# Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation
Matrices

Arieh<EMAIL_ADDRESS>Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3
0WA, United Kingdom. Marcus Webb222Corresponding author,
<EMAIL_ADDRESS>Department of Mathematics, University of
Manchester, Alan Turing Building, Manchester M13 9PL, United Kingdom.

###### Abstract

We introduce and develop a theory of orthogonality with respect to Sobolev
inner products on the real line for sequences of functions with a tridiagonal,
skew-Hermitian differentiation matrix. While a theory of such
$\mathrm{L}_{2}$-orthogonal systems is well established, Sobolev orthogonality
requires new concepts and their analysis. We characterise such systems
completely as appropriately weighed Fourier transforms of orthogonal
polynomials and present a number of illustrative examples, inclusive of a
Sobolev-orthogonal system whose leading $N$ coefficients can be computed in
${\cal O}\\!\left(N\log N\right)$ operations.

Keywords Orthogonal systems, Sobolev orthogonality, spectral methods,
Malmquist–Takenaka functions
Mathematics Subject Classification 42C05, 42C10, 42C30, 65M12, 65M70

##  1 Introduction

###  1.1 Orthonormal systems on the real line

The theory of $\mathrm{L}_{2}$-orthonormal systems on the real line with a
tridiagonal differentiation matrix has been developed in (?, ?, ?, ?). In its
simplest (real) version, let $w\geq 0$ be an absolutely continuous, nonzero
weight function, symmetric with respect to the origin, and
$\\{p_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$ the underlying system of orthonormal
polynomials, which must satisfy

$b_{n}p_{n+1}(\xi)=\xi p_{n}(\xi)-b_{n-1}p_{n-1}(\xi),\qquad n\in\mbox{\Bbb
Z}_{+}.$

for some real numbers $\\{b_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$. Setting

$\varphi_{n}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}p_{n}(\xi)\sqrt{w(\xi)}\mathrm{e}^{{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
x\in\mbox{\Bbb R},\quad n\in\mbox{\Bbb Z}_{+},$ (1.1)

we obtain by Parseval’s theorem an orthonormal system of functions in
$\mathrm{L}_{2}(\mbox{\Bbb R})$. Moreover, under the mild assumption that
polynomials are dense in $\mathrm{L}_{2}(\mbox{\Bbb R};w)$, this system is
dense in $\mathrm{L}_{2}(\mbox{\Bbb R})$ if the support of $w$ is all of ℝ,
otherwise its closure is the Paley–Wiener space
$\mathcal{PW}_{\mathrm{supp}\,w}(\mbox{\Bbb R})$ of all
$\mathrm{L}_{2}(\mbox{\Bbb R})$ functions whose Fourier transform is supported
on $\mathrm{supp}\,w$. Moreover, $\Phi=\\{\varphi_{n}\\}_{n\in\mbox{\sBbb
Z}_{+}}$ obeys

$\varphi_{n}^{\prime}(x)=-b_{n-1}\varphi_{n-1}(x)+b_{n}\varphi_{n+1}(x),\qquad
n\in\mbox{\Bbb Z}_{+}.$ (1.2)

In vector form, (1.2) is
$\mbox{\boldmath$\varphi$\unboldmath}^{\prime}=\mathcal{D}\mbox{\boldmath$\varphi$\unboldmath}$,
where $\mathcal{D}$ is the _differentiation matrix_ of the system, which in
this case is tridiagonal and skew-symmetric. Skew symmetry and the tridiagonal
form provide important advantages on the design of spectral methods with the
basis $\Phi$ (?).

In this paper we generalise the theory to the case of Sobolev-orthogonal
systems, where the Sobolev inner product is of the form

$\langle\varphi,\psi\rangle_{v}=\sum_{\ell=0}^{\infty}v_{\ell}\int_{-\infty}^{\infty}\varphi^{(\ell)}(x)\overline{\psi^{(\ell)}(x)}\,\,\mathrm{d}x,$

defined by the non-zero, non-negative sequence
$\\{v_{\ell}\\}_{\ell\in\mbox{\sBbb Z}_{+}}\subset[0,\infty)$. The
$\mathrm{H}^{s}(\mbox{\Bbb R})$ norm, where $s\in\mbox{\Bbb Z}_{+}$
corresponds to $v_{\ell}=1$ for $\ell=0,1\ldots,s$ and $v_{\ell}=0$ otherwise.

Besides the resulting theory being of interest in its own right, we can
motivate our exploration in the context of spectral methods for PDEs using the
example of the Ornstein–Uhlenbeck process,

$\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial
x^{2}}-a\frac{\partial}{\partial x}\left(xu\right),\qquad x\in\mbox{\Bbb
R},\quad t\geq 0,$ (1.3)

with coefficient of friction described by the positive constant $a$ (?, ?).
Solutions to this PDE satisfy

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{-\infty}^{\infty}u_{x}^{2}(x)+u^{2}(x)\,\,\mathrm{d}x=-\int_{-\infty}^{\infty}2u_{xx}^{2}(x)+\left(2+3a\right)u_{x}^{2}(x)+au^{2}(x)\,\,\mathrm{d}x,$
(1.4)

which shows that the solution decays monotonically to zero in the
$\mathrm{H}^{1}(\mbox{\Bbb R})$ norm. In fact, we can drop some terms and show
that $\frac{\,\mathrm{d}}{\,\mathrm{d}t}\langle
u,u\rangle_{H^{1}}\leq-a\langle u,u\rangle_{H^{1}}$, and hence the norm
decreases at least exponentially with rate dependent on $a$.

Now, consider semi-discretising equation (1.3) in space by spectral method
$u(x,t)\approx u_{N}(x,t):=\sum_{n=0}^{N}a_{n}(t)\varphi_{n}(x)$, where
$\Phi=\\{\varphi_{n}\\}_{n\in\mbox{\sBbb
Z}_{+}}\subset\mathrm{L}_{2}(\mbox{\Bbb R})$ are orthonormal with respect to
the $\mathrm{H}^{1}(\mbox{\Bbb R})$ inner product. If a Galerkin scheme is
used with respect to the $\mathrm{H}^{1}(\mbox{\Bbb R})$ inner product (i.e.
the residual of the PDE at each time $t$ is orthogonal to
$\mathrm{span}\\{\varphi_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$), then the
inequality (1.4) is also satisfied by $u_{N}$ (cf. (?, Ch. 8)). It therefore
follows that any A-stable discretisation in time will be stable.

The plan of this paper is as follows. In Section 2, basing ourselves upon our
earlier theory on $\mathrm{L}_{2}$ inner products, we present a complete
framework for the construction of Sobolev-orthogonal systems on the real line
with a tridiagonal differentiation matrix. This leads to two alternatives
towards the construction of $\mathrm{H}^{s}(\mbox{\Bbb R})$-orthogonal
systems, which are debated in Section 3: the first is the arguably more
obvious approach, yet it leads to formulæ which typically are impossible to
express explicitly, while the second, less natural, results in a more
constructive approach. Section 4 is concerned with systems based upon the
familiar Hermite weight and Section 5 with bilateral (i.e., symmetrised with
respect to the origin) Laguerre weights. In Section 6 we discuss Bessel-like
orthogonal systems originating in various ultraspherical weights: in that case
the closure of the orthogonal system is not $\mathrm{H}^{s}(\mbox{\Bbb R})$
but a relevant Paley–Wiener space. Section 7 generalises the discourse to non-
symmetric measures. In that instance our orthogonal systems are complex-valued
but the approach confers some important advantages. In particular, it allows
us to generalise the Malmquist–Takenaka system to Sobolev setting while
retaining the most welcome feature of this system, namely that the
coefficients can be computed rapidly with Fast Fourier Transform. Finally, in
Section 8 we present brief conclusions.

###  1.2 Sobolev norms beyond this paper

As an aside, our original interest in orthonormal systems (1.1) has been
motivated in (?) by the numerical solution of the linear Schrödinger equation
in the semi-classical regime,

${\mathrm{i}}\varepsilon\frac{\partial u}{\partial
t}=-\varepsilon^{2}\frac{\partial^{2}u}{\partial x^{2}}+V(x)u,\qquad
x\in\mbox{\Bbb R},\quad t\geq 0,$

given with an initial condition at $t=0,x\in\mbox{\Bbb R}$. Here
$0<\varepsilon\ll 1$, while the interaction potential $V$ is real. The
solution of this equation conserves the standard $\mathrm{L}_{2}$ norm (which
motivates the use of $\mathrm{L}_{2}$-orthogonal systems), but it also has
another important invariant: its Hamiltonian,

$H(u)=\int_{-\infty}^{\infty}[\varepsilon|u_{x}(x)|^{2}+\varepsilon^{-1}V(x)|u(x)|^{2}]\,\mathrm{d}x,$

is conserved. This might be viewed as a conservation of a non-standard Sobolev
norm (if $V$ is positive). While the design of Hamiltonian methods for the
Schrödinger equation is still an open problem, it motivates the work reported
in this paper.

We mention in passing another example in which nonstandard Sobolev norms are
non-increasing, the diffusion equation,

$\frac{\partial u}{\partial t}=\frac{\partial}{\partial
x}\left[a(x)\frac{\partial u}{\partial x}\right],\qquad x\in\mbox{\Bbb
R},\quad t\geq 0,$

where $a(x)>a_{\mathrm{min}}>0$ for all $x\in\mbox{\Bbb R}$, given with an
initial condition for $t=0,x\in\mbox{\Bbb R}$. It is readily shown that norm
induced by the following nonstandard Sobolev inner product is non-increasing
as a function of time,

$\langle
u,u\rangle_{a}:=\int_{-\infty}^{\infty}[a(x)u_{x}^{2}(x)+u^{2}(x)]\,\mathrm{d}x.$

We do not pursue these general Sobolev inner products in this paper, but
anticipate reporting on such results in the future.

###  1.3 Related work

Sobolev orthogonality: Polynomials orthogonal with respect to Sobolev measures
have been considered for a long while but the subject received considerable
impetus with the introduction of coherent pairs in (?) and has been surveyed
in (?, ?). Natural questions, given the constructs (1.1) and (1.2) are,
firstly, how to generate Sobolev-orthogonal systems on the real line and,
secondly, is a Fourier integral of an orthogonal polynomial system scaled by
any reasonable function orthogonal with respect to some inner product, whether
in a classical or Sobolev sense, in line with the $\mathrm{L}_{2}$ theory as
briefly reviewed in Section 2. These related questions are the focus of this
paper. Intriguingly, as things stand, the theory in this paper is heavily
based on the theory of classical orthogonal polynomials (as distinct from
Sobolev-orthogonal polynomials).

Fourier–Bessel functions (?, ?): Given a Borel measure $\,\mathrm{d}\mu$ and
the underlying orthonormal system $\\{p_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$, we
define

$\varphi_{n}(x)=\int_{-\infty}^{\infty}p_{n}(\xi)\mathrm{e}^{-{\mathrm{i}}x\xi}\,\mathrm{d}\mu(\xi)$
(1.5)

as the $n$th Fourier–Bessel function: the name is motivated by the Legendre
measure $\,\mathrm{d}\mu(x)=\chi_{(-1,1)}(x)\,\mathrm{d}x$, whereby
$\varphi_{n}(x)=\sqrt{2\pi/x}\mathrm{J}_{n+\frac{1}{2}}(x)$. Note the
similarity between (1.1) and (1.5) (disregarding the normalising factor and
the sign in the exponential, neither of which is of much importance), namely
that both are Fourier transforms of $p_{n}$ with added scaling function:
$\sqrt{w}$ in the first instance, $w$ in the second.

Further variation on this theme is the identity

$\int_{-1}^{1}\mathrm{T}_{n}(\xi)\mathrm{e}^{{\mathrm{i}}x\xi}\frac{\,\mathrm{d}\xi}{\sqrt{1-\xi^{2}}}=\pi{\mathrm{i}}^{n}\mathrm{J}_{n}(x),\qquad
n\in\mbox{\Bbb Z}_{+},$ (1.6)

where $\mathrm{T}_{n}$ is the $n$th Chebyshev polynomial (?). In that case the
weight function has disappeared altogether in the Fourier integral. Note that,
unlike (1.1), Fourier–Bessel functions need not be orthogonal although,
interestingly enough, disregarding signs and normalising constants, the two
formulæ concide (and orthogonality is recovered) for the Legendre measure.

##  2 Characterisation of Sobolev-orthogonal systems

Let us first state the desiderata. We are interested in functions
$\Phi=\\{\varphi_{n}\\}_{n\in\mbox{\sBbb
Z}_{+}}\subset\mathrm{L}_{2}(\mbox{\Bbb R})$ such that _both_ of the following
properties hold.

  1. (A)

There exists sequences $\\{b_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}\subset\mbox{\Bbb
C}\setminus\\{0\\}$ and $\\{c_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}\subset\mbox{\Bbb
R}$ such that

$\varphi_{n}^{\prime}(x)=-\overline{b_{n-1}}\varphi_{n-1}(x)+\mathrm{i}c_{n}\varphi_{n}(x)+b_{n}\varphi_{n+1}(x)$
(2.1)

for $n=0,1,\ldots$ (with $b_{-1}=0$ by convention);

  2. (B)

$\Phi$ is an orthonormal sequence with respect to the Sobolev inner product

$\langle\varphi,\psi\rangle_{v}=\sum_{\ell=0}^{\infty}v_{\ell}\int_{-\infty}^{\infty}\varphi^{(\ell)}(x)\overline{\psi^{(\ell)}(x)}\,\mathrm{d}x,$
(2.2)

defined by the non-zero, non-negative sequence
$\\{v_{\ell}\\}_{\ell\in\mbox{\sBbb Z}_{+}}\subset[0,\infty)$ such that
$\sum_{\ell=0}^{\infty}v_{\ell}>0$.

######  Theorem 1 ((?, ?))

A sequence $\Phi=\\{\varphi_{n}\\}_{n\in\mbox{\sBbb
Z}_{+}}\subset\mathrm{L}_{2}(\mbox{\Bbb R})$ satisfies criterion $(A)$ if and
only if

$\varphi_{n}(x)=\frac{\mathrm{e}^{{\mathrm{i}}\theta_{n}}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{{\mathrm{i}}x\xi}p_{n}(\xi)g(\xi)\,\mathrm{d}\xi,$
(2.3)

where

  * •

$P=\\{p_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$ is an orthonormal polynomial system
on the real line with respect to a probability measure on the real line with
all moments finite and with infinitely many points of increase;

  * •

$\Theta=\\{\theta_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}\subset[0,2\pi)$;

  * •

$g\in\mathrm{L}_{2}(\mbox{\Bbb R})$ satisfies
$\lim_{\xi\to\pm\infty}|\xi^{k}g(\xi)|=0\text{ for }k=0,1,2,\ldots$. We call
such functions _mollifiers_.

###### Remark 2

It is possible to ensure that the paramters $\\{b_{n}\\}_{n\in\mbox{\sBbb
Z}_{+}}$ satisfy $b_{n}>0$ without any genuine loss of generality. This is
achieved by simply setting
$\mathrm{e}^{{\mathrm{i}}\theta_{n}}={\mathrm{i}}^{n}$. We henceforth assume
that $b_{n}>0$.

###### Remark 3

Under the assumption of Remark 2, the functions $\Phi$ are real if and only if
$g(\xi)$ has even real part and odd imaginary part, and $P$ is orthonormal
with respect to an even measure (i.e. it is symmetric about the axis). In this
case, $b_{n}>0$ and $c_{n}=0$ for all $n$.

Theorem 1 and Remarks 2 and 3 were proved by the present authors in (?, ?)
along with results characterising when such systems are orthogonal with
respect to the standard inner product on $\mathrm{L}_{2}(\mbox{\Bbb R})$. The
following Theorem generalises these orthogonality results to the Sobolev inner
products in equation (2.2).

###### Theorem 4

Let $\varphi$ satisfy criterion $(A)$, which implies that (2.3) holds. Then
$\varphi$ also satisfies criterion $(B)$ if and only if the mollifier $g$
satisfies

$w(\xi)=v(\xi)|g(\xi)|^{2},$ (2.4)

where $w(\xi)$ is the positive weight function with respect to which the
polynomials $P$ are orthonormal, and
$v(\xi)=\sum_{\ell=0}^{\infty}v_{\ell}\xi^{2\ell}$. In particular, it is
necessary for the non-negative sequence $\\{v_{\ell}\\}_{\ell\in\mbox{\sBbb
Z}_{+}}$ to decay sufficiently fast that $v(\xi)$ is finite on the support of
$w$.

Proof By Parseval’s Theorem,

$\int_{-\infty}^{\infty}\varphi_{n}(x)\overline{\varphi_{m}(x)}\,\,\mathrm{d}x=(-{\mathrm{i}})^{m-n}\int_{-\infty}^{\infty}p_{n}(\xi)p_{m}(\xi)|g(\xi)|^{2}\,\,\mathrm{d}\xi.$

Furthermore, since
$\widehat{\varphi^{(\ell)}}(\xi)=(-{\mathrm{i}}\xi)^{\ell}\hat{\varphi}(\xi)$
(where $\hat{\varphi}$ denotes the Fourier transform of $\varphi$) we have

$\int_{-\infty}^{\infty}\varphi^{(\ell)}_{n}(x)\overline{\varphi^{(\ell)}_{m}(x)}\,\,\mathrm{d}x=(-{\mathrm{i}})^{m-n}\int_{-\infty}^{\infty}p_{n}(\xi)p_{m}(\xi)\xi^{2\ell}|g(\xi)|^{2}\,\,\mathrm{d}\xi.$

Therefore,

$\langle\varphi_{n},\varphi_{m}\rangle_{v}=(-{\mathrm{i}})^{m-n}\int_{-\infty}^{\infty}p_{n}(\xi)p_{m}(\xi)v(\xi)|g(\xi)|^{2}\,\,\mathrm{d}\xi$

This makes it clear that $\varphi$ is orthonormal with respect to the Sovolev
inner product if and only if $P$ is orthonormal with respect to the measure
$v(\xi)|g(\xi)|^{2}\mathrm{d}\xi$. $\Box$

###### Remark 5

There are infinitely many choices of $g$ which satisfy (2.4), namely

$g(\xi)=\sqrt{\frac{w(\xi)}{v(\xi)}}\mathrm{e}^{{\mathrm{i}}\vartheta(\xi)},$

for any measurable real-valued function $\vartheta$. Our canonical choice is
$\vartheta\equiv 0$, although we know of no good reason, except for
simplicity, why this might be superior to other choices.

It is important to answer what space the resulting orthonormal system is dense
in: ideally this is the inner product space

$\mathrm{H}_{v}(\mbox{\Bbb R}):=\left\\{\psi\in\mathrm{L}_{2}(\mbox{\Bbb
R}):\langle\psi,\psi\rangle_{v}<\infty\right\\},$ (2.5)

endowed with the inner product $\langle\cdot,\cdot\rangle_{v}$, but this need
not be the case.

######  Theorem 6 (Orthogonal bases of Paley–Wiener spaces)

Let $\Phi=\\{\varphi_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$ satisfy the requirements
of Theorem 4 with weight function $w(\xi)$ such that polynomials are dense in
$\mathrm{L}_{2}(\mbox{\Bbb R};w(\xi)\mathrm{d}\xi)$. Then $\Phi$ forms a basis
for the closure (in $\mathrm{H}_{v}(\mbox{\Bbb R})$) of the Paley–Wiener space
$\mathcal{PW}_{\Omega}(\mbox{\Bbb R})$, where $\Omega$ is the support of $w$.

A proof of Theorem 6 can be obtained by modifying Theorem 9 from (?). The key
corollary is that for a basis $\Phi$ satisfying the requirements of Theorem 4
to be complete in $\mathrm{L}_{2}(\mbox{\Bbb R})$, it is necessary that the
polynomial basis $P$ is orthogonal with respect to a measure which is
supported on the whole real line.

##  3 Sobolev cascades

In this section we derive two methods for producing orthonormal systems in the
Sobolev space $\mathrm{H}^{s}(\mbox{\Bbb R})$ where $s=0,1,2,\ldots$.

###  3.1 Cascades of first and second kind

For a weight function $w$ and $s\in\mbox{\Bbb Z}_{+}$ we can define the
following two sequences of bases:

$\varphi_{n}^{\langle
s\rangle}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{{\mathrm{i}}x\xi}\,p_{n}(\xi)\,\sqrt{\frac{w(\xi)}{\sum_{k=0}^{s}\xi^{2k}}}\,\mathrm{d}\xi,$
(3.1)

where $P=\\{p_{n}\\}_{n\in\mbox{\sBbb Z}_{+}}$ are orthonormal polynomials
with respect to $w(\xi)$, and

$\varphi_{n}^{[s]}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{{\mathrm{i}}x\xi}\,p^{[s]}_{n}(\xi)\,\sqrt{w(\xi)}\,\mathrm{d}\xi,$
(3.2)

where $P^{[s]}=\\{p_{n}^{[s]}\\}_{n\in\mbox{\sBbb Z}_{+}}$ are orthonormal
polynomials with respect to the weight

$w^{[s]}(\xi)=\left(\sum_{k=0}^{s}\xi^{2k}\right)\\!w(\xi)=\frac{1-\xi^{2(s+1)}}{1-\xi^{2}}w(\xi).$

By the theory described in Section 2, both systems $\Phi^{\langle
s\rangle}=\\{\varphi_{n}^{\langle s\rangle}\\}_{n\in\mbox{\sBbb Z}_{+}}$ and
$\Phi^{[s]}=\\{\varphi_{n}^{[s]}\\}_{n\in\mbox{\sBbb Z}_{+}}$ have skew-
Hermitian tridiagonal differentiation matrices and both are orthonormal
systems with respect to the standard $\mathrm{H}^{s}(\mbox{\Bbb R})$ Sobolev
inner product described in the introduction. Furthermore, all of these systems
are bases for (closure of) the Paley–Wiener space
$\mathcal{PW}_{\Omega}(\mbox{\Bbb R})$, where $\Omega$ is the support of $w$.

We call the sequence $\Phi^{\langle 0\rangle},\Phi^{\langle
1\rangle},\Phi^{\langle 2\rangle}\ldots$ a Sobolev cascade of the first kind
for the weight function $w$, and $\Phi^{[0]},\Phi^{[1]},\Phi^{[2]}\ldots$ a
Sobolev cascade of the second kind for the weight function $w$. Note that
$\varphi_{n}^{[0]}=\varphi_{n}^{\langle 0\rangle}$.

While a cascade of the first kind is perhaps a more natural generalisation of
$\mathrm{L}_{2}$-orthogonality, it is also more problematic. Typically the
polynomials $p_{n}$ might be already known, however the explicit form of the
integrals (3.1), hence of the $\varphi_{n}^{\langle s\rangle}$s, is often
unknown, even for $s=1$. The issue with cascades of the second kind is
different: the polynomials $P^{[s]}$ are usually unknown for $s\in\mbox{\Bbb
N}$ even for the most familiar measures like Legendre or Hermite. On the other
hand, once we know $p_{n}^{[s]}$ and can compute $\varphi_{n}^{[0]}$
explicitly, the closed form of $\varphi_{n}^{[s]}$ is available for all
$n,s\in\mbox{\Bbb Z}_{+}$ through the integral (3.2). Note that to compute
(1.1) in a closed form we need to be able to integrate explicitly Fourier
transforms of $p\sqrt{w}$ for polynomials $p$: exactly the same is required
for the computation of (3.2).

###  3.2 Sobolev cascades of the second kind

Orthogonal systems in a cascade of the second kind have a simple relationship.

###### Theorem 7

Let $s\in\mbox{\Bbb Z}_{+}$. There exists an infinite, lower triangular matrix
$C^{[s]}$ which has bandwidth $2s$, such that

$\boldsymbol{\varphi^{[0]}}=C^{[s]}\boldsymbol{\varphi^{[s]}}.$ (3.3)

Proof Since $p_{n}^{[0]}$ and $p_{n}^{[s]}$ are polynomials of degree $n$ (for
every $n$), there exists an upper triangular _connection coefficient matrix_
$C^{[s]}$ such that

$p_{n}^{[0]}=\sum_{j=0}^{n}\tilde{C}^{[s]}_{n,j}p_{j}^{[s]}.$ (3.4)

Since $P^{[s]}$ is an orthonormal basis with respect to the weight function
$\left(\sum_{k=0}^{s}\xi^{2k}\right)w(\xi)$, we have the formula

$\tilde{C}^{[s]}_{n,j}=\int_{-\infty}^{\infty}p_{n}^{[0]}(\xi)p_{j}^{[s]}(\xi)\left(\sum_{k=0}^{s}\xi^{2k}\right)\\!w(\xi)\,\mathrm{d}\xi.$
(3.5)

Since $p_{j}^{[s]}(\xi)\left(\sum_{k=0}^{s}\xi^{k}\right)$ is a polynomial of
degree at most $j+2s$, and $P^{[0]}$ is orthonormal with respect to $w$, we
have that $C^{[s]}_{n,j}=0$ if $j\leq n-2s-1$, which proves the desired
bandwidth of the matrix. The proof is completed by multiplying equation (3.4)
by $\sqrt{w(\xi)}$ and taking the inverse Fourier transform:

$\varphi_{n}^{[0]}=\sum_{j=0}^{n}C^{[s]}_{n,j}\varphi_{j}^{[s]},\text{ where
}C^{[s]}_{n,j}={\mathrm{i}}^{n-j}\tilde{C}^{[s]}_{n,j}.$ (3.6)

$\Box$

Note further that if the weight function $w$ is symmetric then all the
polynomials $p_{n}^{[s]}$ maintain the parity of $n$ and it follows easily
that $C_{n,j}^{[s]}=0$ for $n+j$ odd.

Theorem 7 has two consequences. Firstly, if one can calculate
$\\{\varphi_{0}^{[0]},\varphi_{1}^{[0]},\ldots,\varphi_{N}^{[0]}\\}$, then it
is possible to calculate
$\\{\varphi_{0}^{[s]},\varphi_{1}^{[s]},\ldots,\varphi_{N}^{[s]}\\}$ in
$\mathcal{O}(N)$ operations by applying forward substitution to the banded
lower triangular system with matrix $C^{[s]}$.

Secondly, given $N+1$ expansion coefficients in the basis $\Phi^{[0]}$, we can
compute the equivalent expansion coefficients in the basis $\Phi^{[s]}$ in
$\mathcal{O}(N)$ operations. Specifically, if

$\sum_{n=0}^{N}a_{n}^{[0]}\varphi_{n}^{[0]}(x)=\sum_{n=0}^{N}a_{n}^{[s]}\varphi_{n}^{[s]}(x),$

then

${C^{[s]}}^{\top}\\!\boldsymbol{a^{[s]}}=\boldsymbol{a^{[0]}},$ (3.7)

which can be solved in $\mathcal{O}(N)$ operations by back substitution.

In general, it appears that the nonzero entries of $C_{n,j}^{[s]}$ obey no
recognisable numerical relations: for example, the $6\times 6$ principal minor
of $C^{[1]}$ for the Hermite weight is

$\left[\begin{array}[]{cccccc}\sqrt{\frac{3}{2}}&0&0&0&0&0\\\
0&\sqrt{\frac{5}{2}}&0&0&0&0\\\
\sqrt{\frac{1}{3}}&0&\sqrt{\frac{19}{6}}&0&0&0\\\
0&\sqrt{\frac{3}{5}}&0&\sqrt{\frac{39}{10}}&0&0\\\
0&0&\sqrt{\frac{18}{19}}&0&\sqrt{\frac{173}{38}}&0\\\
0&0&0&\sqrt{\frac{819}{407}}&0&\sqrt{\frac{407}{78}}\end{array}\right]\\!.$

It is difficult to discern a pattern: numerical experiments for large values
of $n$ indicate that both $C_{n,n}^{[1]}$ and $C_{n+2,n}^{[1]}$ grow like
${\cal O}\\!\left(\sqrt{n}\right)$.

All this does not rule out computing the $C^{[s]}_{n,j}$s numerically.
Modifying a weight by a quadratic factor and computing the connection
coefficients is discussed in (?) and (?). We won’t go into the details of this
in this paper.

##  4 Hermite-type systems

###  4.1 The Hermite–Sobolev cascade of the first kind

A natural starting point is the Hermite weight $w(\xi)=\mathrm{e}^{-\xi^{2}}$,
$\xi\in\mbox{\Bbb R}$, and $s=1$. The mollifier, by the definitions in Section
3, is $g(\xi)=\mathrm{e}^{-\xi^{2}}/(1+\xi^{2})^{1/2}$, so

$\varphi_{n}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{\mathrm{H}}_{n}(\xi)\sqrt{\frac{\mathrm{e}^{-\xi^{2}}}{1+\xi^{2}}}\mathrm{e}^{{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+},$ (4.1)

where

$\tilde{\mathrm{H}}_{n}(\xi)=\frac{1}{\sqrt{2^{n}n!\sqrt{\pi}}}\mathrm{H}_{n}(\xi),\qquad
n\in\mbox{\Bbb Z}_{+},$

are the orthonormalised Hermite polynomials. Unfortunately, the integrals
(4.1) are not known in an explicit form, not even $\varphi_{0}$.

Figure 4.1: The first six functions $\varphi_{n}^{\langle 1\rangle}$ defined
by (4.1), with corresponding Hermite functions $\varphi_{n}^{\langle
0\rangle}(x)=(-1)^{n}\mathrm{e}^{-x^{2}/2}\tilde{\mathrm{H}}_{n}(x)$ in darker
shade.

In Fig. 4.1 we display the functions $\varphi_{n}$, $n=0,\ldots,5$, computed
by brute-force numerical quadrature. In the background, in fainter colour, we
display the familiar Hermite functions which follow from (1.1) and are
orthonormal in $\mathrm{L}_{2}(\mbox{\Bbb R})$ (while, by Theorem 2, the
$\varphi_{n}$s are orthonormal in $\mathrm{H}^{1}(\mbox{\Bbb R})$).

###  4.2 The Hermite–Sobolev cascade of the second kind

While the polynomials $p_{n}^{[s]}$ from Subsection 3.1 are unknown for
$s\in\mbox{\Bbb N}$, it is possible to generate them, as explained in Section
3 or directly from the moments: in the simplest nontrivial case, $s=1$, the
moments are

$\mu_{n}=\int_{-\infty}^{\infty}\xi^{n}(1+\xi^{2})\mathrm{e}^{-\xi^{2}}\,\mathrm{d}\xi=\left\\{\begin{array}[]{ll}\displaystyle\frac{\sqrt{\pi}(2n)!(2n+3)}{2^{2n+1}n!},&n\mbox{\
even,}\\\\[4.0pt] 0,&n\mbox{\ odd,}\end{array}\right.$

and the first few $p_{n}^{[1]}$s are

$\displaystyle p_{0}^{[1]}(\xi)$ $\displaystyle\equiv$
$\displaystyle\frac{\sqrt{6}}{3\pi^{1/4}},$ $\displaystyle p_{1}^{[1]}(\xi)$
$\displaystyle=$ $\displaystyle\frac{2\sqrt{5}}{5\pi^{1/4}}\xi,$
$\displaystyle p_{2}^{[1]}(\xi)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{57}}{19\pi^{1/4}}\left(\xi^{2}-\frac{5}{6}\right)\\!,$
$\displaystyle p_{3}^{[1]}(\xi)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{130}}{39\pi^{1/4}}\left(\xi^{3}-\frac{21}{10}\xi\right)\\!,$
$\displaystyle p_{4}^{[1]}(\xi)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{9861}}{519\pi^{1/4}}\left(\xi^{4}-\frac{75}{19}\xi^{2}+\frac{117}{76}\right)\\!,$
$\displaystyle p_{5}^{[1]}(\xi)$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{52910}}{2035\pi^{1/4}}\left(\xi^{5}-\frac{245}{39}\xi^{3}+\frac{335}{52}\xi\right)$

and so on. Likewise, it is possible to compute recurrence coefficients,

$b_{0}=\sqrt{\frac{5}{6}},\;b_{1}=\sqrt{\frac{19}{15}},\;b_{2}=\sqrt{\frac{315}{190}},\;b_{3}=\sqrt{\frac{1730}{741}},\;b_{4}=\sqrt{\frac{38665}{13494}},\;b_{5}=\sqrt{\frac{236925}{70411}}$

etc. but difficult to discern any pattern except for the obvious, $b_{n}={\cal
O}\\!\left(n^{1/2}\right)$, $n\gg 1$, a consequence of the proof of the Freud
conjecture in (?). Likewise, we can compute $p_{n}^{[s]}$ for $s\geq 2$: Fig.
4.2 displays $p_{n}^{[s]}$ for $n=2,3,4,5$ and $s=0,1,2,3,4$.

Figure 4.2: The polynomials $p_{n}^{[s]}$ for $n=2,3,4,5$ and $s=0,1,2,4$
(darker hue corresponds to larger $s$).

Computing the $\varphi_{n}^{[1]}$s in line with (1.5) is straightforward:

$\displaystyle\varphi_{0}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}\pi^{-1/4}\mathrm{e}^{-x^{2}/2},$
$\displaystyle\varphi_{1}^{[1]}(x)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{4}{5}}\pi^{-1/4}x\mathrm{e}^{-x^{2}/2},$
$\displaystyle\varphi_{2}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{57}}\pi^{-1/4}(6x^{2}-1)\mathrm{e}^{-x^{2}/2},$
$\displaystyle\varphi_{3}^{[1]}(x)$ $\displaystyle=$
$\displaystyle-\sqrt{\frac{2}{585}}\pi^{-1/4}(10x^{3}-9x)\mathrm{e}^{-x^{2}/2},$
$\displaystyle\varphi_{4}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{39444}}\pi^{-1/4}(76x^{4}-156x^{2}+45)\mathrm{e}^{-x^{2}/2},$
$\displaystyle\varphi_{5}^{[1]}(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{476190}}\pi^{-1/4}(156x^{5}-580x^{3}+405x)\mathrm{e}^{-x^{2}/2}$

and so on.

Figure 4.3: The first six functions $\varphi_{n}^{[1]}$ with corresponding
functions $\varphi_{n}^{[0]}$, which are orthogonal in
$\mathrm{L}_{2}(\mbox{\Bbb R})$, in darker shade.

Fig. 4.3 displays the above functions $\varphi_{n}^{[1]}$ and, in fainter
colour, the functions $\varphi_{n}^{\langle 0\rangle}$ based on the same
weight $w(\xi)=(1+\xi^{2})\mathrm{e}^{-\xi^{2}}$ and defined by (1.1).

###### Lemma 8

For every $n\in\mbox{\Bbb Z}_{+}$ we have
$\varphi_{n}(x)=\lambda_{n}(x)\mathrm{e}^{-x^{2}/2}$, where $\lambda_{n}$ is
an $n$th-degree polynomial.

Proof It is enough to prove that

$\sigma_{n}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\xi^{n}\mathrm{e}^{-\xi^{2}/2+{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+},$

is of the asserted form, i.e. an $n$th degree polynomial times
$\mathrm{e}^{-x^{2}/2}$. This follows readily by induction on $n$ from
$\sigma_{0}(x)=\mathrm{e}^{-x^{2}/2}$ and $\sigma_{n}^{\prime}=\sigma_{n+1}$
because, letting $\sigma_{n}(x)=\alpha_{n}(x)\mathrm{e}^{-x^{2}/2}$, we obtain
$\alpha_{n+1}(x)=\alpha_{n}^{\prime}(x)-x\alpha_{n}(x)$. $\Box$

Alternatively, substituting into (2.1), it is easy to see that

$\lambda_{n}^{\prime}(x)=-b_{n-1}\lambda_{n-1}(x)+x\lambda_{n}(x)+b_{n}\lambda_{n+1}(x),\qquad
n\in\mbox{\Bbb Z}_{+}.$

The proof that $\lambda_{n}$ is an $n$th degree polynomial follows at once by
induction on this differential recurrence, since $b_{n}>0$, $n\in\mbox{\Bbb
N}$.

The bad news is that the $\lambda_{n}$s are not known and, as is trivial to
verify, they do not obey a three-term recurrence relation (hence, by the
Favard theorem, cannot be orthogonal with respect to any Borel measure).
Having said so, the result is highly interesting. It has been proved in (?)
that there exists a unique $\mathrm{L}_{2}$-orthonormal system on the real
line which obeys (1.2) and where each function is a polynomial multiple of the
same $\mathrm{L}_{2}$ function, specifically Hermite functions (or
$\varphi_{n}^{[0]}=\varphi_{n}^{\langle 0\rangle}$ in present notation). The
functions $\\{\varphi_{n}^{[1]}\\}_{n\in\mbox{\sBbb Z}_{+}}$, though, are
$\mathrm{H}^{1}$-orthonormal, they obey (1.2) and
$\varphi_{n}^{[1]}(x)=\mathrm{e}^{-x^{2}/2}\lambda_{n}(x)$.

###### Lemma 9

The only $\mathrm{H}_{v}(\mbox{\Bbb R})$-orthonormal systems (see equation
(2.5)) with a tridiagonal differentiation matrix which are of the form
$\varphi_{n}(x)=G(x)\lambda_{n}(x)$, $n\in\mbox{\Bbb Z}_{+}$, for some
function $G\in\mathrm{L}_{2}(\mbox{\Bbb R})$, $G>0$ (and $G(0)=1$ without loss
of generality), where each $\lambda_{n}$ is a polynomial of degree $n$,
correspond to

$G(x)=\exp\left(-\gamma x^{2}+\delta x\right)$ (4.2)

for some constants $\gamma>0$ and $\delta\in\mbox{\Bbb R}$. The corresponding
weight of orthonormality for $P$ in Theorem 1 is

$w(\xi)\propto v(\xi)\mathrm{e}^{-\xi^{2}/(2\gamma)}.$ (4.3)

Proof We substitute $\varphi_{n}(x)=G(x)\lambda_{n}(x)$ into (2.1), bearing in
mind that $G>0$, to obtain,

$\lambda_{n}^{\prime}(x)=-b_{n-1}\lambda_{n-1}(x)+\left({\mathrm{i}}c_{n}-\frac{G^{\prime}(x)}{G(x)}\right)\lambda_{n}(x)+b_{n}\lambda_{n+1}(x),\qquad
n\in\mbox{\Bbb Z}_{+}.$

Since $\deg\lambda_{m}=m$ by assumption, we deduce, comparing degrees, that
$G^{\prime}/G$ is a linear polynomial, and hence that $G(x)$ is the
exponential of a quadratic polynomial. We can set the constant term in this
quadratic to zero since $G(0)=1$ without loss of generality, so we obtain
equation (4.2).

Inverting the representation in Theorem 1, we have

$p_{n}(\xi)g(\xi)=\frac{(-{\mathrm{i}})^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\varphi_{n}(x)\mathrm{e}^{-{\mathrm{i}}x\xi}\,\mathrm{d}x=\frac{(-{\mathrm{i}})^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\lambda_{n}(x)\mathrm{e}^{-\gamma
x^{2}\xi+\delta x-{\mathrm{i}}x\xi}\,\mathrm{d}x.$

The case $n=0$ tells us that

$g(\xi)\propto\exp(-(\xi-{\mathrm{i}}\delta)^{2}/(4\gamma)).$ (4.4)

Theorem 2 tells us that for $\mathrm{H}_{v}(\mbox{\Bbb R})$ orthonormality we
require

$w(\xi)=v(\xi)|g(\xi)|^{2},$ (4.5)

which completes the proof of necessity of the forms of $G$ and $w$.

Now we prove sufficiency. Let
$w(\xi)=C^{2}v(\xi)\mathrm{e}^{-\xi^{2}/2\gamma}$ where $C$ ensures $w$ has
unit integral, $g(\xi)=C\mathrm{e}^{-(\xi-{\mathrm{i}}\delta)^{2}/(4\gamma)}$,
and define $\Phi$ as in Theorem 1. By Theorem 4, $\Phi$ is an
$\mathrm{H}_{v}(\mbox{\Bbb R})$-orthonormal system, so all that remains to
prove is that $\varphi_{n}(x)=G(x)\lambda_{n}(x)$ where $\lambda_{n}$ is a
polynomial of degree $n$. It is sufficient to show that
$\rho_{n}(x)=\int_{-\infty}^{\infty}\xi^{n}g(\xi)\mathrm{e}^{{\mathrm{i}}x\xi}\,\,\mathrm{d}\xi$
is $G(x)$ times a polynomial of degree $n$, which can be readily shown by
induction starting from $\rho_{0}(x)\propto G(x)$ and leveraging
$\rho_{n+1}(x)=-{\mathrm{i}}\rho_{n}^{\prime}(x)$. $\Box$

###  4.3 An $\mathrm{H}^{\infty}(\mbox{\Bbb R})$ system based on the Hermite
weight

Let $\sigma\in(0,1)$, $w(\xi)=\mathrm{e}^{-\xi^{2}}$ (i.e. the standard
Hermite weight) and $v(\xi)=\mathrm{e}^{\sigma\xi^{2}}$, $\xi\in\mbox{\Bbb
R}$. Therefore, by Theorem 3, the functions $\varphi_{n}$, as defined by
(2.3), are orthogonal with respect to the infinite Sobolev inner product

$\langle
f,g\rangle_{v}=\sum_{\ell=0}^{\infty}\frac{\sigma^{\ell}}{\ell!}\int_{-\infty}^{\infty}f^{(\ell)}(x)g^{(\ell)}(x)\,\mathrm{d}x.$
(4.6)

In this case $p_{n}$s are scaled Hermite polynomials and $\varphi_{n}$s can be
computed explicitly.

###### Theorem 10

The Hermite weight $w(\xi)=\mathrm{e}^{-\xi^{2}}$, $x\in\mbox{\Bbb R}$,
generates the $\mathrm{H}^{\infty}(\mbox{\Bbb R})$ system

$\varphi_{n}^{[\infty]}(x)=\frac{1}{\sqrt{1+\sigma}}\left(\frac{1-\sigma}{1+\sigma}\right)^{\\!n/2}\varphi_{n}^{[0]}\\!\left(\frac{x}{\sqrt{1-\sigma^{2}}}\right)\\!\exp\\!\left(\frac{\sigma
x^{2}}{2(1-\sigma^{2})}\right)\\!,\qquad n\in\mbox{\Bbb Z}_{+},$ (4.7)

where $\varphi_{n}^{[0]}$ is the standard $n$th Hermite function.

Proof Let

$\tilde{\varphi}_{n}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{H}_{n}(\xi)\mathrm{e}^{-\frac{1}{2}(1+\sigma)\xi^{2}+{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+},$

whereby, orthonormalising Hermite polynomials, (2.3) yields
$\varphi_{n}(x)=\tilde{\varphi}_{n}(x)/\sqrt{2^{n}n!\sqrt{\pi}}$. Using the
standard generating function for Hermite polynomials,

$\displaystyle\sum_{n=0}^{\infty}\frac{\tilde{\varphi}_{n}(x)}{n!}t^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\left[\sum_{n=0}^{\infty}\frac{\mathrm{H}_{n}(\xi)}{n!}({\mathrm{i}}t)^{n}\right]\\!\exp\\!\left(-\frac{1+\sigma}{2}\xi^{2}+{\mathrm{i}}x\xi\right)\\!\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp\\!\left(2{\mathrm{i}}\xi
t+t^{2}-\frac{1+\sigma}{2}\xi^{2}+{\mathrm{i}}x\xi\right)\\!\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\sigma}}\exp\\!\left(-\frac{x^{2}}{2(1+\sigma)}\right)\\!\exp\\!\left(-\frac{2xt}{1+\sigma}-\frac{1-\sigma}{1+\sigma}t^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\sigma}}\exp\\!\left(-\frac{x^{2}}{2(1+\sigma)}\right)\\!\exp\\!\left(-\frac{2x}{\sqrt{1-\sigma^{2}}}\left(\sqrt{\frac{1\\!-\\!\sigma}{1\\!+\\!\sigma}}t\right)-\left(\sqrt{\frac{1\\!-\\!\sigma}{1\\!+\\!\sigma}}t\right)^{\\!2}\right)$

and, using the same generating function in the opposite direction,

$\displaystyle\sum_{n=0}^{\infty}\frac{\tilde{\varphi}_{n}(x)}{n!}t^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\sigma}}\exp\\!\left(-\frac{x^{2}}{2(1+\sigma)}\right)\\!\sum_{n=0}^{\infty}\frac{1}{n!}\mathrm{H}_{n}\\!\left(-\frac{x}{\sqrt{1-\sigma^{2}}}\right)\\!\left(\sqrt{\frac{1-\sigma}{1+\sigma}}t\right)^{\\!n}.$

Therefore

$\tilde{\varphi}_{n}(x)=\frac{(-1)^{n}}{\sqrt{1+\sigma}}\left(\frac{1-\sigma}{1+\sigma}\right)^{\\!n/2}\mathrm{H}_{n}\\!\left(\frac{x}{\sqrt{1-\sigma^{2}}}\right)\\!\exp\\!\left(-\frac{x^{2}}{2(1+\sigma)}\right)\\!.$

Normalising,

$\displaystyle\varphi_{n}^{[\infty]}(x)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n}}{\sqrt{(1+\sigma)2^{n}n!\sqrt{\pi}}}\left(\frac{1-\sigma}{1+\sigma}\right)^{\\!n/2}\mathrm{H}_{n}\\!\left(\frac{x}{\sqrt{1-\sigma^{2}}}\right)\\!\exp\\!\left(-\frac{x^{2}}{2(1+\sigma)}\right)\\!,$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\sigma}}\left(\frac{1-\sigma}{1+\sigma}\right)^{\\!n/2}\varphi_{n}^{[0]}\\!\left(\frac{x}{\sqrt{1-\sigma^{2}}}\right)\\!\exp\\!\left(\frac{\sigma
x^{2}}{2(1-\sigma^{2})}\right)\\!,\qquad n\in\mbox{\Bbb Z}_{+},$

and (4.7) is true. $\Box$

Figure 4.4: The first six $\mathrm{H}^{\infty}$ Hermite-type functions
$\varphi_{n}^{[\infty]}$ for $\sigma=\frac{1}{4},\frac{1}{2},\frac{3}{4}$ in
progressively darker hues.

Fig. 4.4 displays the functions $\varphi_{n}^{[\infty]}$, $n=0,\ldots,5$, for
three different values of $\sigma\in(0,1)$. The zeros of $\varphi_{n}$ are
scaled zeros of a Hermite polynomial and, the scaling being monotone, the
zeros are ‘squeezed’ in a uniform manner for increasing $\sigma$, as evident
in the figure.

##  5 Bilateral Laguerre-type weights

Deferring the standard Laguerre weight (which is not symmetric) to Section 7,
we let $w(\xi)=(1+\xi^{2})\mathrm{e}^{-|\xi|}$, $\xi\in\mbox{\Bbb R}$. Note
that the underlying orthogonal polynomials are unknown explicitly, yet can be
computed. The $\varphi_{n}^{[1]}$s are

$\displaystyle\varphi_{0}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}\sqrt{\pi}}\frac{1}{1+4x^{2}},$
$\displaystyle\varphi_{1}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{16}{\sqrt{26}\sqrt{\pi}}\frac{x}{(1+4x^{2})^{2}},$
$\displaystyle\varphi_{2}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{1167}\sqrt{\pi}}\frac{1+248x^{2}+208x^{4}}{(1+4x^{2})^{3}},$
$\displaystyle\varphi_{3}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{16}{\sqrt{23179}\sqrt{\pi}}\frac{-21x+456x^{3}+496x^{5}}{(1+4x^{2})^{4}},$
$\displaystyle\varphi_{4}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{309347971}\sqrt{\pi}}\frac{2925-128784x^{2}+1703264x^{4}+3029760x^{6}+1369344x^{8}}{(1+4x^{2})^{5}},$
$\displaystyle\varphi_{5}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\frac{16}{\sqrt{22678864934}\sqrt{\pi}}\frac{25749x\\!-\\!1017424x^{3}\\!+\\!5715040x^{5}\\!+\\!13510400x^{7}\\!+\\!7744768x^{9}}{(1+4x^{2})^{6}}.$

The general formula is a polynomial of degree $2n-[1-(-1)^{n}]/2$ in $x$ (of
the same parity as $n$), divided by $(1+4x^{2})^{n+1}$. This can be easily
verified because by (2.3) and $g(\xi)=\mathrm{e}^{-|\xi|/2}$ each
$\varphi_{n}^{[1]}$ is a linear combination of
$\lambda_{n},\lambda_{n-2},\ldots$, where

$\lambda_{n}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\xi^{n}\mathrm{e}^{-|\xi|^{2}+{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+}$

and $\lambda_{n}^{\prime}(x)=\lambda_{n+1}(x)$ implies that

$\lambda_{n}(x)=\lambda^{(n)}_{0}(x)=\frac{2\sqrt{2}}{\sqrt{\pi}}\frac{\,\mathrm{d}^{n}}{\,\mathrm{d}x^{n}}\frac{1}{1+4x^{2}},\qquad
n\in\mbox{\Bbb Z}_{+}.$

##  6 Bessel-like functions

###  6.1 Transformation of Chebyshev polynomials

We rewrite (1.6) in the form (2.3),

$\varphi_{n}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-1}^{1}\tilde{\mathrm{T}}_{n}(\xi)\mathrm{e}^{{\mathrm{i}}x\xi}\frac{\,\mathrm{d}\xi}{\sqrt{1-\xi^{2}}}=(-1)^{n}\mathrm{J}_{n}(x),$

where $\tilde{\mathrm{T}}_{n}=\sqrt{2/\pi}\mathrm{T}_{n}$ (except that
$\tilde{\mathrm{T}}_{0}\equiv\mathrm{T}_{0}/\sqrt{\pi}$) are orthonormal
Chebyshev polynomials. This does not fit within the framework of Theorem 2,
because $v(\xi)=(1-\xi^{2})^{-3/2}\chi_{(-1,1)}(\xi)$ (derived using (2.3)) is
not $\mathrm{L}_{2}$. Moreover, it is easy to verify directly that the
$\varphi_{n}$s cannot be bounded in any Sobolev norm because the
Weber–Schafheitlin formula (?, 10.22.57) implies that for
$\mathrm{Re}\,\lambda>0$

$\int_{-\infty}^{\infty}\frac{\varphi_{n}^{2}(x)\,\mathrm{d}x}{x^{\lambda}}=\int_{-\infty}^{\infty}\frac{\mathrm{J}_{n}^{2}(x)\,\mathrm{d}x}{x^{\lambda}}=\frac{\mathrm{\Gamma}(n+\frac{1}{2})\mathrm{\Gamma}(\lambda)}{2^{\lambda-1}\mathrm{\Gamma}^{2}(\frac{1}{2}\lambda+\frac{1}{2})\mathrm{\Gamma}(\frac{1}{2}\lambda+n+\frac{1}{2})}\stackrel{{\scriptstyle\lambda\rightarrow
0}}{{\longrightarrow}}\infty.$

If instead of a Chebyshev measure we use the Legendre measure,
$w(\xi)=\chi_{(-1,1)}(\xi)$, the state of affairs is different:
$g(\xi)=\chi_{(-1,1)}$ results in

$\varphi_{n}(x)=(-1)^{n}\sqrt{\frac{n+\frac{1}{2}}{x}}\mathrm{J}_{n+\frac{1}{2}}(x),\qquad
x\in\mbox{\Bbb R},$ (6.1)

and the $\varphi_{n}$s are integrable on ℝ.

###  6.2 The Legendre weight

The most obvious example of an $\mathrm{H}^{1}(\mbox{\Bbb R})$ system is based
on the Legendre weight $w(\xi)=\chi_{(-1,1)}(\xi)$, in which the orthonormal
polynomials are $p_{n}=\sqrt{n+\frac{1}{2}}\mathrm{P}_{n}$. Thus,

$\varphi_{n}^{\langle
1\rangle}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\sqrt{n+\frac{1}{2}}\int_{-1}^{1}\frac{\mathrm{P}_{n}(\xi)}{\sqrt{1+\xi^{2}}}\mathrm{e}^{{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+}.$

While $\\{\varphi_{n}^{\langle 1\rangle}\\}_{n\in\mbox{\sBbb Z}_{+}}$ is
orthonormal in $\mathrm{H}^{1}(\mbox{\Bbb R})$, it is not a complete basis
because all Fourier spectra are restricted to $[-1,1]$, yet it might be of an
independent interest. Perhaps more vexing issue is that above integrals are
not available in an explicit form. This is not an insurmountable problem in
the computation of the $\varphi_{n}^{\langle 1\rangle}$s which we can compute
for individual values of $x$ using a Fast Fourier Transform (?, ?), although
it presents an obvious obstacle to their analysis.

Figure 6.1: The first six functions $\varphi_{n}^{\langle 1\rangle}$ for the
Legendre weight.

In Fig. 6.1 we have computed $\varphi_{0}^{\langle
1\rangle},\ldots,\varphi_{5}^{\langle 1\rangle}$ numerically. Like other
transformed functions (1.1) or (2.3), the $\varphi_{n}^{\langle 1\rangle}$s
seem to be endowed with a wealth of structural features and regularities which
have been discussed briefly (for (1.1)) in (?) but overall are a subject for
future research.

###  6.3 Sobolev–Legendre cascades

We revisit the essence of Subsections 3.1.1–2, except that the range of
integration is now $[-1,1]$. Firstly, we set $w=\chi_{(-1,1)}$, let $p_{n}$ be
the (orthonormalised) Legendre polynomials and, for every $s\in\mbox{\Bbb
Z}_{+}$ set

$\varphi_{n}^{\langle
s\rangle}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-1}^{1}p_{n}(\xi)\left(\sum_{\ell=0}^{s}\xi^{2\ell}\\!\right)^{\\!-1/2}\mathrm{e}^{{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+}.$ (6.2)

Secondly, we might define
$w_{s}(\xi)=\chi_{(-1,1)}(\xi)\sum_{\ell=0}^{s}\xi^{2\ell}$, $s\in\mbox{\Bbb
Z}_{+}$, and set

$\varphi_{n}^{[s]}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-1}^{1}p^{[s]}_{n}(\xi)\mathrm{e}^{{\mathrm{i}}x\xi}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+},$ (6.3)

where $\\{p_{n}^{[s]}\\}_{n\in\mbox{\sBbb Z}_{+}}$ is the orthonormal
polynomial system corresponding to the weight $w_{s}$. It follows at once from
Theorem 2 that

$\sum_{\ell=0}^{s}\int_{-\infty}^{\infty}\frac{\,\mathrm{d}^{\ell}\varphi_{m}^{\langle
s\rangle}(x)}{\,\mathrm{d}x^{\ell}}\frac{\,\mathrm{d}^{\ell}\varphi_{n}^{\langle
s\rangle}(x)}{\,\mathrm{d}x^{\ell}}\,\mathrm{d}x=\sum_{\ell=0}^{s}\int_{-\infty}^{\infty}\frac{\,\mathrm{d}^{\ell}\varphi_{m}^{[s]}(x)}{\,\mathrm{d}x^{\ell}}\frac{\,\mathrm{d}^{\ell}\varphi_{n}^{[s]}(x)}{\,\mathrm{d}x^{\ell}}\,\mathrm{d}x=\delta_{m,n}$

for all $m,n\in\mbox{\Bbb Z}_{+}$ and both $\\{\varphi_{n}^{\langle
s\rangle}\\}_{n\in\mbox{\sBbb Z}_{+}}$ and
$\\{\varphi_{n}^{[s]}\\}_{n\in\mbox{\sBbb Z}_{+}}$ are orthonormal sets in
$\mathrm{H}^{s}(\mbox{\Bbb R})$. Of course, neither is dense in the Sobolev
space because their Fourier spectra are restricted to $[-1,1]$ – they are
dense in an obvious generalisation of Paley–Wiener spaces to the realm of
Sobolev spaces. The systems (6.2) and (6.3) are the Sobolev–Legendre cascades
of the first and the second kind, respectively.

We recall a major practical difference between the two cascades: except for
the case $s=0$ (when $\varphi^{\langle 0\rangle}_{n}=\varphi^{[0]}_{n}$ has
been given in (6.1)), $\varphi_{n}^{\langle s\rangle}$ is unknown in an
explicit form while $\varphi_{n}^{[s]}$, being an integral in $[-1,1]$ of a
polynomial times $\mathrm{e}^{{\mathrm{i}}x\xi}$, can be computed at great
ease and is a linear combination of spherical Bessel functions. Consequently,
in the sequel we focus on the Sobolev–Lagendre cascade of the second kind.

Figure 6.2: The Sobolev–Legendre cascade of the second kind: The first six
functions $\varphi_{n}^{[s]}$ for $s=0,1,2$: increasing $s$ corresponds to
increasing line thickness and darker hue.

Fig. 6.2 displays the beginning (that is, $s=0,1,2$) of the cascade of the
second kind. The obvious observation is that $\varphi^{[s]}_{0}$ is a scalar
multiple of $\varphi^{[0]}_{0}$ and the same is true for $\varphi_{1}^{[s]}$
and $\varphi_{1}^{[0]}$, respectively. This follows from (6.3) because
$p_{0}^{[s]}$ is a constant, while $p_{1}^{[s]}$ is a scalar multiple of
$\xi$. Another obvious indication is that, as $s$ grows, $\varphi_{n}^{[s]}$
converges pointwise to a function $\varphi_{n}^{[\infty]}$ yet this might be
less interesting than it appears. In particular,

###### Lemma 11

$\varphi_{n}^{[\infty]}\equiv 0$.

Proof Let

$u_{s}=\int_{-1}^{1}\sum_{\ell=0}^{s}\xi^{2\ell}\,\mathrm{d}\xi,\qquad
s\in\mbox{\Bbb Z}_{+}.$

Then $p_{0}^{[0]}\equiv 1/\sqrt{u_{2}}$ and, by (6.3),

$\varphi_{0}^{[s]}(x)=\sqrt{\frac{2}{\pi u_{s}}}\frac{\sin x}{x}.$

But

$u_{s}=\sum_{\ell=0}^{s}\frac{1}{\ell+\frac{1}{2}}\stackrel{{\scriptstyle\ell\rightarrow\infty}}{{\longrightarrow}}\infty$

and the lemma follows. $\Box$

Alternatively,
$\lim_{s\rightarrow\infty}w^{[s]}(\xi)=(1-\xi^{2})^{-1}\chi_{(-1,1)}\not\in\mathrm{L}_{2}(\mbox{\Bbb
R})$. An obvious remedy, which we do not pursue in this paper, is to consider
the weight $w_{s}(\xi)=\sum_{\ell=0}^{s}\sigma^{\ell}\xi^{2\ell}$ for some
$\sigma\in(0,1)$.

###  6.4 The Sobolev-ultraspherical cascade of the second kind

We construct a cascade of the second kind based on the ultraspherical weight
$w^{[0]}(\xi)=(1-\xi^{2})\chi_{(-1,1)}(\xi)$. Therefore

$w^{[s]}(\xi)=w^{[0]}(\xi)\sum_{\ell=0}^{s}\xi^{2\ell}=(1-\xi^{2s+2})\chi_{(-1,1)},\qquad
s\in\mbox{\Bbb Z}_{+}$

and, as $s\rightarrow\infty$, $w^{[s]}$ converges weakly to the Legendre
weight.

Figure 6.3: The Sobolev–ultrasperical cascade: The first six functions
$\varphi_{n}^{[s]}$ for $s=0,1,2,3$: increasing $s$ corresponds to darker hue.

In Fig. 6.3 we display the functions
$\varphi_{n}^{[0]},\ldots,\varphi_{n}^{[3]}$ for $n=0,1,\ldots,5$. While the
convergence to a limit in each figure is quite persuasive, we must beware of
‘proof by picture’: convergence is equally pictorially persuasive in Fig. 6.2
where, as we have already seen, it need not take place.

##  7 Non-symmetric measures

The most obvious non-symmetric weight function is the Laguerre weight
$w(\xi)=\mathrm{e}^{-\xi}\chi_{[0,\infty)}(\xi)$. In that case the
$\varphi_{n}$s are the Malmquist–Takenaka functions, which have a particularly
neat form,

$\varphi_{n}(x)=\sqrt{\frac{2}{\pi}}{\mathrm{i}}^{n}\frac{(1+2{\mathrm{i}}x)^{n}}{(1-2{\mathrm{i}}x)^{n+1}},\qquad
n\in\mbox{\Bbb Z}_{+},$ (7.1)

(?), and they are dense in $\mathcal{PW}_{[0,\infty)}(\mbox{\Bbb R})$. It is
possible to extend them to a system dense in all of $\mathrm{L}_{2}(\mbox{\Bbb
R})$ by melding them with another system, generated by the mirror image of the
Laguerre weight, $\mathrm{e}^{\xi}\chi_{(-\infty,0]}(\xi)$: together we obtain
the same system as (7.1), except that now $n$ ranges over all of ℤ.

It is, of course, perfectly possible for a system with a non-symmetric measure
to be dense in $\mathrm{L}_{2}(\mbox{\Bbb R})$, provided that the support of
$w$ is all of ℝ: an example is the Hermite-type weight
$w(\xi)=(1-\xi)^{2}\mathrm{e}^{-\xi^{2}}$.

###  7.1 Shifted Hermite weight

Letting $\rho\in\mbox{\Bbb R}$, we consider the weight
$w(\xi)=\mathrm{e}^{-(\xi-\rho)^{2}}$. The underlying orthonormal polynomials
are $p_{n}(\xi)=\tilde{\mathrm{H}}_{n}(\xi-\rho)$, where
$\tilde{\mathrm{H}}_{n}$ is the orthonormalised Hermite polynomial,
$\tilde{\mathrm{H}}_{n}=\mathrm{H}_{n}/\sqrt{2^{n}n!\sqrt{\pi}}$. Therefore,
seeking $\mathrm{H}^{1}(\mbox{\Bbb R})$ orthogonality,

$\displaystyle\varphi_{n}^{\langle 0\rangle}(x,\rho)$ $\displaystyle=$
$\displaystyle\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{\mathrm{H}}_{n}(\xi-\rho)\mathrm{e}^{-(\xi-\rho)^{2}/2+{\mathrm{i}}x\xi}\,\mathrm{d}\xi=\mathrm{e}^{{\mathrm{i}}\rho
x}\varphi_{n}^{\langle 0\rangle}(x,0),$ $\displaystyle\varphi_{n}^{\langle
1\rangle}(x,\rho)$ $\displaystyle=$
$\displaystyle\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{\mathrm{H}}_{n}(\xi-\rho)\frac{\mathrm{e}^{-(\xi-\rho)^{2}/2+{\mathrm{i}}x\xi}}{\sqrt{1+\xi^{2}}}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+}.$

It is easy to verify that

$\varphi_{n}^{\langle 0\rangle}(x,\rho)=\mathrm{e}^{{\mathrm{i}}\rho
x}\varphi_{0}^{\langle 0\rangle}(x,0),\qquad x,\rho\in\mbox{\Bbb R}.$

Thus, $\varphi_{n}^{\langle 0\rangle}=\varphi_{n}^{[0]}$ is merely a complex-
valued rotation of the standard Hermite function. The situation is more
intriguing with regard to $\varphi_{n}^{\langle 1\rangle}$. Shifting the
variable of integration,

$\varphi_{n}^{\langle
1\rangle}(x,\rho)=\frac{{\mathrm{i}}^{n}\mathrm{e}^{{\mathrm{i}}\rho
x}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{\mathrm{H}}_{n}(\xi)\sqrt{\frac{\mathrm{e}^{-\xi^{2}}}{1+(\sigma+\xi)^{2}}}\mathrm{e}^{{\mathrm{i}}x\xi}\,\mathrm{d}\xi.$

On the face of it, we recover an expression similar to (2.3), except that
$v(\xi)=1+(\sigma+\xi)^{2}$ is not an even function and does not define a
Sobolev inner product.

Figure 7.1: Shifted Sobolev–Hermite functions: $|\varphi_{n}^{\langle
1\rangle}(x,1)|$ in thicker line and darker colour and
$\mathrm{Re}\,\varphi_{n}^{\langle 1\rangle}(x,1)$, for $n=0,\ldots,5$.

Fig. 7.1 displays the absolute and real values of the complex-valued functions
$\varphi_{n}^{\langle 1\rangle}$.

###  7.2 The Laguerre weight

####  7.2.1 Sobolev–Laguerre functions of first kind

Let $w(\xi)=\mathrm{e}^{-\xi}\chi_{[0,\infty)}(\xi)$, a Laguerre weight. Thus,
the $\varphi_{n}^{\langle 0\rangle}$s are Malmquist–Takenaka functions (7.1),
which we need to complement with their ‘reflections’ for $n\in-\mbox{\Bbb N}$
to form a system dense in $\mathrm{L}_{2}(\mbox{\Bbb R})$. By similar token,
we need to complement $\varphi_{n}^{\langle 1\rangle}$s, $n\in\mbox{\Bbb
Z}_{+}$, with functions generated with
$w(\xi)=\mathrm{e}^{\xi}\chi_{(-\infty,0]}(\xi)$ (and indexed by
$n\in-\mbox{\Bbb N}$) to attain completeness in $\mathrm{H}^{1}(\mbox{\Bbb
R})$.

It is possible to compute individual $\varphi_{n}^{\langle 1\rangle}$s
explicitly in terms of Bessel functions of the second kind (a.k.a. Weber
functions) $\mathrm{Y}_{n}$ (?, 10.2.4) and Struve functions ${\bf H}_{n}$ (?,
11.2.1). We first let

$z=\frac{1}{2}-{\mathrm{i}}x,\qquad g(z)=\mathrm{Y}_{0}(z)-{\bf H}_{0}(z).$

The functions $\varphi_{n}$ can be represented explicitly as linear
combinations of derivatives of the function $g$.

###### Lemma 12

The explicit form of the functions $\varphi_{n}^{\langle 1\rangle}$ is

$\varphi_{n}^{\langle
1\rangle}(x)=-\frac{\sqrt{2\pi}}{4}{\mathrm{i}}^{n}\sum_{\ell=0}^{n}{n\choose\ell}\frac{1}{\ell!}g^{(\ell)}\\!\\!\left(\frac{1}{2}-{\mathrm{i}}x\right)\\!,\qquad
n\in\mbox{\Bbb Z}_{+},$ (7.2)

while $\varphi_{-n-1}^{\langle
1\rangle}(x)=(-1)^{n+1}\overline{\varphi_{n}^{\langle 1\rangle}(x)}$,
$n\in\mbox{\Bbb Z}_{+}$.

Proof Letting

$\eta_{n}(x)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\xi^{n}\frac{\mathrm{e}^{-\xi/2+{\mathrm{i}}x\xi}}{\sqrt{1+\xi^{2}}}\,\mathrm{d}\xi,\qquad
n\in\mbox{\Bbb Z}_{+},$

we compute the generating function

$\displaystyle G(x,T)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{\eta_{n}(x)}{n!}T^{n}=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}\frac{1}{\sqrt{1+\xi^{2}}}\exp\\!\left(-\frac{\xi}{2}+T\xi+{\mathrm{i}}x\xi\right)\\!\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{2\pi}}{4}\left[\mathrm{Y}_{0}\\!\left(\frac{1}{2}-{\mathrm{i}}x-T\right)-{\bf
H}_{0}\\!\left(\frac{1}{2}-{\mathrm{i}}x-T\right)\\!\right]\\!.$

Therefore

$\eta_{n}(x)=\frac{\partial^{n}G(x,T)}{\partial
T^{n}}\,\rule[-8.0pt]{0.75pt}{22.0pt}_{\,T=0}=(-1)^{n+1}\frac{\sqrt{2\pi}}{4}g^{(n)}\\!\\!\left(\frac{1}{2}-{\mathrm{i}}x\right)\\!,\qquad
n\in\mbox{\Bbb Z}_{+}.$

Laguerre polynomials $\mathrm{L}_{n}$ are orthonormal and

$\mathrm{L}_{n}(x)=\sum_{\ell=0}^{n}(-1)^{\ell}{n\choose\ell}\frac{x^{\ell}}{\ell!},\qquad
n\in\mbox{\Bbb Z}_{+}$

(?, 18.5.12) and it follows from Theorem 1 that

$\varphi_{n}^{\langle
1\rangle}(x)=\frac{{\mathrm{i}}^{n}}{\sqrt{2\pi}}\int_{0}^{\infty}\mathrm{L}_{n}(\xi)\frac{\mathrm{e}^{-\xi/2+{\mathrm{i}}x\xi}}{\sqrt{1+\xi^{2}}}\,\mathrm{d}\xi={\mathrm{i}}^{n}\sum_{\ell=0}^{n}(-1)^{\ell}{n\choose\ell}\frac{\eta_{\ell}(x)}{\ell!},$

thereby proving (7.2) upon the substitution of the explicit form of
$\eta_{\ell}$.

Extending this to $n\leq-1$ is trivial. $\Box$

###### Corollary 1

The functions $\varphi_{n}$ for $n\in\mbox{\Bbb Z}_{+}$ have the generating
function

$\sum_{n=0}^{\infty}\frac{\varphi_{n}^{\langle
1\rangle}(x)}{n!}t^{n}=-\frac{\sqrt{2\pi}}{4}\mathrm{e}^{{\mathrm{i}}t}\sum_{\ell=0}^{\infty}\frac{({\mathrm{i}}t)^{\ell}}{\ell!^{2}}g^{(\ell)}\\!\\!\left(\frac{1}{2}-{\mathrm{i}}x\right)\\!.$
(7.3)

The proof is elementary, using (7.2). Moreover, (7.3) can be easily extended
to

$\sum_{n=-\infty}^{\infty}\\!\frac{\varphi_{n}^{\langle
1\rangle}(x)}{|n|!}\zeta^{n}=\frac{\sqrt{2\pi}}{4}\\!\left[\mathrm{e}^{-{\mathrm{i}}\zeta^{-1}}\\!\sum_{\ell=0}^{\infty}\frac{(-{\mathrm{i}}\zeta^{-1})^{\ell}}{\ell!^{2}}g^{(\ell)}\\!\left(\frac{1}{2}\\!+\\!{\mathrm{i}}x\\!\right)-\mathrm{e}^{{\mathrm{i}}\zeta}\sum_{\ell=0}^{\infty}\frac{({\mathrm{i}}\zeta)^{\ell}}{\ell!^{2}}g^{(\ell)}\\!\left(\frac{1}{2}\\!-\\!{\mathrm{i}}x\\!\right)\\!\right]\\!,$

which makes sense for $|\zeta|=1$.

Since

$\displaystyle
z^{2}\mathrm{Y}_{n}^{\prime\prime}(z)+z\mathrm{Y}_{n}^{\prime}(z)+(z^{2}-n^{2})\mathrm{Y}_{n}(z)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle z^{2}{\bf
H}_{n}^{\prime\prime}(z)+z{\bf H}_{n}^{\prime}(z)+(z^{2}-n^{2}){\bf H}_{n}(z)$
$\displaystyle=$
$\displaystyle\frac{z^{n+1}}{2^{n-1}\sqrt{\pi}\mathrm{\Gamma}(n+\frac{1}{2})}$

(?, 11.10.5 & 11.2.7), it follows that $g$ obeys

$zg^{\prime\prime}(z)+g^{\prime}(z)+zg(z)=-\frac{2}{\sqrt{\pi}\mathrm{\Gamma}(\frac{1}{2})}=-\frac{2}{\pi}$

and we can express $g^{(\ell)}$ as a linear combination of $g$ and
$g^{\prime}$ with rational coefficients. We do not pursue further this course
of action. Functions $\varphi_{n}^{\langle s\rangle}$ for $s\geq 2$ (or even
the underlying orthogonal polynomials $p_{n}^{\langle s\rangle}$) are no
longer available in an explicit form.

####  7.2.2 Sobolev–Laguerre functions of the second kind

An alternative is to consider the Sobolev–Laguerre cascade of the second kind.
While the orthogonal polynomials $p_{n}^{[s]}$ for the weight
$w_{s}(\xi)=\mathrm{e}^{-\xi}\chi_{[0,\infty)}\sum_{\ell=0}^{s}\xi_{\ell}^{2}$
are unknown for $s\in\mbox{\Bbb N}$, the underlying moments are trivial to
compute and such polynomials can be generated at will. Also the computation of
the $\varphi_{n}^{[s]}$s does not present a problem: for example

$\displaystyle\varphi_{0}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3\pi}}\frac{1}{1-2{\mathrm{i}}x},$
$\displaystyle\varphi_{1}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{87\pi}}{\mathrm{i}}\left[-\frac{4}{1-2{\mathrm{i}}x}+\frac{3(1+2{\mathrm{i}}x)}{(1-2{\mathrm{i}}x)^{2}}\right]\\!,$
$\displaystyle\varphi_{2}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{16211\pi}}{\mathrm{i}}^{2}\left[\frac{34}{1-2{\mathrm{i}}x}-\frac{40(1+2{\mathrm{i}}x)}{(1-2{\mathrm{i}}x)^{2}}+\frac{29(1+2{\mathrm{i}}x)^{2}}{(1-2{\mathrm{i}}x)^{3}}\right]\\!,$
$\displaystyle\varphi_{3}^{[1]}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{9812127\pi}}{\mathrm{i}}^{3}\left[-\frac{480}{1-2{\mathrm{i}}x}+\frac{762(1+2{\mathrm{i}}x)}{(1-2{\mathrm{i}}x)^{2}}-\frac{804(1+2{\mathrm{i}}x)^{2}}{(1-2{\mathrm{i}}x)^{3}}+\frac{559(1+2{\mathrm{i}}x)^{3}}{(1-2{\mathrm{i}}x)^{4}}\right]$

and so on: all this seems very similar to (7.1) and for a good reason: for any
$s\in\mbox{\Bbb Z}_{+}$ we can expand the relevant orthonormal polynomials in
the Laguerre basis,

$p_{n}^{[s]}(x)=\sum_{j={0}}^{n}\gamma_{n,j}^{[s]}\mathrm{L}_{j}(x)$

(cf. (3.5)), whereby it follows from (3.2) that

$\varphi_{n}^{[s]}(x)=\sum_{j=0}^{n}\frac{\gamma_{n,j}^{[s]}}{\sqrt{2\pi}}\int_{0}^{\infty}\\!\mathrm{L}_{j}(\xi)\mathrm{e}^{-\xi/2+{\mathrm{i}}x\xi}\,\mathrm{d}\xi=\sum_{j=0}^{n}\gamma_{n,j}^{[s]}\varphi_{j}^{[0]}(x).$

Note that the matrix $\\{\gamma_{n,j}^{[s]}\\}_{n,j\in\mbox{\sBbb Z}_{+}}$ is
the inverse of the banded connection matrix ${C^{[s]}}^{\top}$ from Theorem 7.
A similar construction applies also to $\varphi_{n}^{[s]}$ for $n\leq-1$.

The most remarkable feature of the Malmquist–Takenaka system is that the
expansion coefficients $\hat{f}^{[0]}_{n}=\langle f,\varphi_{n}^{[0]}\rangle$
can be computed for $-N+1\leq n\leq N$ in ${\cal O}\\!\left(N\log N\right)$
operations using the Fast Fourier Transform. By (3.7), however, once
$\hat{\mbox{\boldmath$f$\unboldmath}}^{[0]}$ is known and assuming that the
requisite derivatives of $f$ are available, it costs just ${\cal
O}\\!\left(N\right)$ operations to compute

$f_{n}^{[s]}=\sum_{\ell=0}^{s}\int_{-\infty}^{\infty}f^{(\ell)}(x)\overline{{\varphi_{n}^{[s]}}^{(\ell)}(x)}\,\mathrm{d}x,\qquad-N+1\leq
n\leq N.$

(The derivatives of $\varphi_{n}^{[s]}$s can be computed similarly to the
functions themselves,
${\mbox{\boldmath$\varphi$\unboldmath}_{n}^{[s]}}^{(\ell)}={C^{[s]}}^{-\top}{\mbox{\boldmath$\varphi$\unboldmath}^{[0]}}^{(\ell)}$.)
Altogether, the cost scales as ${\cal O}\\!\left(sN\log N\right)$.

##  8 Conclusion

In a sequence of previous papers (?, ?, ?, ?) the current authors have
sketched different aspects of an overarching theory of
$\mathrm{L}_{2}$-orthonormal systems on the real line with a tridiagonal
differentiation matrix. In this paper we extend the framework to orthogonality
with respect to Sobolev spaces. Unlike in the case of orthogonal polynomials,
where Sobolev orthogonality is of a completely different flavour to
orthogonality with respect to a Borel measure (?, ?, ?), in our case we can
leverage many elements of the “$\mathrm{L}_{2}$ theory” to a Sobolev setting:
the connection to standard orthogonal polynomials via a weighed Fourier
transform, density in Paley–Wiener spaces and fast computation of certain
expansions. Other aspects of the theory are new, in particular the existence
of two cascades, the latter of which can be ascended by banded triangular
connection coefficients.

The work of this paper is a stepping stone toward the development of spectral
methods on the real line that respect a wide range of invariants that can be
expressed as conservation of a variable-weight Sobolev norm: a couple of
examples have been given in Section 1. We expect to return to this issue in a
forthcoming paper.

## Acknowedgements

The authors are grateful for very useful and enlightening correspondence with
Enno Diekma, Erik Koelink and Tom Koornwinder. We gratefully acknowledge the
partial support by the Simons Foundation Award No 663281 granted to the
Institute of Mathematics of the Polish Academy of Sciences for the years
2021–2023. MW ackowledges support by Computational Mathematics in Quantum
Mechanics, Grant of the National Science Centre (SONATA-BIS), project no.
2019/34/E/ST1/00390.

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