Generalized functions and Laguerre expansions

Abstract

In this paper we study and characterize expansions of distributions in the Zemanian spaces, ${\mathcal {H}}_{\mu }$ and its dual ${\mathcal {H}}_{\mu }^{\prime }$ ($\mu \ge -\frac{1}{2}$) with respect to Laguerre functions. We obtain as applications of this result, the kernel Theorem and a structure Theorem for ${\mathcal {H}}_{\mu }^{\prime }$. We also introduce a new algebra of generalized functions in the sense of J. F. Colombeau such that it satisfies interesting properties involving the Hankel transformation and Hankel convolution.

Appendix

1.1 Boundedness of Laguerre polynomials

Duran [7, Corollary 2.2] gives the following bound for generalized Laguerre polynomials. Let $\mu > -2$ and $n \in {\mathbf {N}}_0$. Then

$$\begin{aligned} \left| L_{n}^{\mu }(t)\right| \le 2 \frac{(n+[\mu ]+2)!}{n!\,\,([\mu ]+2)!}\,\,e^{\frac{t}{2}} \end{aligned}$$

(29)

where [x] is the integer part of x.

1.2 Boundedness of $c_{\mu ,k}$

Proposition 6

Let $c_{\mu ,k}=\Bigl (\frac{2\Gamma (k+1)}{\Gamma (k+\mu +1)}\Bigr )^{\frac{1}{2}}$. Then there exists a constant $C(\mu )$ depending only of $\mu$, such that for $\mu > -1$ and for all $k \in {\mathbf {N}}_0$,

$$\begin{aligned} c_{\mu , k} \le C(\mu )(k+\mu + 1)^{\frac{1}{2}}. \end{aligned}$$

(30)

Proof

We observe that for $\mu > 0$,

$$\begin{aligned} c_{\mu ,k}= & {} \left( \frac{\Gamma (k+1)\Gamma (\mu +1)}{\Gamma (k+\mu +1)}\right) ^{\frac{1}{2}} \,\left( \frac{2}{\Gamma (\mu +1)}\right) ^{\frac{1}{2}}\\= & {} (B(k+1,\mu +1))^{\frac{1}{2}}\left( \frac{2(k+\mu +1)}{\Gamma (\mu +1)}\right) ^{\frac{1}{2}} \\\le & {} \left( \frac{2}{\Gamma (\mu +1)}\int _0^1(1-t)^{\mu -1}t^kdt\right) ^{\frac{1}{2}}\left( k+\mu +1\right) ^{\frac{1}{2}} \end{aligned}$$

where $B{:}\,{\mathbf {R}}_+ \times {\mathbf {R}}_+ \rightarrow {\mathbf {R}}$ is the beta function defined by

$$\begin{aligned} B(x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\, dt. \end{aligned}$$

(31)

By other hand, if $-1<\mu <0$,

$$\begin{aligned} c_{\mu ,k}= & {} \left( \frac{\Gamma (k+1)\Gamma (\mu )}{\Gamma (k+\mu +1)}\right) ^{\frac{1}{2}} \,\left( \frac{2}{\Gamma (\mu )}\right) ^{\frac{1}{2}} \\= & {} \left( \frac{\Gamma (k+1)\Gamma (\mu +1)(k+\mu +1)}{\Gamma (k+\mu +2)}\right) ^{\frac{1}{2}} \,\left( \frac{2}{\Gamma (\mu +1)}\right) ^{\frac{1}{2}} \\= & {} \big (B(k+1,\mu +1)\big )^{\frac{1}{2}} \,\left( \frac{2(k+\mu +1)}{\Gamma (\mu +1)}\right) ^{\frac{1}{2}} \\\le & {} \big (\frac{2}{\Gamma (\mu + 1)} \int _0^1(1-t)^{\mu }dt \big )^{\frac{1}{2}} (k +\mu + 1)^{\frac{1}{2}} \end{aligned}$$

The case $\mu =0$ is obvious. Then for $\mu > -1$,

$$\begin{aligned} c_{\mu ,k}\le C(\mu )(k+\mu +1)^{\frac{1}{2}}. \end{aligned}$$

(32)

$\square$

1.3 ${\mathcal {H}}_{\mu } \subset L^2({\mathbf {R}}_+)$, for $\mu \ge -\frac{1}{2}$

Proposition 7

Let $\mu \ge -\frac{1}{2}$ and $m>2\mu +2$. Then for $\varphi \in {\mathcal {H}}_{\mu }$,

$$\begin{aligned} \Vert \varphi \Vert _{2}\le C_{1}\Vert \varphi \Vert _{0,0,\mu ,\infty }+C_{2}\Vert \varphi \Vert _{m,0,\mu ,\infty }, \end{aligned}$$

(33)

for certain constants $C_{1}$ and $C_{2}$.

Proof

We observe that

$$\begin{aligned} \Vert \varphi \Vert _{2}= & {} \Bigl \{\int _{0}^{\infty }|\varphi |x^{-\mu -\frac{1}{2}}|\varphi |x^{\mu +\frac{1}{2}}\,dx\Bigr \}^{\frac{1}{2}} \\\le & {} \Bigl \{\Vert \varphi \Vert _{0,0,\mu ,\infty }\Bigr \}^{\frac{1}{2}}\Bigl \{\int _{0}^{\infty }|\varphi |x^{\mu +\frac{1}{2}}\,dx\Bigr \}^{\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned} \int _{0}^{\infty }|\varphi |x^{\mu +\frac{1}{2}}\,dx= & {} \int _{0}^{1}|\varphi |x^{\mu +\frac{1}{2}}\,dx\\&+\,\int _{1}^{\infty }(1+x^{2})(1+x^{2})^{-1}x^{2\mu +1}x^{-\mu -\frac{1}{2}}|\varphi |\,dx \\\le & {} C \Vert \varphi \Vert _{0,0,\mu ,\infty }+C_{2}\Vert \varphi \Vert _{m,0,\mu ,\infty }. \end{aligned}$$

From elementary computation we conclude the Proposition $\square$

1.4 Equivalency between the families of seminorms $\{\Vert \cdot \Vert _{m,r,\mu ,2}{:}\,m, r \in {\mathbf {N}}_0 \}$ and $\{\Vert \cdot \Vert _{m,r,\mu ,\infty }{:}\,m, r \in {\mathbf {N}}_0\}$

We recall the seminorms introduced in (1),

$$\begin{aligned} \Vert \varphi \Vert _{m, r, \mu , \infty }:= \sup _{x \in {\mathbf {R}}_+}\left| (1+x^2)^m (x^{-1}D)^r \left[ x^{-\mu -\frac{1}{2}}\varphi (x)\right] \right| \end{aligned}$$

and the seminorms introduced in (20),

$$\begin{aligned} \parallel \varphi \parallel _{m,r,\mu ,2}=\parallel (1+x^{2})^{m}(x^{-1}D)^{r}\left[ x^{-\mu -\frac{1}{2}}\varphi (x)\right] \parallel _{2}. \end{aligned}$$

Proposition 8

$\{\Vert \cdot \Vert _{m,r,\mu ,2}{:}\,m, r \in {\mathbf {N}}_0 \}$ is a family of seminorms equivalent to $\{\Vert \cdot \Vert _{m,r,\mu ,\infty }{:}\,m, r \in {\mathbf {N}}_0\}$.

Proof

Let $\varphi \in {\mathcal {H}}_{\mu }$. We have that $(1+x^{2})^{m}(x^{-1}D)^{r}[x^{-\mu -\frac{1}{2}}\varphi ] \in L^{2}({\mathbf {R}}_+)$, for all $m, r \in {\mathbf {N}}_0$. Then,

$$\begin{aligned} \Vert \varphi \Vert _{m,r,\mu ,2}= & {} \Bigl \{\int _{0}^{\infty }\bigl |(1+x^{2})^{-1}(1+x^{2})(1+x^{2})^{m} \nonumber \\&\cdot \,(x^{-1}D)^{r}\bigl [x^{-\mu -\frac{1}{2}}\varphi \bigr ]\bigr |^{2}\,dx\Bigr \}^{\frac{1}{2}} \nonumber \\\le & {} \Bigl \{\int _{0}^{\infty }|1+x^{2}|^{-2}\,dx\Bigr \}^{\frac{1}{2}} \nonumber \\&\cdot \,\Bigl \{\sup _{x\in (0,\infty )}\bigl |(1+x^{2})^{m+1}(x^{-1}D)^{r}\bigl [x^{-\mu -\frac{1}{2}}\varphi \bigr ]\bigr |^{2}\Bigr \}^{\frac{1}{2}} \nonumber \\\le & {} C \Vert \varphi \Vert _{m+1,r,\mu ,\infty }. \end{aligned}$$

(34)

It follows easily that $\{\Vert \cdot \Vert _{m,r,\mu ,\infty }{:}\,m, r \in {\mathbf {N}}_0 \}$ is equivalent to $\{\Vert \cdot \Vert ^{\prime }_{m,r,\mu ,\infty }{:}\,m, r \in {\mathbf {N}}_0 \}$, where $\Vert \cdot \Vert ^{\prime }_{m,r,\mu ,\infty }$ are the seminorms of ${\mathcal {H}}_{\mu }$ given by

$$\begin{aligned} \Vert \varphi \Vert '_{m,r,\mu ,\infty }=\sup _{x\in {\mathbf {R}}_+}\left| x^{m}(x^{-1}D)^{r}\left[ x^{-\mu -\frac{1}{2}}\varphi (x)\right] \right| . \end{aligned}$$

(35)

In a similar way, $\{\Vert \cdot \Vert _{m,r,\mu ,2}{:}\,m, r \in {\mathbf {N}}_0 \}$ is equivalent to $\{\Vert \cdot \Vert ^{\prime }_{m,r,\mu ,2}{:}\,m, r \in {\mathbf {N}}_0 \}$, where $\Vert \cdot \Vert ^{\prime }_{m,r,\mu ,2}$ are the seminorms of ${\mathcal {H}}_{\mu }$ given by

$$\begin{aligned} \Vert \varphi \Vert '_{m,r,\mu ,2}=\left\| x^{m}(x^{-1}D)^{r}\left[ x^{-\mu -\frac{1}{2}}\varphi (x)\right] \right\| _{2}. \end{aligned}$$

(36)

We observe that if $f\in C^{1}({\mathbf {R}}_+)$ such that $f'$ is of rapid descent (i.e a function g is of rapid descent if g verifies that $\sup _{x\in (0,\infty )}\Vert (1+x^{2})^{m}g(x)\Vert _{\infty }<\infty$ for all integer non negative m), then

$$\begin{aligned} f(x)=-\int _{x}^{\infty }f'(t)\,dt, \quad x>0 \end{aligned}$$

(37)

and

$$\begin{aligned} \Vert f\Vert _{\infty }\le \Vert f'\Vert _{1}\le \Vert (1+x^{2})f'\Vert _{2}\Vert (1+x^{2})^{-1}\Vert _{2}. \end{aligned}$$

(38)

Let $\varphi \in {\mathcal {H}}_{\mu }$. We observe that $\gamma _{\mu , m, r}(x):= x^{m}(x^{-1}D)^{r}[x^{-\mu -\frac{1}{2}}\varphi (x)]$ verifies the above conditions and

$$\begin{aligned} \gamma _{\mu , m, r}^{\prime } = m\gamma _{\mu , m-1, r} +\gamma _{\mu , m+1, r+1}. \end{aligned}$$

In case that $m=0$, applying (38) we have that

$$\begin{aligned} \Vert \varphi \Vert '_{0,r,\mu ,\infty }= & {} \bigl \Vert (x^{-1}D)^{r}\bigl [x^{-\mu -\frac{1}{2}}\varphi (x)\bigr ]\bigr \Vert _{\infty } \nonumber \\\le & {} C\bigl \Vert (1+x^{2})\bigl \{ (x^{-1}D)^{r}\bigl [x^{-\mu -\frac{1}{2}}\varphi (x)\bigr ]\bigr \}^{\prime }\bigr \Vert _{2} \nonumber \\\le & {} C\bigl \{\Vert \varphi \Vert '_{1,r+1,\mu ,2}+\Vert \varphi \Vert '_{3,r+1,\mu ,2}\bigr \}. \end{aligned}$$

(39)

In case that $m \ge 1$,

$$\begin{aligned} \Vert \varphi \Vert '_{m,r,\mu ,\infty }= & {} \bigl \Vert x^{m}(x^{-1}D)^{r}\bigl [x^{-\mu -\frac{1}{2}}\varphi (x)\bigr ]\bigr \Vert _{\infty } \nonumber \\\le & {} C\bigl \Vert (1+x^{2})D\bigl \{ x^{m}(x^{-1}D)^{r}\bigl [x^{-\mu -\frac{1}{2}}\varphi (x)\bigr ]\bigr \}\bigr \Vert _{2} \nonumber \\\le & {} C\bigl \{m\Vert \varphi \Vert '_{m-1,r,\mu ,2}+\Vert \varphi \Vert '_{m+1,r+1,\mu ,2}+m\Vert \varphi \Vert _{m+1,r,\mu ,2} \nonumber \\&+\,\Vert \varphi \Vert '_{m+3,r+1,\mu ,2}\bigr \} \end{aligned}$$

(40)

Combining (34), (39) and (40) we completes the proof. $\square$

Remark 5

The above Proposition remains valid in ${\mathbb {R}}_{+}^{n}$ and the proof is similar to one dimensional case, noting that the equality (37) becomes

$$\begin{aligned} f(x)=(-1)^{n}\int _{x_{1}}^{\infty }\ldots \int _{x_{n}}^{\infty }D_{1}\ldots D_{n}f(t_{1},\ldots ,t_{n})\,dt_{1},\ldots ,dt_{n}, \end{aligned}$$

for $x=(x_{1},\ldots ,x_{n})\in {\mathbb {R}}_{+}^{n}$, valid to $f\in C^{n}({\mathbf {R}}_+)$ such that $D_{1}\ldots D_{n}f$ is of rapid descent. Moreover, for T given by (25) and taking into account the commutativity of $T_{i}$ we obtain that

$$\begin{aligned}&D_{i} T^{k}\Bigl \{x^{-\mu -\frac{1}{2}}\phi (x)\Bigr \}=D_{i}\Bigl \{T_{n}^{ k_{n}} \ldots T_{1}^{k_{1}}\Bigr \}\Bigl \{x^{-\mu -\frac{1}{2}}\phi (x)\Bigr \}\\&\quad = x_{i}\Bigl \{T_{n}^{ k_{n}} \ldots T_{i}^{k_{i}+1}\ldots T_{1}^{ k_{1}}\Bigr \}\Bigl \{x^{-\mu -\frac{1}{2}}\phi (x)\Bigr \}. \end{aligned}$$

So,

$$\begin{aligned} D_{1}\ldots D_{n} T^{k}\Bigl \{x^{-\mu -\frac{1}{2}}\phi (x)\Bigr \}=x_{1}\ldots x_{n} T^{k+1}\Bigl \{x^{-\mu -\frac{1}{2}}\phi (x)\Bigr \}, \end{aligned}$$

where $k+1=( k_{1}+1, \ldots , k_{n}+1)$.