## Tohoku Mathematical Journal

### On connections between Hankel, Laguerre and Jacobi transplantations

Krzysztof Stempak

#### Abstract

Proved are two results showing connections between the Hankel transplantation and a transplantation for a certain kind of Laguerre and Jacobi expansions. An asymptotic formula of Hilb's type for Laguerre and Jacobi polynomials is used. As an application of this link we obtain an extension of Guy's transplantation theorem for the Hankel transform to the case $\alpha,\gamma>-1$ also with more weights allowed. This is done by transferring a corresponding transplantation result for Jacobi expansions which was proved by Muckenhoupt. In the case when $\alpha,\gamma\ge-1/2$ the same is obtained by using Schindler's explicit kernel formula for the transplantation operator.

#### Article information

Source
Tohoku Math. J. (2), Volume 54, Number 4 (2002), 471-493.

Dates
First available in Project Euclid: 11 April 2005

https://projecteuclid.org/euclid.tmj/1113247646

Digital Object Identifier
doi:10.2748/tmj/1113247646

Mathematical Reviews number (MathSciNet)
MR1936265

Zentralblatt MATH identifier
1040.42022

#### Citation

Stempak, Krzysztof. On connections between Hankel, Laguerre and Jacobi transplantations. Tohoku Math. J. (2) 54 (2002), no. 4, 471--493. doi:10.2748/tmj/1113247646. https://projecteuclid.org/euclid.tmj/1113247646

#### References

• R. Askey, A transplantation theorem for Jacobi series, Illinois J. Math. 13 (1969), 583–590.
• D. L. Guy, Hankel multiplier transformations and weighted $p$-norms, Trans. Amer. Math. Soc. 95 (1960), 137–189.
• S. Igari, On the multipliers of Hankel transform, Tôhoku Math. J. 24 (1972), 201–206.
• Y. Kanjin, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc. 103 (1988), 1063–1069.
• Y. Kanjin, A transplantation theorem for Laguerre series, Tôhoku Math. J. 43 (1991), 537–555.
• N. N. Lebedev, Special functions and their applications, Dover Publications, New York, 1972.
• B. Muckenhoupt, Transplantation theorems and multiplier theorems for Jacobi series, Mem. Amer. Math. Soc. 64 (1986), no. 356.
• B. Muckenhoupt, Mean convergence of Hermite and Laguerre series. II, Trans. Amer. Math. Soc. 147 (1970), 433–460.
• S. Schindler, Explicit integral transform proofs of some transplantation theorems for the Hankel transform, SIAM J. Math. Anal. 4 (1973), 367–384.
• K. Stempak, On connections between Hankel, Laguerre and Heisenberg multipliers, J. London Math. Soc. 51 (1995), 286–298.
• K. Stempak, Transplanting maximal inequalities between Laguerre and Hankel multipliers, Monatsh. Math. 122 (1996), 187–197.
• K. Stempak and W. Trebels, On weighted transplantation and multipliers for Laguerre expansions, Math. Ann. 300 (1994), 203–219.
• G. Szegö, Orthogonal Polynomials, Colloquium Publications, American Mathematical Society, New York, 1966.
• S. Thangavelu, Transplantation, summability and multipliers for multiple Laguerre expansions, Tôhoku Math. J. 44 (1992), 279–298.