On polynomials orthogonal with respect to certain Sobolev inner products. (English) Zbl 0734.42016

Orthogonal polynomials \(p^{\lambda}_ n(x)\) for the Sobolev inner product \[ <f,g>=\int f(x)g(x)d\phi (x)+\lambda \int f'(x)g'(x)d\psi (x) \] are investigated. An interesting notion of coherent pairs of Borel measures \(\phi\) and \(\psi\) is introduced. If \(p_ n(x)\) are the orthogonal polynomials with respect to the measure \(\phi\), then \((\phi,\psi)\) is a coherent pair when there exist non-zero constants \(C_ k\) such that \[ \int p'_ n(x)p'_ m(x)d\psi (x)=\frac{d_{\min (m,n)}}{C_ mC_ n}. \] It is shown that for a coherent pair (\(\phi\),\(\psi\)) the Sobolev orthogonal polynomials \(p^{\lambda}_ n(x)\) can be expanded in terms of the orthogonal polynomials \(p_ n(x)\) in such a way that the expansion coefficients (except the last) are independent of n and themselves orthogonal polynomials in the variable \(\lambda\). Several examples are worked out and an application involving the evaluation of Sobolev-Fourier coefficients is given.


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] Althammer, P., Eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen und deren Anwendung auf die beste Approximation, J. Reine Angew. Math., 211, 192-204 (1962) · Zbl 0108.27204
[2] Askey, R.; Ismail, M., Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc., 319 (1984) · Zbl 0548.33001
[3] Bavinck, H.; Meijer, H. G., Orthogonal polynomials with respect to a symmetric inner product involving derivatives, Appl. Anal., 33, 103-117 (1989) · Zbl 0648.33007
[4] Brenner, J., Über eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen in einer und zwei Variablen, (Ph.D. thesis (1969), Techn. Hochschule: Techn. Hochschule Stuttgart) · Zbl 0234.33016
[5] Brenner, J., Über eine Erweiterung des Orthogonalitätsbegriffes bei Polynomen, (Constructive Theory of Functions (1972), Akadémiai Kiadó: Akadémiai Kiadó Budapest) · Zbl 0234.33016
[6] Canuto, C.; Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38, 67-88 (1982) · Zbl 0567.41008
[7] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon & Breach: Gordon & Breach New York · Zbl 0389.33008
[8] Cohen, E. A., Zero distribution and behavior of orthogonal polynomials in the Sobolev space \(W^{1,2}\)[−1, 1], SIAM J. Math. Anal., 6, 105-116 (1975) · Zbl 0272.42013
[9] Gautschi, W., Minimal solutions of three-term recurrence relations and orthogonal polynomials, Math. Comp., 36, 547-554 (1981) · Zbl 0466.33008
[10] Gröbner, W., Orthogonale Polynomsysteme, die gleichzeitig mit \(f(x)\) auch deren Ableitung \(f\)′\((x)\) approximieren, (Collatz, L.; Meinardus, G.; Unger, H., Funktionalanalysis, Approximationstheorie, Numerische Mathematik (1965), Birkhäuser-Verlag: Birkhäuser-Verlag Basel) · Zbl 0188.14001
[11] Hahn, W., Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Z., 39, 634-638 (1935)
[12] Hahn, W., Über Orthogonalpolynome mit besonderen Eigenschaften, (Butzer, P. L.; Fehér, F., E. B. Christoffel (1981), Birkhäuser-Verlag: Birkhäuser-Verlag Basel) · Zbl 0473.33008
[13] Hahn, W., Zur Theorie der Orthogonalpolynome, Forschungszentrum Graz, Bericht Nr. 190 (1983) · Zbl 0505.33009
[14] Iserles, A.; Koch, P. E.; Nørsett, S. P.; Sanz-Serna, J. M., Orthogonality and approximation in a Sobolev space, (Mason, J. C.; Cox, M. G., Algorithms for Approximation (1990), Chapman & Hall: Chapman & Hall London) · Zbl 0749.41030
[15] Kumar, R., A class of quadrature formulas, Math. Comp., 28, 769-778 (1974) · Zbl 0291.65005
[16] Kumar, R., Certain Gaussian quadratures, J. IMA, 14, 175-182 (1974) · Zbl 0287.65015
[17] Koekoek, R., Generalizations of Laguerre Polynomials, Univ. of Delft Tech. Rep. 89-51 (1989)
[18] Lewis, D. C., Polynomial least square approximations, Amer. J. Math., 69, 273-278 (1947) · Zbl 0033.35603
[19] Littlejohn, L. L.; Everitt, W. N.; Williams, S. C., Orthogonal polynomials in weighted Sobolev spaces, (Vinuesa, J., Orthogonal Polynomials and Their Applications (1989), Dekker: Dekker New York) · Zbl 0676.33005
[20] Marcellan, F.; Ronveaux, A., On a Class of Polynomials Orthogonal with Respect to a Discrete Sobolev Inner Product, Univ. of Namur Tech. Report (1990) · Zbl 0732.42016
[21] Maroni, P., Sur la suite des polynômes orthogonaux associée à la forme \(u = δc + λ(x\) − \(c)^{−1}L\), (Cachafeiro, A.; Godoy, E., Actas V Simposium sobre Polinomios Ortogonales y Aplication Vigo (1988)) · Zbl 0767.42007
[22] Price, T. E., Orthogonal polynomials for nonclassical weight functions, SIAM J. Numer. Anal., 16, 999-1006 (1979) · Zbl 0426.65008
[23] Rainville, E. D., Special Functions (1960), Macmillan: Macmillan New York · Zbl 0050.07401
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