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Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier expansion of $M$?

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. The case $n=3$ is already interesting to me. In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?

Clarification: Note that $M$ is a function of $n$ variables so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$. These functions form an orthonormal basis with respect to the Gaussian measure on $\mathbb R^n$.

Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier expansion of $M$?

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. The case $n=3$ is already interesting to me. In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?

Clarification: Note that $M$ is a function of $n$ variables so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$. These functions form an orthonormal basis with respect to the Gaussian measure on $\mathbb R^n$.

Small Addition

A similar question can be asked for $MAX (x_1,x_2,\dots,x_n)$. Here $n$ can be even and even for $n=2$ is of interest. For the case $n=3$ it could be interesting to compare between the Hermite-Fourier expansion of $M(x_1,x_2,x_3)$ and of $MAX (x_1,x_2,x_3)$.

Consider an odd number of real numbers $x_1,x_2,\dots x_n$. Let $M$ be their median. What can be said about the Hermite–Fourier expansion of $M=M(x_1,x_2,\dots,x_n)$?

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. The case $n=3$ is already interesting to me. In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?

Clarification: Note that $M$ is a function of $n$ variables so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$. These functions form an orthonormal basis with respect to the Gaussiamn measure on $\mathbb R^n$.

Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier expansion of $M$?

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. The case $n=3$ is already interesting to me. In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?

Clarification: Note that $M$ is a function of $n$ variables so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$. These functions form an orthonormal basis with respect to the Gaussian measure on $\mathbb R^n$.

Consider an odd number of random i.i.d. Gaussian real numbers $x_1,x_2,\dots x_n$. Let $M$ be their median. What can be said about the Hermite–Fourier expansion of $M=M(x_1,x_2,\dots,x_n)$?

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. The case $n=3$ is already interesting to me. In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?

Clarification: Note that $M$ is a function of $n$ variables so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$.

Consider an odd number of real numbers $x_1,x_2,\dots x_n$. Let $M$ be their median. What can be said about the Hermite–Fourier expansion of $M=M(x_1,x_2,\dots,x_n)$?

Of course, an actual formula will be of interest but I will be happy to see also some numerical data. The case $n=3$ is already interesting to me. In particular, what is the contribution (sum of squares) of the coefficients for degree-$k$ terms?

Clarification: Note that $M$ is a function of $n$ variables so we need to find the inner product of $M$ with $H_{k_1}(x_1)H_{k_2}(x_2)\dotsm H_{k_n}(x_n)$. These functions form an orthonormal basis with respect to the Gaussiamn measure on $\mathbb R^n$.