Volume 34, Issue 4
Harmonic Polynomials via Differentiation

Anal. Theory Appl., 34 (2018), pp. 336-347.

Published online: 2018-11

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• Abstract

It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}\Upsilon(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $\Upsilon$ is a harmonic homogeneous polynomial of degree $k,$ $\Upsilon\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributional derivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $\Upsilon$, but that the delta part is independent of $\Upsilon$, that is, it depends only on $p−\Upsilon$. We also show that the exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α\neq 2−n$.
Furthermore, we establish that harmonic polynomials appear as multiples of $r^{2−n−2k−2k'}$ when $p(∇)$ is applied to harmonic multipoles of the form $\Upsilon'(x)r^{2−n−2k'}$ for some $\Upsilon' \in \mathcal{H}_k$.

• Keywords

Harmonic functions, harmonic polynomials, distributions, multipoles.

46F10, 33C55

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@Article{ATA-34-336, author = {}, title = {Harmonic Polynomials via Differentiation}, journal = {Analysis in Theory and Applications}, year = {2018}, volume = {34}, number = {4}, pages = {336--347}, abstract = {

It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}\Upsilon(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $\Upsilon$ is a harmonic homogeneous polynomial of degree $k,$ $\Upsilon\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributional derivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $\Upsilon$, but that the delta part is independent of $\Upsilon$, that is, it depends only on $p−\Upsilon$. We also show that the exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α\neq 2−n$.
Furthermore, we establish that harmonic polynomials appear as multiples of $r^{2−n−2k−2k'}$ when $p(∇)$ is applied to harmonic multipoles of the form $\Upsilon'(x)r^{2−n−2k'}$ for some $\Upsilon' \in \mathcal{H}_k$.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2017-0062}, url = {http://global-sci.org/intro/article_detail/ata/12847.html} }
TY - JOUR T1 - Harmonic Polynomials via Differentiation JO - Analysis in Theory and Applications VL - 4 SP - 336 EP - 347 PY - 2018 DA - 2018/11 SN - 34 DO - http://doi.org/10.4208/ata.OA-2017-0062 UR - https://global-sci.org/intro/article_detail/ata/12847.html KW - Harmonic functions, harmonic polynomials, distributions, multipoles. AB -

It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}\Upsilon(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $\Upsilon$ is a harmonic homogeneous polynomial of degree $k,$ $\Upsilon\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributional derivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $\Upsilon$, but that the delta part is independent of $\Upsilon$, that is, it depends only on $p−\Upsilon$. We also show that the exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α\neq 2−n$.
Furthermore, we establish that harmonic polynomials appear as multiples of $r^{2−n−2k−2k'}$ when $p(∇)$ is applied to harmonic multipoles of the form $\Upsilon'(x)r^{2−n−2k'}$ for some $\Upsilon' \in \mathcal{H}_k$.

Ricardo Estrada. (1970). Harmonic Polynomials via Differentiation. Analysis in Theory and Applications. 34 (4). 336-347. doi:10.4208/ata.OA-2017-0062
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