Riesz and Boas Interpolation Formulas

• Riesz and Boas interpolation formulas are explicit methods to represent derivatives of bandlimited functions and trigonometric polynomials with guaranteed convergence.
• They have been extended to abstract Banach spaces, Mellin analysis, and discrete settings, providing robust error estimates and universal coefficient frameworks.
• These formulas underpin modern sampling theory and spectral analysis, enabling precise derivative reconstruction in signal processing and on manifolds.

Riesz and Boas interpolation formulas provide explicit series or finite-sum representations for derivatives of functions in classes of bandlimited signals or trigonometric polynomials, further admitting operator extensions in abstract Banach-space settings. These formulas are foundational in harmonic analysis, sampling theory, operator theory, and their applications in analysis on manifolds, Mellin analysis, and discrete transforms.

1. Definitions and Classical Formulations

Let $B_\sigma^p$ denote the Bernstein space of entire functions of exponential type $\sigma$ bounded on the real axis and lying in $L^p(\mathbb{R})$, equivalently characterized by $$\mathrm{supp}\,\mathcal F f \subset [-\sigma,\sigma]$$. The Bernstein inequality asserts:

$$\|f^{(m)}\|_p \leq \sigma^m \, \|f\|_p, \qquad m = 0,1,2,\ldots.$$

The classical Riesz interpolation formula yields the derivative of a $2\pi$-periodic trigonometric polynomial $P_N$ of degree $N$ as

$$P_N'(x) = \frac{1}{4N} \sum_{j=1}^{2N} (-1)^{j+1} \frac{P_N\left(x+\frac{2j-1}{2N}\,\pi\right)}{\sin^2\left(\frac{2j-1}{4N}\,\pi\right)} \,,\quad P_N \in B_N^\infty.$$

Boas’s formula (1937) gives, for $f \in B_\sigma^p$,

$$f'(x) = \frac{4 \sigma}{\pi^2} \sum_{k\in\mathbb{Z}} \frac{(-1)^{k-1}}{(2k-1)^2} \, f\left(x + \frac{\pi}{\sigma}(k-\tfrac{1}{2})\right),$$

with convergence in $L^p(\mathbb{R})$ or uniformly for $p=\infty$ (Pesenson, 24 Dec 2025).

2. Abstract Operator Extensions in Banach Spaces

Given a complex Banach space $E$ and a one-parameter uniformly bounded $C_0$-group $\{e^{tD}\}_{t\in\mathbb{R}}$ of bounded linear operators (with generator $D$), the Bernstein vectors are defined by

$$\mathbf{B}_\sigma(D) = \left\{ f \in \bigcap_{k\geq0} \mathrm{Dom}(D^k) : \|D^k f\| \leq \sigma^k \|f\| \ \forall k \right\}.$$

For such $f$, the trajectory $t\mapsto e^{tD} f$ extends to an $E$-valued entire function of exponential type $\le \sigma$. In this abstract framework, the Boas-type operator interpolation formulas read:

$D^r f = \mathcal{B}_D^{(r)}(\sigma) f,$

where, for odd and even orders, respectively: \begin{align*} \mathcal{B}D^{(2m-1)}(\sigma) &= \left(\frac{\sigma}{\pi}\right)^{2m-1} \sum{k \in \mathbb{Z}} (-1)^{k+1} A_{m,k} e^{\frac{\pi}{\sigma}(k-\frac{1}{2}) D} f, \ \mathcal{B}D^{(2m)}(\sigma) &= \left(\frac{\sigma}{\pi}\right)^{2m} \sum{k \in \mathbb{Z}} (-1)^{k+1} B_{m,k} e^{\frac{\pi k}{\sigma} D} f, \end{align*} with universal coefficients $A_{m,k}, B_{m,k}$ as in the classical formulas, ensuring norm convergence in $E$ (Pesenson, 24 Dec 2025, Pesenson, 2013).

Equivalence Theorems

Under isometry hypotheses for $e^{tD}$, the following conditions are equivalent for $f \in E$ (Pesenson, 2013):

1. $f$ is a Bernstein vector: $f \in B_\omega(D)$.
2. The trajectory $t \mapsto e^{tD} f$ is entire of exponential type $\omega$ and bounded on $\mathbb{R}$.
3. The Boas-type interpolation formulas (above) hold.

3. Generalizations: Mellin and Discrete Settings

Mellin Analysis

In Lebesgue-Mellin spaces $X_p(\mathbb{R}^+)$, consider the Mellin translation group $(U(t)f)(x) = f(e^t x)$ with generator $O f(x) = x \frac{d}{dx} f(x)$. For $f \in B^p_\omega(O)$, the Mellin analogues of the Riesz–Boas interpolation formulas are (Pesenson, 2021): \begin{align*} O^{2m-1} f(x) &= \pi^{2m-1} \sum_{k\in\mathbb{Z}} (-1)^{k+1} A_{m,k} f(e^{(k - \tfrac{1}{2}) x}), \ O^{2m} f(x) &= \pi^{2m} \sum_{k\in\mathbb{Z}} (-1)^{k+1} B_{m,k} f(e^{k x}), \end{align*} with absolute and uniform convergence on compact subsets and applicable in numerical Mellin inversion and scale-invariant signal processing.

Discrete Hilbert and Kak-Hilbert Transforms

For the discrete Hilbert transform $H$ on $\ell^2(\mathbb{Z})$ with its one-parameter group $e^{tH}$,

$$H^{2m-1} a = \sum_{k\in\mathbb{Z}} A_{m,k} e^{(k-\tfrac{1}{2})H} a, \quad H^{2m} a = \sum_{k\in\mathbb{Z}} B_{m,k} e^{kH} a,$$

where coefficients $A_{m,k}, B_{m,k}$ involve derivatives of the sinc function at translated nodes (Pesenson, 2021). The Kak-Hilbert transform $K$ provides analogous series expansions.

Setting	Operator ($D$)	Sampling grid	Interpolation formula
Classical	$\tfrac{d}{dx}$	$x + \frac{\pi}{\sigma}(k-\tfrac{1}{2})$	Boas (infinite series) for $f'$, Riesz (finite sum) for $P'$
Banach space	generator $D$	$e^{tD}$ trajectories	Boas-type operator formulas as above
Mellin	$O = x \tfrac{d}{dx}$	$e^{(k-\tfrac{1}{2})x}$	Mellin Boas-type formulas
Discrete ($\ell^2$)	Hilbert, Kak-Hilbert	$e^{(k-\tfrac{1}{2})H}$	Series in powers of $H$ or $K$

4. Relation to Sampling Theorems

Shannon-type sampling reconstructs $f$ from its point-samples. In contrast, Riesz and Boas formulas reconstruct $f^{(m)}$ (derivatives) from linear combinations of $f$ at equally spaced nodes—critical for derivative estimation and function approximation in signal processing. Sampling formulas and interpolation formulas are complementary: the former recover the function, the latter its derivatives (Pesenson, 24 Dec 2025).

In abstract Banach spaces, sampling theorems for $e^{tD}f$ trajectories also appear:

$$\langle e^{zD} f, g \rangle = \sum_{k\in\mathbb{Z}} \langle e^{\gamma k D} f, g \rangle \mathrm{sinc} \Bigl(\frac{z}{\gamma} - k \Bigr),\quad z \in \mathbb{C},$$

for $0 < \gamma < 1,$ $g \in E^*$ (Pesenson, 24 Dec 2025).

5. Applications to Manifolds, Lie Groups, and Spectral Analysis

On compact homogeneous manifolds $M = G/K$, the infinitesimal generators $D_j$ of Lie algebra elements $X_j$ induce isometric $C_0$-groups on $L^p(M)$. For $D = \sum_{j=1}^d a_j D_j$ and $f$ in the Bernstein subspace $B_\omega(D)$, Boas-type operator formulas provide spectral reconstruction and derivative sampling:

$$D^r f = B^{(r)}(\omega) f,\qquad r \in \mathbb{N} \quad [1311.5995].$$

This framework generalizes to the Heisenberg group and the Schrödinger representation in $L^2(\mathbb{R})$ with generators $D = \frac{1}{i} \frac{d}{dx}, X = x \cdot$, further extending to combinations $pD+qX$:

$$(pD + qX)^r f = B^{(r)}(\omega) f, \quad r \in \mathbb{N}.$$

6. Convergence, Error Estimates, and Universality of Coefficients

All Boas-type infinite series converge absolutely (in $L^p$, $X_p$, $\ell^2$ norm) and uniformly on compact sets (Pesenson, 2021, Pesenson, 2021). For numerical implementation, truncation at $|k| \leq N$ yields errors of $O(N^{-2})$, uniformly bounded on compacts (Pesenson, 2021). The universal coefficients $A_{m,k}, B_{m,k}$ in all settings—classical, abstract, Mellin, discrete—arise independently of the operator, provided the underlying group is isometric (Pesenson, 2013).

7. Broader Impact and Theoretical Significance

Riesz and Boas interpolation formulas, as extended and unified in operator-theoretic contexts, establish a template for reconstructing derivatives or operator powers via sampled function values or orbits. Their abstract generalizations under $C_0$-group actions underpin spectral analysis on manifolds and Lie groups, inform finite/infinite difference schemes in numerical approximation, and facilitate sampling in non-classical domains such as Mellin analysis and discrete transforms. The framework is robust against changes in underlying space, operator, or domain, contingent only on analytic and boundedness properties of the associated trajectories (Pesenson, 24 Dec 2025, Pesenson, 2013, Pesenson, 2021, Pesenson, 2021).