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Hypercomplex systems originating from orthogonal polynomials

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Literature cited

  1. Yu. M. Berezanskii and S. G. Krein, “Hypercomplex systems with continual bases,” Usp. Mat. Nauk,12, No. 1, 147–152 (1957).

    Google Scholar 

  2. Yu. M. Berezanskii and A. A. Kaluzhnyi, “Hypercomplex systems with a locally compact basis” [in Russian], Preprint, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, 82.40, Kiev (1982) (English translation in Selecta Mathematica Sovietica,4, No. 2, 151–200 (1985)).

  3. Yu. M. Berezanskii and A. A. Kaluzhnyi, “A nuclear function space based on a hypercomplex system,” Ukr. Mat. Zh.,35, No. 1, 9–17 (1983).

    Google Scholar 

  4. Yu. M. Berezanskii, “Hypercomplex systems with discrete basis,” Dokl. Akad. Nauk SSSR,81, No. 3, 329–332 (1951).

    Google Scholar 

  5. Yu. M. Berezanskii, “Some normed rings originating from orthogonal polynomials,” Ukr. Mat. Zh.,3, No. 4, 412–432 (1951).

    Google Scholar 

  6. Yu. M. Berezanskii, “Almost-periodic generalized functions and sequences connected with differential and difference equations,” Mat. Sb.,32, No. 1, 157–194 (1953).

    Google Scholar 

  7. I. I. Hirschman, Jr., “Harmonic analysis and ultraspherical polynomials,” Symp. on Harmonic Anal. and Related Integral transforms, Cornell Univ. (1956).

  8. G. Gasper, “Linearization of the product of Jacobi polynomials. II,” Can. J. Math.,22, No. 3, 532–543 (1970).

    Google Scholar 

  9. R. Askey and G. Gasper, “Linearization of the product of Jacobi polynomials. III,” ibid.,23, No. 2, 332–338 (1971).

    Google Scholar 

  10. S. Igari and Y. Uno, “Banach algebra related to the Jacobi polynomials,” Tohoku Math. J., No. 4, 668–673 (1969).

    Google Scholar 

  11. G. Gasper, “Banach algebras for Jacobi series and positivity of a kernel,” Ann. Math.,95, No. 2, 261–280 (1972).

    Google Scholar 

  12. T. P. Laine, “The product formula and convolution structure for the generalized Chebyshev polynomials,” SIAM J. Math. Anal.,11, No. 1, 133–146 (1980).

    Google Scholar 

  13. A. Schwartz, “Generalized convolutions and positive definite functions associated with general orthonormal series,” Pac. J. Math.,55, No. 2, 565–582 (1974).

    Google Scholar 

  14. Yu. M. Berezanskii, “Hypercomplex systems with compact and discrete basis,” Thesis, Author's Abstract of Candidate's Dissertation, Physicomathematical Sciences, Kiev (1950).

    Google Scholar 

  15. R. Lasser, “Orthogonal polynomials and hypergroups,” Rend. Math., Ser. 7,3, 185–209 (1983).

    Google Scholar 

  16. R. Lasser, “Bochner theorems for hypergroups and their applications to orthogonal polynomial expansions,” J. Approx. Theory,37, No. 4, 311–325 (1983).

    Google Scholar 

  17. R. Lasser, “Fourier-Stiltjes transforms on hypergroups,” Analysis,2, No. 14, 281–303 (1982).

    Google Scholar 

  18. R. Lasser, “On the problem of modified moments,” Proc. Am. Math. Soc.,90, No. 3, 360–362 (1984).

    Google Scholar 

  19. R. I. Jewett, “Spaces with an abstract convolution of measures,” Adv. Math.,18, No. 1, 1–101 (1975).

    Google Scholar 

  20. K. A. Ross, “Hypergroups and centers of measure algebras,” Instit. Nazionale Alta Mat.; Symposia Math.,22, 189–203 (1977).

    Google Scholar 

  21. C. F. Dunkl and D. E. Ramires, “Kravtchouk polynomials and the symmetrization of hypergroups,” SIAM J. Math. Anal.,5, 351–366 (1974).

    Google Scholar 

  22. L. I. Vainerman, “The duality principle for finite hypercomplex systems,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 12–21 (1985).

    Google Scholar 

  23. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York (1978).

    Google Scholar 

  24. Yu. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc. (1968).

  25. R. Lasser, “Linearization of the product of associated Legendre polynomials,” SIAM J. Math. Anal.,14, No. 2, 403–408 (1983).

    Google Scholar 

  26. D. M. Bressond, “Linearization and related formulas for q-ultraspherical polynomials,” ibid.,12, No. 2, 161–168 (1981).

    Google Scholar 

  27. P. Cartier, “Harmonic analysis on trees,” Proc. Symp. Pure Math.,26, Providence, R. I., AMS (1974), pp. 419–424.

    Google Scholar 

  28. J. P. Arnaud, “Fonctions sphériques et fonctions définies positives sur l'arbre homogène,” G. R. Acad. Sci. A,290, No. 2, 99–101 (1980).

    Google Scholar 

  29. G. Szegö, Orthogonal Polynomials, AMS Coll. Publ. No. 23 (1959).

  30. S. Wolfenstetter, “Jacobi-Polynome und Bessel-Funktionen unter dem Gesichtpunkt der harmonischen Analyse,” Diss. Dokt. Naturwiss., Fak. Math. Inf. Techn. Univ., München (1984).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev

    Yu. M. Berezanskii & A. A. Kalyuzhnyi

  2. Kiev Technical Institute, USSR

    Yu. M. Berezanskii & A. A. Kalyuzhnyi

Authors
  1. Yu. M. Berezanskii
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  2. A. A. Kalyuzhnyi
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 38, No. 3, pp. 275–284, May–June, 1986.

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Berezanskii, Y.M., Kalyuzhnyi, A.A. Hypercomplex systems originating from orthogonal polynomials. Ukr Math J 38, 237–245 (1986). https://doi.org/10.1007/BF01056816

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  • DOI: https://doi.org/10.1007/BF01056816

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