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Search: MSC category 42C10 ( Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) )

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1. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$.

Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension
Categories:42B35, 42C10, 30D50, 28A80

2. CJM 2004 (vol 56 pp. 431)

Rosenblatt, Joseph; Taylor, Michael
Group Actions and Singular Martingales II, The Recognition Problem
We continue our investigation in [RST] of a martingale formed by picking a measurable set $A$ in a compact group $G$, taking random rotates of $A$, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing $A$ from a small amount of data from this martingale. This leads to problems in harmonic analysis on $G$, including an analysis of integrals of products of Gegenbauer polynomials.

Categories:43A77, 60B15, 60G42, 42C10

3. CJM 1998 (vol 50 pp. 1236)

Kalton, N. J.; Tzafriri, L.
The behaviour of Legendre and ultraspherical polynomials in $L_p$-spaces
We consider the analogue of the $\Lambda(p)-$problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the ``best possible'' result that if $2
Categories:42C10, 33C45, 46B07

4. CJM 1997 (vol 49 pp. 175)

Xu, Yuan
Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres
Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

Keywords:Orthogonal polynomials in several variables, sphere, h-harmonics
Categories:33C50, 33C45, 42C10

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