Celebratio Mathematica

R. Askey and S. Wainger: “On the behavior of special classes of ultraspherical expansions, II,” J. Analyse Math. 15 (1965), pp. 193–220. article

R. Askey and S. Wainger: “Mean convergence of expansions in Laguerre and Hermite series,” Am. J. Math. 87 : 3 (July 1965), pp. 695–708. MR 182834 Zbl 0125.31301 article

R. Askey and S. Wainger: “On the behavior of special classes of ultraspherical expansions, I,” J. Analyse Math. 15 (1965), pp. 193–220. MR 193290 Zbl 0132.29403 article

R. Askey and S. Wainger: “Integrability theorems for Fourier series,” Duke Math. J. 33 : 1 (1966), pp. 223–228. MR 192260 Zbl 0136.36501 article

R. Askey and S. Wainger: “A transplantation theorem between ultraspherical series,” Ill. J. Math. 10 : 2 (1966), pp. 322–344. MR 211187 Zbl 0135.27603 article

R. Askey and S. Wainger: “A transplantation theorem for ultraspherical coefficients,” Pac. J. Math. 16 : 3 (1966), pp. 393–405. MR 217508 Zbl 0136.37201 article

Let \( f(\theta) \) be integrable on \( (0,\pi) \) and define \begin{align*} a_n &= \int_0^{\pi} f(\theta)\cos n\theta \,d\theta,\\ b_n &= n^{1/2}\int_0^{\pi} f(\theta)P_n(\cos \theta)(\sin \theta)^{1/2} \,d\theta \end{align*} where \( P_n(x) \) is the Legendre polynomial of degree \( n \). Then \begin{equation*}\tag{1} c \leq \sum_{n=0}^{\infty}|a_n|^p (n+1)^{\alpha}/\sum_{n=0}^{\infty}|b_n|^p(n+1)^{\alpha} \leq C \end{equation*} for \( 1 < p < \infty \), \( -1 < \alpha < p - 1 \), where \( C \) and \( c \) depend on \( p \) and \( \alpha \) but not on \( f \). From this we obtain a form of the Marcinkiewicz multiplier theorem for Legendre coefficients. Also an analogue of the Hardy–Littlewood theorem on Fourier coefficients of monotone coefficients is obtained. In fact, any norm theorem for Fourier functions can be transplanted by (1) to a corresponding theorem for Legendre coefficients.

Actually, the main theorem of this paper deals with ultraspherical coefficients and (1) is just a typical special case, which is stated as above for simplicity.

R. Askey and S. Wainger: “A dual convolution structure for Jacobi polynomials,” pp. 25–36 in Orthogonal expansions and their continuous analogues (Edwardsville, IL, 27–29 April 1967). Edited by D. T. Haimo. Southern Illinois University Press (Carbondale, IL), 1968. MR 232027 Zbl 0174.36305 incollection

Deborah Tepper Haimo	Related
Stephen Wainger	Related

R. Askey and S. Wainger: “A convolution structure for Jacobi series,” Am. J. Math. 91 : 2 (April 1969), pp. 463–485. MR 264132 Zbl 0186.12303 article

For ultraspherical polynomials and in particular for Legendre polynomials there is an addition formula which has as a consequence a type of convolution. This was introduced by Gelfand [1950] and by Bochner [1954] and has a number of very interesting consequences. It has been used to obtain the analogue of the Hardy–Littlewood theorem on fractional integration [Askey and Wainger 1965; Muckenhoupt and Stein 1965] and to obtain a simple proof of Kogbetliantz’s theorem on uniform \( (C,\alpha) \) summability from the end point result that Szegő has in his book [1959]. For Jacobi polynomials the addition theorem has not been found yet, and so the analogous convolution structure is not known. We find a form of this convolution which is sufficient to obtain the above results as well as other applications.

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Richard Allen Askey

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