The stability property of representations of the Fourier sums on a quadratic curve is measured through a condition number. We analyze the conditioning of the continuous Fourier sums and the discrete Fourier sums on the unit circle and the standard parabola. We give an upper bound for the conditioning in terms of norms of two families of orthogonal polynomials in one variable. We establish explicit bounds for the conditioning of some special Fourier sums. In particular, we show that the conditioning of nth partial Fourier sums on the circle and the standard parabola corresponding to special weight functions grows at most like O(n).

中文翻译：

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二次曲线上傅立叶和的调节

二次曲线上傅里叶和表示的稳定性是通过条件数来测量的。我们分析了单位圆和标准抛物线上连续傅里叶和和离散傅里叶和的条件。我们根据一个变量中两个正交多项式族的范数给出条件的上限。我们为一些特殊傅里叶和的条件建立了明确的界限。特别地，我们证明了圆上的第n个部分傅里叶和以及对应于特殊权函数的标准抛物线的条件最多增长为O ( n )。