\alpha,\beta}(\theta)$ uniformly on intervals $[-\pi+\varepsilon,-\varepsilon]$ and $[\varepsilon,\pi-\varepsilon]$. For even function $F$ similar results were obtained by G. Szegő and Ye. A. Pleshchyova.

Détail

Keywords: trigonometric Jacobi polynomials, Fourier sums
Mots-clés : equiconvergens.

Bibliographie

[1] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962, 500 pp.

[2] Pleschëva E. A., “Teoremy ravnoskhodimosti dlya ryadov Fure–Yakobi”, Sovremennye metody kraevykh zadach, Materialy Voronezh. vesen. mat. shk. “Pontryaginskie chteniya – XV”, Izd-vo VGU, Voronezh, 2004, 169

[3] Jackson D., “Orthogonal trigonometric sums”, Ann. Math. (2), 34 (1933), 799–814 | DOI | MR | Zbl

[4] Szegö G., “On bi-orthogonal systems of trigonometric polynomials”, Magy. tud. akad. Mat. kut. intéz. közl., 8:3 (1964), 255–273 | MR | Zbl

[5] Badkov V. M., “Priblizhenie funktsii v ravnomernoi metrike summami Fure po ortogonalnym polinomam”, Tr. MIAN, 145, 1980, 20–62 | MR | Zbl

[6] Badkov V. M., “Ravnoskhodimost s obychnym ryadom Fure summiruemoi funktsii ee ryada Fure po trigonometricheskim ortogonalnym polinomam”, Priblizhenie funktsii. Teoreticheskie i prikladnye aspekty, Cb. st., MIET, M., 2003, 69–75

[7] Badkov V. M., “Priblizhenie funktsii chastnymi summami ryada Fure po obobschennym mnogochlenam Yakobi”, Mat. zametki, 3:6 (1968), 671–682 | MR | Zbl

[8] Badkov V. M., “Ravnoskhodimost ryadov Fure po ortogonalnym mnogochlenam”, Mat. zametki, 5:3 (1969), 285–295 | MR | Zbl

[9] Badkov V. M., “Equiconvergence of Fourier sums in orthogonal polynomials”, Proc. Steklov Inst. Math., 10, 2004, S101–S127 | MR

[10] Badkov V. M., Vvedenie v edinuyu teoriyu algebraicheskikh i trigonometricheskikh ortogonalnykh polinomov, Izd-vo Ural. un-ta, Ekaterinburg, 2006, 132 pp. | MR

[11] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsiya Lezhandra, Nauka, M., 1965, 296 pp.

[12] Badkov V. M., “Otsenki funktsii Lebega i ostatka ryada Fure–Yakobi”, Sib. mat. zhurn., 9:6 (1968), 1263–1283 | MR | Zbl

[13] Bari N. K., Trigonometricheskie ryady, GIFML, M., 1961, 936 pp. | MR