# GOALS-JWST: Resolving the Circumnuclear Gas Dynamics in NGC 7469 in the Mid-
Infrared

Vivian U Department of Physics and Astronomy, 4129 Frederick Reines Hall,
University of California, Irvine, CA 92697, USA  Thomas Lai IPAC, California
Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125  Marina
Bianchin Universidade Federal de Santa Maria, Departamento de Física, Centro
de Ciências Naturais e Exatas, 97105-900, Santa Maria, RS, Brazil  Department
of Physics and Astronomy, 4129 Frederick Reines Hall, University of
California, Irvine, CA 92697, USA  Raymond P. Remigio Department of Physics
and Astronomy, 4129 Frederick Reines Hall, University of California, Irvine,
CA 92697, USA  Lee Armus IPAC, California Institute of Technology, 1200 E.
California Blvd., Pasadena, CA 91125  Kirsten L. Larson AURA for the European
Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Tanio Díaz-Santos Institute of Astrophysics,
Foundation for Research and Technology-Hellas (FORTH), Heraklion, 70013,
Greece  School of Sciences, European University Cyprus, Diogenes street,
Engomi, 1516 Nicosia, Cyprus  Aaron Evans National Radio Astronomy
Observatory, 520 Edgemont Rd, Charlottesville, VA, 22903, USA  Department of
Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA
22903, USA  Sabrina Stierwalt Physics Department, 1600 Campus Road, Occidental
College, Los Angeles, CA 90041, USA  David R. Law Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Matthew A. Malkan
Department of Physics & Astronomy, 430 Portola Plaza, University of
California, Los Angeles, CA 90095, USA  Sean Linden Department of Astronomy,
University of Massachusetts at Amherst, Amherst, MA 01003, USA  Yiqing Song
Department of Astronomy, University of Virginia, 530 McCormick Road,
Charlottesville, VA 22903, USA  National Radio Astronomy Observatory, 520
Edgemont Rd, Charlottesville, VA, 22903, USA  European Southern Observatory,
Alonso de Cordova 3107, Santiago, Chile  Paul P. van der Werf Leiden
Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
Tianmu Gao Department of Astronomy, Beijing Normal University, Beijing 100875,
China  George C. Privon National Radio Astronomy Observatory, 520 Edgemont Rd,
Charlottesville, VA, 22903, USA  Department of Astronomy, University of
Florida, P.O. Box 112055, Gainesville, FL 32611, USA  Anne M. Medling
Department of Physics & Astronomy and Ritter Astrophysical Research Center,
University of Toledo, Toledo, OH 43606,USA  ARC Centre of Excellence for All
Sky Astrophysics in 3 Dimensions (ASTRO 3D); Australia  Loreto Barcos-Muñoz
National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville, VA,
22903, USA  Department of Astronomy, University of Virginia, 530 McCormick
Road, Charlottesville, VA 22903, USA  Christopher C. Hayward Center for
Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY
10010, USA  Hanae Inami Hiroshima Astrophysical Science Center, Hiroshima
University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
Jeff Rich The Observatories of the Carnegie Institution for Science, 813 Santa
Barbara Street, Pasadena, CA 91101  Susanne Aalto Department of Space, Earth
and Environment, Chalmers University of Technology, 412 96 Gothenburg, Sweden
Philip Appleton IPAC, California Institute of Technology, 1200 E. California
Blvd., Pasadena, CA 91125  Thomas Bohn Hiroshima Astrophysical Science Center,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526,
Japan  Torsten Böker European Space Agency, Space Telescope Science Institute,
Baltimore, MD 21218, USA  Michael J. I. Brown School of Physics and Astronomy,
Monash University, Clayton, VIC 3800, Australia  Vassilis Charmandaris
Department of Physics, University of Crete, Heraklion, 71003, Greece
Institute of Astrophysics, Foundation for Research and Technology-Hellas
(FORTH), Heraklion, 70013, Greece  School of Sciences, European University
Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus  Luke Finnerty
Department of Physics & Astronomy, 430 Portola Plaza, University of
California, Los Angeles, CA 90095, USA  Justin Howell IPAC, California
Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125  Kazushi
Iwasawa Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona
(IEEC-UB), Martí i Franquès, 1, 08028 Barcelona, Spain  ICREA, Pg. Lluís
Companys 23, 08010 Barcelona, Spain  Francisca Kemper Institut de Ciencies de
l’Espai (ICE, CSIC), Can Magrans, s/n, 08193 Bellaterra, Barcelona, Spain
ICREA, Pg. Lluís Companys 23, Barcelona, Spain  Institut d’Estudis Espacials
de Catalunya (IEEC), E-08034 Barcelona, Spain  Jason Marshall 4Glendale
Community College, 1500 N. Verdugo Rd., Glendale, CA 91208  Joseph M.
Mazzarella IPAC, California Institute of Technology, 1200 E. California Blvd.,
Pasadena, CA 91125  Jed McKinney Department of Astronomy, University of
Massachusetts, Amherst, MA 01003, USA.  Department of Astronomy, The
University of Texas at Austin, 2515 Speedway Blvd Stop C1400, Austin, TX
78712, USA  Francisco Muller-Sanchez Department of Physics and Materials
Science, The University of Memphis, 3720 Alumni Avenue, Memphis, TN 38152, USA
Eric J. Murphy National Radio Astronomy Observatory, 520 Edgemont Road,
Charlottesville, VA 22903, USA  David Sanders Institute for Astronomy,
University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822  Jason Surace
IPAC, California Institute of Technology, 1200 E. California Blvd., Pasadena,
CA 91125

(Received September 2, 2022; Revised September 27, 2022)

###### Abstract

The nearby, luminous infrared galaxy (LIRG) NGC 7469 hosts a Seyfert nucleus
with a circumnuclear star-forming ring and is thus the ideal local laboratory
for investigating the starburst–AGN connection in detail. We present integral-
field observations of the central 1.3 kpc region in NGC 7469 obtained with the
_JWST_ Mid-InfraRed Instrument. Molecular and ionized gas distributions and
kinematics at a resolution of $\sim$100 pc over the $4.9-7.6\mu$m region are
examined to study gas dynamics influenced by the central AGN. The low-
ionization [Fe II] $\lambda$5.34$\mu$m and [Ar II] $\lambda$6.99$\mu$m lines
are bright on the nucleus and in the starburst ring, as opposed to H2 S(5)
$\lambda$6.91$\mu$m which is strongly peaked at the center and surrounding
ISM. The high-ionization [Mg V] line is resolved and shows a broad,
blueshifted component associated with the outflow. It has a nearly face-on
geometry that is strongly peaked on the nucleus, where it reaches a maximum
velocity of $-$650 km s-1, and extends about 400 pc to the East. Regions of
enhanced velocity dispersion in H2 and [Fe II] $\sim$180 pc from the AGN that
also show high $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ and $L$([Fe
II])/$L$(Pf$\alpha$) ratios to the W and N of the nucleus pinpoint regions
where the ionized outflow is depositing energy, via shocks, into the dense
interstellar medium between the nucleus and the starburst ring. These resolved
mid-infrared observations of the nuclear gas dynamics demonstrate the power of
_JWST_ and its high-sensitivity integral-field spectroscopic capability to
resolve feedback processes around supermassive black holes in the dusty cores
of nearby LIRGs.

Luminous infrared galaxies (946) — Galaxy nuclei (609) — Active galactic
nuclei (16) — Galactic winds (572) — Seyfert galaxies (1447)

††journal: ApJL††facilities: _JWST_ (NIRCam and MIRI)††software: astropy
(Astropy Collaboration et al., 2013, 2018), Cosmology calculator (Wright,
2006), _JWST_ Science Calibration Pipeline (Bushouse et al., 2022), ifscube
(Ruschel-Dutra & Dall’Agnol De Oliveira, 2020; Ruschel-Dutra et al., 2021),
BADASS (Sexton et al., 2021), CAFE (Marshall et al., 2007, Dìaz-Santos et al.
2022, in preparation), JDAVis (Lim et al., 2022), PAfit (Krajnović et al.,
2006), pymccorelation (Curran, 2014; Privon et al., 2020)

##  1 Introduction

Accreting supermassive black holes (SMBHs) within active galactic nuclei
(AGNs) are thought to play a prominent role in influencing the interstellar
medium (ISM) of their host through feedback mechanisms such as outflows (see
reviews by Veilleux et al., 2020; Armus et al., 2020). Distinguishing the
drivers of such winds and determining their direct impacts on surrounding star
formation has been difficult due to observational challenges such as heavy
dust obscuration. Investigating the triggering of AGN outflows is most ideally
done close to the launching site which is often obscured by dust in the case
of many Seyfert and luminous infrared galaxies (LIRGs; $L_{\mathrm{IR}}\geq
10^{11}L_{\odot}$), with the latter exhibiting extinction upward of $A_{V}\sim
40$ mag or higher (Mattila et al., 2007; Väisänen et al., 2017; U et al.,
2019; Falstad et al., 2021; Pérez-Torres et al., 2021).

NGC 7469 offers a prime opportunity to study these phenomena in detail. It is
a nearby ($z=0.01627$; $D_{L}=$ 70.6 Mpc)111NED LIRG hosting a bright Seyfert
1.5 nucleus (Landt et al., 2008) surrounded by a starburst ring
($R_{\mathrm{peak}}\sim$ 530 pc; Genzel et al., 1995; Song et al., 2021) with
a bimodal age distribution of stellar populations (Díaz-Santos et al., 2007).
Evidence of a wide-angle E-W biconically illuminated outflow in the near-
infrared coronal line [Si VI] $\lambda$1.96$\mu$m (R[SiVI] = 90 pc) has been
presented using VLT/SINFONI integral field spectroscopy (IFS) (FWHM =
0$\farcs$14, or 46 pc; Müller-Sánchez et al., 2011). The characteristic
outflow structure with broad and blueshifted components along the minor axis
of the galactic disk was not seen in the coarser optical GTC/MEGARA IFS
observations (FWHM = 0$\farcs$93, or 307 pc; Cazzoli et al., 2020). However,
turbulent, non-circular kinematics, detected in a component of the
H$\alpha$-[N II] complex within the central 610 pc, might be associated with
it. More recently, extended circumnuclear outflows were found reaching 531 pc
from the AGN in VLT/MUSE observations of [O III] $\lambda$5007Å (FWHM =
1$\farcs$23, or 406 pc; Robleto-Orús et al., 2021; Xu & Wang, 2022). ALMA
observations of CO($1-0$), CO($2-1$), and [C I] in the inner $\sim$ 2 kpc
region with angular resolutions of 0$\farcs$35 (120 pc) show largely
rotational kinematics (Izumi et al., 2020; Nguyen et al., 2021). Despite the
multiwavelength effort, there is still no clear picture for how the putative
outflow interacts with the circumnuclear gas and starburst ring.

With _JWST_ ’s 6.5-meter mirror and advanced instrument suite, we can now
resolve the dynamics of gas and dust at angular resolution of $\sim$
0$\farcs$2$-$0$\farcs$8 in the mid-infrared wavelengths (5$-$28$\mu$m). In
particular, this _Letter_ investigates the detailed gas kinematics in the
inner 600-pc region (the “inner ISM region” hereafter) of NGC 7469. We first
describe the observations and data processing in Section 2. Section 3
highlights the moment maps of several key ionized and molecular gas emission
lines and demonstrates the richness of features in the mid-infrared spectra
extracted from the inner ISM region. In Section 4, we discuss the properties
of the detected outflow and the shock excitation in the ISM. In several
companion papers, we investigate the infrared spectral properties of the AGN
(Armus et al. 2022, in preparation), the starburst ring (Lai et al., 2022),
and the circumnuclear star-forming regions (Bohn et al., 2022) in NGC 7469.

Throughout the paper, $H_{0}=70$ km s-1 Mpc-1, $\Omega_{\rm m}$ = 0.30, and
$\Omega_{\rm vac}$ = 0.70 have been adopted. At the redshift of $z=0.01627$,
1″= 330 pc (Wright, 2006).

##  2 Observations and Data Reduction

Mid-infrared IFS observations of NGC 7469 were taken with the _JWST_ Mid-
InfraRed Instrument (MIRI; Rieke et al., 2015; Labiano et al., 2021) in Medium
Resolution Spectroscopy (MRS) mode on 2022 July 3-4 UT as part of the Early
Release Science (ERS) Program 1328 (Co-PIs: L. Armus and A. Evans). The
observations covered the full 4.9$-$28.8$\mu$m range using the short (A),
medium (B), and long (C) sub-bands in all four channels. We adopted the FASTR1
readout pattern to optimize the dynamic range expected in the observations.
Using the extended source 4-pt dither pattern, the science exposure time per
sub-band was 444 seconds. Because our source is extended, we linked the
observation to a dedicated background with the same observational parameters
in all three grating settings.

Uncalibrated science and background observations were downloaded using the
MAST Portal and processed with the _JWST_ Science Calibration Pipeline
(Bushouse et al., 2022) version 1.6$+$ in batch mode. The Detector1 pipeline
applies detector-level corrections and ramp fitting to the individual
exposures. The output rate images were subsequently processed outside the
_JWST_ pipeline to flag newly-acquired bad pixels and additional cosmic ray
artifacts, and to remove vertical stripes and zeropoint residuals remaining
after the pipeline dark subtraction. These additional corrections broadly
follow the steps taken for _JWST_ ERO observations as described by Pontoppidan
et al. (2022). The resulting rate files are then processed with the _JWST_
Spec2 pipeline for distortion and wavelength calibration, flux calibration,
and other 2d detector level steps. Residual fringe corrections using prototype
pipeline code have been applied to both the Stage 2 products and to the 1-D
spectra resulting from Stage 3 processing.

Stage 3 processing (Spec3) performs background subtraction before combining
data from multiple exposures into the final data cubes. Background light is
subtracted from the 2-D science images using a master background frame
generated from our associated background observations. The master background
is a 1-D median sigma-clipped spectrum calculated over the field-of-view (FOV)
of the background observations and projected to the entire 2-D detector array.
Since it is a combination of many detector pixels, it does not degrade the
signal-to-noise ratio (SNR) the way a pixel-by-pixel background subtraction
would. The cube-building step in Spec3 assembles a single 3-D data cube from
all of the individual 2-D calibrated detector images, combining data across
individual wavelength bands and channels.

NIRCam’s broadband F150W, F200W, and F444W images of NGC 7469 from our
ERS-1328 Program have been included in this paper for visualization purposes.
Readers are referred to Bohn et al. (2022) for the data processing details.

##  3 Analysis and Results

###  3.1 Emission Line Maps

In this paper, we focus our investigation on the Channel 1 MRS data
(4.89$\mu$m to 7.66$\mu$m), which hosts key diagnostic lines for shock
excitation ([Fe II] and H2), strength of the radiation field ([Ar II]), and
coronal region ([Mg V]) at the highest spatial resolution afforded by MRS at
0$\farcs$3\. Known issues with MRS wavelength solutions (at the time of this
writing) are generally not a concern for Ch 1, where zeropoint variations at
the 0.001$\mu$m level ($\sim$ 60 km s-1) are within the spectral noise at
these wavelengths. We fit single Gaussian profiles to three of the brightest
ionized and molecular lines in Ch 1 ([Fe II] $\lambda$5.34$\mu$m, H2 S(5)
$\lambda$6.91$\mu$m, and [Ar II] $\lambda$6.99$\mu$m) on a spaxel-by-spaxel
basis and computed their corresponding flux, velocity, and velocity dispersion
maps using ifscube (Ruschel-Dutra & Dall’Agnol De Oliveira, 2020; Ruschel-
Dutra et al., 2021). We correct our linewidth measurements for instrumental
broadening ($\sigma_{\mathrm{inst}}$ = 36.5 km s-1) by subtracting it in
quadrature.

As shown in Figure 1, the morphologies of the emission lines differ among
these molecular and low-ionization potential (IP) gases: [Fe II] (IP = 7.9 eV)
and [Ar II] (IP = 15.8 eV) are bright and compact at the nucleus and at
several star-forming clumps in the ring. The starburst ring appears slightly
asymmetric around the central nucleus, where the southeast (SE) inner edge is
closer to the AGN. In contrast, the H2 dominates and appears extended at the
center but relatively weak in the ring. The nuclear H2 emission is
$\sim$1$\farcs$4 across, nearly 5 times larger than the FWHM of the point
spread function (PSF) of 0$\farcs$3 at 7.2$\mu$m continuum. Filamentary
structures extend in the NE and SW directions from the central emission that
appear to be well aligned with high surface brightness CO gas (see Figure 2 in
Izumi et al., 2020) at the sites of the innermost spiral arms.

Figure 1: The distribution of flux (top; in log erg s-1 cm-2 pixel-1),
velocity (middle; in km s-1), and velocity dispersion (bottom; in km s-1) for
several bright emission lines in Ch1: [Fe II] $\lambda$5.34$\mu$m (left), H2
S(5) $\lambda$6.91$\mu$m (middle), [Ar II] $\lambda$6.99$\mu$m (right).
Contours based on the flux distribution are overlaid on the kinematic maps of
the respective gas species. The black cross marks the peak of the H2 emission.
North is up and East is to the left. The distributions of the molecular and
ionized gas emission are strikingly different, with H2 being preferentially
bright at the center while [Fe II] and [Ar II] appear clumpy at the star-
forming ring. H2 and [Fe II] exhibit enhanced dispersion $\sim$ 180 pc off the
nucleus.

The gas kinematics for these lines are largely similar, sharing a bulk
rotational motion around the ring. The kinematic major axis has a position
angle (PA) of 126° as measured using PaFit (Krajnović et al., 2006). The
morphological semi-major axis (2$\farcs$1) and semi-minor axis (1$\farcs$7) of
the ring gives an inclination angle $i=$ 51°. The velocity field is
reminiscent of a tilted rotating ring with the near (far) side in the E/NE
(W/SW). We compute the contribution to the velocity dispersion due to rotation
and beam smearing following the technique from Swinbank et al. (2012); De
Breuck et al. (2014): we measure the luminosity-weighted velocity gradient
across the FWHM of the beam at each spaxel and subtracted it from the velocity
dispersion in quadrature. For all these gases, the median effect was at the 6%
level ($\sim$ 4 km s-1), with 75% of the spaxels affected under 10% ($\sim$ 7
km s-1) and 95% under 20% ($\sim$ 14 km s-1).

The H2 velocity dispersion is slightly suppressed at the center (95 km s-1, or
75 km s-1 intrinsically) but increases toward the NW direction in a cone
shape, similar to the shocked H2 $\lambda$2.12$\mu$m detected in Mrk 273 (U et
al., 2013). The [Fe II] gas, sensitive to strong shocks, shows high velocity
dispersions (reaching as high as 135 km s-1) roughly along the kinematic axis
toward both NW and SE.

###  3.2 Spectral Fits

To quantify the ISM conditions surrounding the AGN, we divide the inner
1$\farcs$8 (600 pc) ISM region within the ring as seen at 7.1$\mu$m, the
observed wavelength of [Ar II], into a 3 $\times$ 3 grid and subsequently
extract 1-D spectra from these bins. Since the central region contains a point
source from the AGN, we apply a wavelength-dependent aperture correction to
the central spectrum. The aperture correction is calculated using the MRS PSF
models from WebbPSF (Perrin et al., 2014) that have been adjusted to match
preliminary in-flight performance during _JWST_ commissioning (private
communication). These models are nearly diffraction limited longward of
8$\mu$m and moderately elliptical at shorter wavelengths. Figure 2 shows the
extraction regions (0$\farcs$6 on a side) and their corresponding spectra
normalized at [Ar II]. The stitched Ch 1 short-medium-long spectra exhibit a
number of strong molecular and fine structure lines, as well as polycyclic
aromatic hydrocarbon (PAH) features. Several of these emission lines were
previously seen with _Spitzer_ /IRS, but with a large aperture that included
the starburst ring and the AGN (Stierwalt et al., 2013; Inami et al., 2013).
The relatively poor spatial resolution of _Spitzer_ precluded studies of the
variations in the strengths and profile shapes of these lines with positions
on these scales, now made possible with _JWST_. Detailed discussions of the
nuclear AGN spectrum and of the dust grains in the star-forming ring are
presented in two companion papers (Armus et al. 2022, in preparation, Lai et
al., 2022, respectively).

Figure 2: (Top left) Extraction grid (cyan) overlaid on the Channel 1 flux
image at 7.1$\mu$m. Each extraction region is 0$\farcs$6 on a side, capturing
the inner ISM region in this 3$\times$3 grid. The PSF FWHM is marked with a
filled circle in the bottom left corner. (Bottom) The Ch 1 short-medium-long
stitched spectra extracted from the grid, each labeled by its direction
relative to the center ‘cen’. Spectral features are labeled; the spectra are
normalized at [Ar II] $\lambda$6.99$\mu$m for ease of comparison. A box around
coronal line [Mg V] $\lambda$5.61$\mu$m is drawn. (Inset) A zoomed-in view of
the spectral region centered on [Mg V]. A vertical line at 5.609$\mu$m as
defined based on the peak of [Mg V] at the central grid indicates the systemic
velocity. [Mg V] is present in most of the inner ISM regions (except for NW
and SW); its blueshifted flux is most pronounced in the E.

A handful of high-ionization lines ([Fe VIII] $\lambda$5.48$\mu$m with IP =
124 eV, [Mg VII] $\lambda$5.50$\mu$m with IP = 186 eV, and [Mg V]
$\lambda$5.61$\mu$m with IP = 109 eV) show clear blue-shifted wings. The
coronal line [Mg V] is the strongest among these features in all the 1-D
spectra. Given their high IPs, photo-ionized coronal lines are typically
produced on several hundred parsecs away from the AGN (Ferguson et al., 1997).
Kinematics of coronal lines often reveal blueshifts indicating that outflows
on these scales are common (e.g. Riffel et al., 2021). While strong coronal
lines might be expected at the nucleus, broad and blueshifted components may
be present in the immediate vicinity of the nucleus if an outflow is present.
Indeed, a close look at our extracted spectra reveals that [Mg V] is most
significantly shifted in the E, 70 km s-1 with respect to the center (Figure 2
inset). The blueshifted asymmetric profile of [Mg V] can be seen in several
other directions as well.

To further investigate the distribution and kinematics of [Mg V], we fit two
Gaussian components using a modified version of the Bayesian AGN Decomposition
Analysis for SDSS Spectra software package (BADASS; Sexton et al., 2021)
spaxel-by-spaxel and generate moment maps for the two components; see Appendix
A. The core, narrow component of [Mg V] is plausibly consistent with rotation
at the same PA as that for the low-ionization and molecular gases. A broad
component is identified in a subset of the central spaxels, encompassing the
blueshifted outflowing gas. The outflow is detected up to a projected distance
of $\sim$420 pc E of the nucleus. Within this physical extent, the gas
exhibits a median line-of-sight velocity of $v_{\mathrm{med}}=-205$ km s-1 but
reaching as high as $-650$ km s-1 close to the AGN. This outflow is also
identified in other high-ionization lines such as [Fe VIII]
$\lambda$5.45$\mu$m (IP = 124 eV) and [Ne VI] $\lambda$7.65$\mu$m (IP = 126
eV), and mid-ionization lines such as [O IV] $\lambda$25.9$\mu$m (IP = 55 eV),
the detailed line profiles for which will be characterized in Armus et al.
2022, in preparation. We do find that the high ionization lines in the nuclear
spectrum exhibit extreme blueshifted velocities upward of 1000 km s-1,
indicating the presence of very fast winds at the center.

Multi-component spectral fitting is performed on the full Ch 1 spectral
coverage of each 1-D spectrum using the Continuum And Feature Extraction
(CAFE; Marshall et al., 2007, Dìaz-Santos et al. 2022, in preparation)
software package originally written in IDL for analyzing _Spitzer_ /IRS data.
While individual atomic or molecular lines can be fit using single or double
Gaussian profiles, CAFE specializes in decomposing the emission in the mid-
infrared regime into AGN, PAHs, dust of different temperatures, and starburst
components, and is ideal for recovering properties of PAH features in 1-D
spectra that often require multiple components to fit correctly. The
development of an improved version of the CAFE code that handles high-
resolution spectral decomposition is fully described in Dìaz Santos et al.
2022, in preparation). Major updates include a fully-functional Python version
applicable to the data format and spectral resolution of _JWST_ IFS data. The
flux densities for the various relevant emission-line features are reported in
Table 1.

Table 1: Emission Line Flux Densities in MRS Channel 1

Location | H2 S(5) | H2 S(6) | H2 S(7) | PAH 6.2$\mu$m | [Fe II] $\lambda$5.34$\mu$m | Pf$\alpha$
---|---|---|---|---|---|---
E | 1.97 $\pm$ 0.05 | 0.42 $\pm$ 0.05 | 0.90 $\pm$ 0.08 | 83.82 $\pm$ 6.49 | 1.28 $\pm$ 0.06 | 0.25 $\pm$ 0.03
N | 3.15 $\pm$ 0.09 | 0.67 $\pm$ 0.05 | 1.99 $\pm$ 0.12 | 93.40 $\pm$ 3.48 | 1.73 $\pm$ 0.09 | 0.28 $\pm$ 0.03
NE | 1.76 $\pm$ 0.06 | 0.29 $\pm$ 0.06 | 0.81 $\pm$ 0.07 | 156.27 $\pm$ 4.84 | 1.95 $\pm$ 0.09 | 0.49 $\pm$ 0.03
cen | 3.70 $\pm$ 0.13 | 0.82 $\pm$ 0.10 | 1.95 $\pm$ 1.09 | 121.82 $\pm$ 32.01 | 2.00 $\pm$ 0.14 | …$\pm$ …
NW | 1.26 $\pm$ 0.06 | 0.24 $\pm$ 0.02 | 0.68 $\pm$ 0.06 | 103.58 $\pm$ 5.36 | 1.49 $\pm$ 0.08 | 0.36 $\pm$ 0.03
S | 4.88 $\pm$ 0.23 | 1.23 $\pm$ 0.19 | 2.50 $\pm$ 0.29 | 235.51 $\pm$ 7.74 | 4.21 $\pm$ 0.26 | 0.86 $\pm$ 0.15
SE | 1.52 $\pm$ 0.09 | 0.27 $\pm$ 0.07 | 0.61 $\pm$ 0.08 | 173.99 $\pm$ 5.47 | 2.79 $\pm$ 0.13 | 0.66 $\pm$ 0.09
SW | 4.46 $\pm$ 0.11 | 0.92 $\pm$ 0.14 | 2.30 $\pm$ 0.20 | 350.55 $\pm$ 7.97 | 5.73 $\pm$ 0.27 | 0.83 $\pm$ 0.11
W | 7.89 $\pm$ 0.18 | 1.75 $\pm$ 0.11 | 4.56 $\pm$ 0.22 | 178.10 $\pm$ 9.80 | 3.77 $\pm$ 0.22 | 0.53 $\pm$ 0.08

Note. — All flux densities are in units of 10-23 W m-2 pc-2. The aperture used
for the grid extraction is 0$\farcs$6 $\times$ 0$\farcs$6\. Pf$\alpha$ is not
well detected at the center and thus its measurement is omitted.

##  4 Discussion

###  4.1 Outflow Characteristics

As briefly introduced in Section 1, previous multiwavelength efforts in the
literature converged on the presence of non-rotational kinematics in the
circumnuclear region of NGC 7469, but the picture of how the non-rotating gas
behaved at different physical scales was incomplete due to mismatches in
observational parameters and data sensitivity (Müller-Sánchez et al., 2011;
Cazzoli et al., 2020; Robleto-Orús et al., 2021). With _JWST’s_ superb
sensitivity, spectral resolution, and integral-field capability in the mid-
infrared wavelengths, we can now incorporate the molecular, low- and high-
ionization gases into a coherent picture with the same data set on the same
physical scales. We see from Figure 2 that the high IP [Mg V] line is
prominently blueshifted, with a nearly face-on geometry and flux extending E
of the nucleus. The intensity of the broad, blueshifted component from Figure
A.2 (lower left panel) is overplotted on the NIRCAM images as contours in
Figure 3 (inset).

Figure 3: (Left) Three-color F150W/F200W/F444W NIRCam full-array image of NGC
7469 showing the large-scale spiral-arm structures. The inner core is
saturated and thus masked out. The overlaid box shows the subarray region.
(Inset) Three-color F150W/F200W/F444W NIRCam subarray image showing the star-
forming ring around the central nucleus. A 1″ (330 pc) scale bar is shown for
comparison. The black contours indicate the intensity of the blueshifted
component of the coronal line [Mg V], most of which is coming toward us from
the central AGN with a slight eastward extension. (Right) A cartoon schematic
showing a nearly face-on outflow that appears one-sided and mostly in
blueshift (blue cone). The inclined star-forming ring (near side in the N and
far side in the S) is rotating clockwise. Gas is shock-heated on the W side
representing the interaction region where the outflow plows through the ISM of
the galaxy. Components are not drawn to scale but merely to illustrate a
plausible scenario based on the observed dynamics of the outflow, the inner
ISM region, and the ring.

Given that the gas is photoionized and [Mg V] is tracing the illumination
pattern, we see primarily the blue-shifted outflowing gas coming toward us
from a nearly face-on galaxy with inclination angle of 45° (Davies et al.,
2004), which is consistent with our measured $i$ of 51° based on the ring
structure. The [Mg V] observations further support the picture in which the
circumnuclear obscuration is roughly in the same plane as the galaxy’s large-
scale disk.

To place our findings in the context of literature results, we consider the
scales of the outflows detected with different instruments and tracers.
Robleto-Orús et al. (2021) found an outflow as characterized by blueshifted [O
III] $\lambda$5007Å extending 531 pc from the nucleus. Resolved at similar
scales as our MIRI/MRS data, the [O III] outflow may be more extended than the
coronal wind we detect in [Mg V] because it is more easily excited (with IP =
35 eV) and more extincted at the center. On the other hand, the biconically-
illuminated [Si VI] $\lambda$1.96$\mu$m outflow presented by Müller-Sánchez et
al. (2011) has an E-W orientation, but at much smaller scale close to the AGN.
The entire [Si VI] emission (IP = 167 eV) falls within the central 1$-$2
spaxels of MIRI/MRS. The mid-infrared coronal outflow we find could have
physical connection with the X-ray warm absorber in this AGN in terms of
location and ionization condition (Blustin et al., 2007). Our results suggest
a scenario where the high-ionization outflow detected by _JWST_ likely bridges
the nuclear-scale ($<$ 100 pc) coronal line region outflow and the larger-
scale narrow line region winds. The outflow appears one-sided because it is
approaching us nearly face-on; the projected size of a potentially receding
red-side cone behind the AGN may be too small to be spatially resolved by MIRI
(Figure 3). Even though there may be a hint of a redshifted wing in the W
spectrum (Figure 2), the SNR is too weak in the individual spaxels for a
robust spectral fit. The redshifted wind may simply be obscured by dense,
intervening ISM. Alternatively, the outflow may also appear asymmetric if an
inclined jet is pushing on the gas in the disk (e.g. in NGC 1052; Fernández-
Ontiveros et al., 2019). While NGC 7469 may host a radio core–jet structure as
resolved by the Very Large Array (Orienti & Prieto, 2010), no evidence of
radio jet–ISM interaction has yet been found in its nuclear region (Xu & Wang,
2022).

###  4.2 ISM excitation

Our MIRI/MRS observations allow us, for the first time, to examine the high-
ionization outflow identified here in [Mg V] at $r<300$ pc in detail. We
search for evidence that indicates how the outflow may impact its
surroundings. The mid-infrared rotational transitions of H2, when coupled with
measurements of the PAH emission, provide a robust diagnostic of the
excitation conditions and help determine the mechanism responsible for
exciting the H2 lines (Higdon et al., 2006; Ogle et al., 2010; Guillard et
al., 2012; Cluver et al., 2013). Young massive stars may heat both PAHs and H2
in photodissociation regions (PDRs), but shocks arising from outflows or X-ray
emission from AGN effectively dissociate PAH molecules and small grains (Jones
et al., 1996). Empirically, low-luminosity star-forming galaxies exhibit a
limited range of luminosity ratio $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ over
several orders of magnitude in $L(\mathrm{H}_{2})$ while those for AGN display
line ratios $\sim$15 times greater (Roussel et al., 2007). Thus, the line
ratio $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ has been used to distinguish between
AGN, stars, and shocks as the driver of molecular emission in extreme
environments such as LIRGs (Stierwalt et al., 2014; U et al., 2019).

Ideally, the total power in the warm H2 and all forms of PAH lines should be
used in any analysis of PDR heating, given that the bulk of the molecular gas
mass may be traced by the lower rotational H2 transitions. However, to take
advantage of the high spatial resolution in MIRI Ch 1 and minimize
contamination in the inner ISM region from the nucleus and the ring, we opt to
confine our analysis of $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ to the lines
present in this channel, recognizing the potential limitations. In Ch 1 MRS
data cube (Figure 2), four H2 transitions are identified: S(5)
$\lambda$6.91$\mu$m is detected and appears prominent in all the extracted
spectra; S(6) $\lambda$6.11$\mu$m, while detected, sits on top of a prominent
PAH feature at 6.2$\mu$m and may be dominated by PAHs at certain locations;
S(7) $\lambda$5.51$\mu$m may be partially blended with [Mg VII]
$\lambda$5.5$\mu$m; and S(8) $\lambda$5.05$\mu$m is comparatively weak. For
these reasons, the S(5) line is the most optimal line among the possibilities.
Its proximity to PAH6.2$\mu$m reduces the risk of potential calibration
differences between sub-bands and differential extinction, so we focus on
using this pair of features for the $L(\mathrm{H}_{2})/L(\mathrm{PAH})$
diagnostic. We note that close to the AGN ($<$ 300 pc), the H2 gas is expected
to be quite hot ($\sim$ 900-1100K; Pereira-Santaella et al., 2022; Lambrides
et al., 2019) and is dominated by AGN heating in the central $\sim$100 pc
(Armus et al. 2022, in preparation), so the S(5) line likely carries much of
the H2 luminosity, justifying our reliance on this line to measure the
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ ratio.

Figure 4 (left) shows the $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ distribution
versus H2 luminosity density for the different locations within the inner ISM.
The first result is that the corner points (NW, NE, SE, and SW) all have lower
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ ratios than the other locations. Examining
the spectra in Figure 2, the PAH features are more prominent at the corner
locations that contain a larger fractional emission from the ring than at the
E-W or N-S locations. This is expected, since PAH emission mostly originates
from PDRs and, for some galaxy populations and/or environments, can be used as
star formation rate tracers (e.g. Peeters et al., 2004). In contrast, the H2
emission is relatively weaker in the ring, as seen in Figure 1, in agreement
with what is expected from actively star-forming regions.

Figure 4: (Left) $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ vs. H2 luminosity density
diagnostics diagram showing the data points corresponding to different parts
of the inner ISM region. The grey points are local LIRGs from Stierwalt et al.
(2014) for which these line transitions are detected and H2 luminosities
normalized by the extraction aperture; upper limits have been excluded. The
inner ISM in NGC 7469 tends to be bright in H2 but spans a range of
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ values nearly representative of the local
LIRGs. (Right) The same $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ line ratios
plotted against $L$([Fe II])/$L$(Pf$\alpha$). The correlation seen here
($\rho$ = 0.62${}^{+0.26}_{-0.40}$ with a $p$-value of 0.10 for all the
points; $\rho$ = 0.86${}^{+0.09}_{-0.19}$, with a $p$-value of 0.01 excluding
the SW outlier) suggests that shocks traced by [Fe II] are correlated with the
shocked H2 gas. Pf$\alpha$ is not well detected in the central spectrum and
thus the center point is excluded from this plot.

For comparison, we have plotted in Figure 4 the integrated _Spitzer_ /IRS data
points for a sample of local LIRGs (Stierwalt et al., 2014) from the Great
Observatories All-sky LIRGs Survey (GOALS; Armus et al., 2009). Because the
angular-size aperture (3$\farcs$7 $\times$ 9$\farcs$5) used to extract the
_Spitzer_ /IRS spectra is much larger than that of our extraction grid size,
we show the H2 luminosity surface density, $\Sigma_{H_{2}}$, in the x-axis
instead of luminosity, for a more direct comparison with the _Spitzer_ /IRS
values. Our data points span a broad range in
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ of the GOALS LIRGs, and are consistent
with the more H2 bright sources for a given
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ value. This shift may be due to the fact
that the global values from _Spitzer_ /IRS are uniformly weighted and
encompass many star-forming regions and obscured AGN sources, while our
resolved data points represent the inner ISM region close to the Seyfert
nucleus in NGC 7469 where H2 is prominently detected.

The inner ISM region does not host bright star-forming regions and therefore
PDR models are not applicable, but X-rays from the AGN and shocks from the
outflow can heat the molecular gas (Petric et al., 2018; Minsley et al.,
2020). Since iron is highly depleted onto grains, [Fe II] is a commonly
adopted tracer of shock-excited gas where grains have been processed by
outflows or other ionizing sources (see references in reviews by Sajina et
al., 2022; U, 2022). Infrared studies of supernova remnants indicate that not
only do they produce H2 lines as luminous as bright star-forming regions, they
are also associated with enhanced [Fe II]/Br$\gamma$ relative to typical H II
regions (Oliva et al., 1989). The latter observation has been predicted by
theoretical models of shock fronts (McKee et al., 1984), where the ratio of
[Fe II]-to-Hydrogen is deemed a useful indicator of the efficiency of the
shock front in destroying dust grains (Hollenbach & McKee, 1989). Here, we
examine the ratio of $L$([Fe II])/$L$(Pf$\alpha$) given that the H
recombination line Pfund $\alpha$ traces star formation, provides a
calibration baseline, and is detected in Ch 1.

Figure 4 (right) shows $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ plotted as a
function of $L$([Fe II])/$L$(Pf$\alpha$), which appears to be well correlated
with the exception of the SW region. We compute a correlation for all the data
points using pymccorrelation (Privon et al., 2020) with 104 bootstrapping
iterations to estimate the uncertainties. We determine the Pearson’s
correlation coefficient $\rho$ = 0.62${}^{+0.26}_{-0.40}$, with a $p$-value of
0.10 driven mostly by the outlier point at SW and the large uncertainties of
Pf$\alpha$. If we exclude SW (which shows the peakiest PAH6.2$\mu$m profile
and the most extreme slope on the red end of the Ch1 spectra where Pf$\alpha$
is located; see Figure 2), the resulting coefficient becomes $\rho$ =
0.86${}^{+0.09}_{-0.19}$, with a $p$-value of 0.01, suggesting that [Fe II]
and H2 at the W and N locations are likely excited by the same shocks driven
by the outflow. This correlation between [Fe II] and H2 in (U)LIRGs has been
studied by Hill & Zakamska (2014), who also inferred that shocks, including
AGN-driven ones, are the likely source of H2 heating.

The most extreme region, W, which partially overlaps the region of high H2 and
[Fe II] velocity dispersion (Figure 1), is, interestingly, on the opposite
side of the center from the blueshifted coronal gas. It is possible that the
H2 and [Fe II] are regions of high velocity dispersion represent regions where
the receding, partially obscured part of the outflow is plowing into the dense
ISM and releasing [Fe II] into the gas phase. A strong correlation between [Fe
II] and H2 may be associated with sites of supernova remnants, where [Fe II]
is produced in radiative shocks after grain destruction as the supernova
remnant propagates (Hill & Zakamska, 2014). Given the extremely high-velocity
coronal line components detected in the nuclear spectrum (Armus et al. 2022,
in preparation), the AGN itself is likely driving the highly-ionized wind
detected here in [Mg V], which deposits energy into the ISM via shocks in the
W and NW regions where the dispersion of the warm molecular gas and the
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ ratios are highest. It is somewhat
surprising, given the location of the high dispersion H2 gas, that the NW
spectrum is low in Figure 4 (left). However, this extraction region is large
and partially intersects the inner edge of the ring, where the H2 is weak and
the PAH is bright. Examining the individual pixels in this region indicates
the $L(\mathrm{H}_{2})/L(\mathrm{PAH})$ ratio changes by a factor of eight as
one moves from the upper right to lower left of this region. Therefore the
area of maximum dispersion in the NW would indeed be located much higher in
Figure 4, close to the position of region W, consistent with our simple model.
A more detailed analysis of these regions will be performed in a subsequent
paper using a spectral decomposition tool designed to produce accurate line
ratio maps on finer scales.

##  5 Summary

In this _Letter_ , we present new _JWST_ MIRI MRS observations of NGC 7469 and
focus on the analysis of the inner ISM region between the central AGN and the
starburst ring. The high spatial- and spectral-resolution available with the
new IFS capability of MIRI enables a detailed investigation of the rotational
H2, low- and high-ionization fine structure lines, and dust features at mid-
infrared wavelengths for the first time. We summarize our findings below.

  * •

The morphology of the low-ionization [Fe II] and [Ar II] lines are bright on
the nucleus and in the starburst ring, showing several star-forming clumps and
regions of enhanced emission. The H2, in contrast, is strongly peaked on the
nucleus and surrounding ISM, and is relatively weak in the starburst ring.

  * •

The [Mg V] emission line is resolved, and shows a broad, blueshifted component
that is likely associated with the coronal line outflow in NGC 7469. The
blueshifted highly-ionized outflow traced by [Mg V], has a nearly face-on
geometry that is strongly peaked on the nucleus, with an extension that
reaches about 400 pc to the East. The [Mg V] gas has a median line-of-sight
velocity $v_{\mathrm{med}}=-205$ km s-1 and reaches as high as $-650$ km s-1
close to the AGN.

  * •

There are regions of enhanced velocity dispersion in H2 and [Fe II] $\sim$180
pc from the AGN that also feature excited $L$([Fe II])/$L$(Pf$\alpha$) and
$L(\mathrm{H}_{2})/L(\mathrm{PAH})$ ratios, most clearly seen to the N and W.
We identify these regions as the locations where the outflow is depositing
energy into the dense interstellar gas via shocks, between the nucleus and the
starburst ring.

Such a detailed view of the mid-infrared gas dynamics within the central
region of a dusty LIRG nucleus is made possible for the first time, fully
demonstrating the high spatial- and spectral-resolution plus high-sensitivity
integral-field capability of _JWST_.

We thank Jim Braatz, Patrick Ogle, Andreea Petric, and the anonymous reviewer
for suggestions that significantly improved the paper. This work is based on
observations made with the NASA/ESA/CSA _JWST_. The research was supported by
NASA grant JWST-ERS-01328. The data were obtained from the Mikulski Archive
for Space Telescopes at the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5-03127 for JWST. These observations are associated
with program #1328 and can be accessed via
http://dx.doi.org/10.17909/0fe2-cf33 (catalog DOI:10.17909/0fe2-cf33). VU
acknowledges funding support from NASA Astrophysics Data Analysis Program
(ADAP) grant 80NSSC20K0450. The Flatiron Institute is supported by the Simons
Foundation. HI and TB acknowledge support from JSPS KAKENHI Grant Number
JP19K23462 and the Ito Foundation for Promotion of Science. AMM acknowledges
support from the National Science Foundation under Grant No. 2009416. ASE and
SL acknowledge support from NASA grant HST-GO15472. YS was funded in part by
the NSF through the Grote Reber Fellowship Program administered by Associated
Universities, Inc./National Radio Astronomy Observatory. SA gratefully
acknowledges support from an ERC Advanced Grant 789410, from the Swedish
Research Council and from the Knut and Alice Wallenberg (KAW) Foundation. KI
acknowledges support by the Spanish MCIN under grant
PID2019-105510GB-C33/AEI/10.13039/501100011033. F.M-S. acknowledges support
from NASA through ADAP award 80NSSC19K1096. Finally, this research has made
use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration.

##  Appendix A Fitting the Coronal Line

As mentioned in Section 2, we conduct an in-depth investigation of the coronal
line [Mg V] over the entire Ch1 FOV using the BADASS (Sexton et al., 2021)
software package. We fit one- and two-component Gaussian profiles to the line
spaxel-by-spaxel (e.g. Figure A.1) and generate their resulting flux,
velocity, and velocity dispersion maps (Figure A.2). The velocity dispersion
maps have been corrected for instrumental broadening
($\sigma_{\mathrm{inst}}=36.5$ km s-1). The flux map of the secondary broad
component of [Mg V] (bottom left panel of Figure A.2) is overplotted on NIRCam
images as contours in Figure 3 to indicate the high-ionization outflow.

We note that this two-component spectral fitting was only applied to [Mg V] on
a per-spaxel basis in this work to generate the moment maps. In a companion
paper (Armus et al. 2022, in preparation), multi-component fitting is applied
to all the emission lines in the full MIRI/MRS nuclear spectrum extracted from
a small aperture on the AGN.

Figure A.1: (Top) Example two-component Gaussian fit to the [Mg V] from one
spaxel at the center. Two Gaussian components — a main (green dashed) and a
broad (red dashed) — are needed to fit the “outflow” region of the map. The
residual resulting from subtracting Model (blue) from Data (black) is shown in
the bottom plot.  Figure A.2: Flux (left, in log 10-17 erg cm-2 s-1 px-1),
velocity (middle, in km s-1), and velocity dispersion (right, in km s-1) maps
of the core, narrow Gaussian component (top) and of the second, broad,
blueshifted component (bottom) of [Mg V]. The cross marks the location of the
central AGN defined at the continuum. The velocity dispersion maps have been
corrected for instrumental broadening. Note that the colorbar on these maps
are all different in order to optimize visualizing the range of values spanned
by each of the maps. North is up and East is to the left.

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# Complete interpolating sequences for the Gaussian shift-invariant space

Anton Baranov  Department of Mathematics and Mechanics
St. Petersburg State University
St. Petersburg, Russia<EMAIL_ADDRESS>,  Yurii Belov  Department
of Mathematics and Computer Science
St. Petersburg State University
St. Petersburg, Russia<EMAIL_ADDRESS>and  Karlheinz Gröchenig
Faculty of Mathematics
University of Vienna
Oskar-Morgenstern-Platz 1
A-1090 Vienna, Austria<EMAIL_ADDRESS>
###### Abstract.

We give a full description of complete interpolating sequences for the shift-
invariant space generated by the Gaussian. As a consequence, we rederive the
known density conditions for sampling and interpolation.

###### Key words and phrases:

Sampling, interpolation, Riesz bases, small Fock spaces, shift-invariant
space, Avdonin-type condition

###### 2000 Mathematics Subject Classification:

Primary 30H20; Secondary 30D10, 30E05, 42C15, 94A20

A. B. and Yu. B. were supported by the Ministry of Science and Higher
Education of the Russian Federation, agreement No 075-15-2021-602, and by the
Russian Foundation for Basic Research grant 20-51-14001-ANF-a. K. G. was
supported in part by the project P31887-N32 of the Austrian Science Fund (FWF)

##  1\. Main results

Consider the shift-invariant space of functions on $\mathbb{R}$ with Gaussian
generator $g(z)=e^{-az^{2}}$ for $a>0$ defined as

$V^{2}=\bigg{\\{}f(z)=\sum_{n\in\mathbb{Z}}c_{n}e^{-a(z-n)^{2}}:\
(c_{n})\in\ell^{2}(\mathbb{Z})\bigg{\\}}.$

We consider the space $V^{2}$ as a subspace of $L^{2}(\mathbb{R})$ with the
usual $L^{2}$-norm.

The space $V^{2}$ belongs to the general family of shift-invariant spaces.
Given a generator $g\in L^{2}(\mathbb{R})$, such a space is defined as

$V^{2}(g)=\\{f(z)=\sum_{n\in\mathbb{Z}}c_{n}g(z-n):\
(c_{n})\in\ell^{2}(\mathbb{Z})\\}\subseteq L^{2}(\mathbb{R}).$

The primary example is the classical Paley–Wiener space $PW=\\{f\in
L^{2}(\mathbb{R}):\mathrm{supp}\,\hat{f}\subseteq[-1/2,1/2]\\}$, which, by the
sampling theorem of Shannon–Whittaker–Kotelnikov, can be identified with the
shift-invariant space $V^{2}(\frac{\sin\pi x}{\pi x})$. In signal processing
[1] shift-invariant spaces are often taken as a substitute for the
Paley–Wiener space. A unifying feature of both $V^{2}$ (Gaussian generator)
and $PW$ (sinc-generator) is the fact that both spaces can be viewed as spaces
of entire functions by interpreting the variable $z$ to be in $\mathbb{C}$.

It is easy to see that the norm equivalence
$\|f\|_{L^{2}(\mathbb{R})}\asymp\|(c_{n})\|_{\ell^{2}(\mathbb{Z})}$ holds on
$V^{2}$ with some absolute constants. In what follows it will be convenient
for us to work with the second quantity and so we put
$\|f\|_{V^{2}}:=\|(c_{n})\|_{\ell^{2}(\mathbb{Z})}$, and we often identify
$V^{2}$ with $\ell^{2}(\mathbb{Z})$ via the mapping $(c_{n})\mapsto
f(z)=\sum_{n\in\mathbb{Z}}c_{n}e^{-(z-n)^{2}}$.

A sequence $\Lambda=\\{\lambda_{n}\\}_{n\in\mathbb{Z}}\subset\mathbb{R}$ is
said to be sampling for $V^{2}$, if

$\sum_{n\in\mathbb{Z}}|f(\lambda_{n})|^{2}\asymp\|f\|^{2}_{V^{2}},\qquad f\in
V^{2},$

and we say that $\Lambda$ is interpolating for $V^{2}$ if for every
$(a_{n})\in\ell^{2}(\mathbb{Z})$ there exists $f\in V^{2}$ such that
$f(\lambda_{n})=a_{n}$. If $\Lambda$ is both sampling and interpolating (or,
equivalently, the solution of the interpolation problem is unique), we say
that $\Lambda$ is a complete interpolating sequence.

Sampling and interpolation in the space $V^{2}$ and in more general shift-
invariant spaces were studied in [8, 9]. The main result can be formulated as
follows. Recall that $\Lambda$ is said to be separated if
$\inf_{\lambda,\lambda^{\prime}\in\Lambda,\lambda\neq\lambda^{\prime}}|\lambda-\lambda^{\prime}|>0$,
and denote by $D^{+}({\Lambda})$ and $D^{-}({\Lambda})$ the usual upper and
lower Beurling densities

$D^{+}(\Lambda)=\lim_{r\to\infty}\frac{\sup_{x\in\mathbb{R}}{\rm
card}\,(\Lambda\cap[x,x+r])}{r},\quad
D^{-}(\Lambda)=\lim_{r\to\infty}\frac{\inf_{x\in\mathbb{R}}{\rm
card}\,(\Lambda\cap[x,x+r])}{r}.$

######  Theorem 1.1.

  * (i)

Sufficiency: Every separated sequence $\Lambda\subset\mathbb{R}$ with
$D^{-}({\Lambda})>1$ is sampling for $V^{2}$. Moreover, it contains a complete
interpolating sequence.

  * (ii)

Necessity: If $\Lambda$ is sampling for $V^{2}$, then $\Lambda$ is a finite
union of separated subsequences and $\Lambda$ contains a separated sequence
$\widetilde{\Lambda}$ such that $D^{-}(\widetilde{\Lambda})\geq 1$.

######  Theorem 1.2.

  * (i)

Every separated sequence $\Lambda$ with $D^{+}({\Lambda})<1$ is a set of
interpolation for $V^{2}$. Moreover, it can be enlarged to a complete
interpolating sequence.

  * (ii)

If $\Lambda$ is a set of interpolation for $V^{2}$, then $\Lambda$ is
separated and $D^{+}({\Lambda})\leq 1$.

One of the main insights obtained in [8, 9] is the similarity of $V^{2}$ with
the Paley–Wiener space $PW$ with respect to sampling and interpolation. In
particular, for both $V^{2}$ and $PW$ the same density conditions hold. For
$PW$ these go back to Beurling, Kahane, and Landau [6, 14, 15, 18].

Theorems 1.1 and 1.2 provide an almost characterization of sampling sets and
of interpolating sets, but they leave open the case of critical density. If a
set is simultaneously sampling and interpolating, then $D(\Lambda)=1$. The
case of the critical density is much more subtle, because anything may happen.
For $PW$ the problem of complete interpolating sequences was solved in [12]
and [16], for $V^{2}$ it was open so far.

Our main result is a complete and explicit description of complete
interpolating sequences for $V^{2}$. Moreover, this characterization can be
extended to the larger class of complex Gaussians

$g_{c}(z)=e^{-cz^{2}},\qquad c=a+ib,\ \ a>0,\ \ b\in\mathbb{R}.$

We denote the space $V^{2}(g_{c})$ by $V_{c}^{2}$.

It is easy to see that for $f(z)=\sum_{n\in\mathbb{Z}}c_{n}e^{-c(z-n)^{2}}$ we
still have $\|f\|_{L^{2}(\mathbb{R})}\asymp\|(c_{n})\|_{\ell^{2}(\mathbb{Z})}$
and so we can define the norm in $V_{c}^{2}$ by
$\|f\|_{V^{2}_{c}}:=\|(c_{n})\|_{\ell^{2}(\mathbb{Z})}$.

######  Theorem 1.3.

Given $c\in\mathbb{C}$ with $a=\mathrm{Re}\,c>0$, an increasing sequence
$\Lambda\subset\mathbb{R}$ is a complete interpolating sequence for
$V^{2}_{c}$, if and only if $\Lambda$ is separated and there exists an
enumeration $\Lambda=\\{\lambda_{n}\\}_{n\in\mathbb{Z}}$,
$\lambda_{n}=n+\delta_{n}$, $n\in\mathbb{Z}$, such that

  1. (a)

$(\delta_{n})_{n\in\mathbb{Z}}\in\ell^{\infty}$;

  2. (b)

there exists $N\geq 1$ and $\delta>0$ such that

$\sup_{n\in\mathbb{Z}}\frac{1}{N}\Big{|}\sum_{k=n+1}^{n+N}\delta_{k}\Big{|}\leq\delta<\frac{1}{2}.$

Theorem 1.3 is stronger than the results in [8]. Indeed, as a corollary, we
will deduce the density results for sampling or interpolation obtained
previously in [8] and extend them to the spaces generated by complex
Gaussians.

At least to us, Theorem 1.3 is quite surprising, as it shows a marked
difference between $V^{2}_{c}$ and $PW$. The characterizing conditions go back
to Avdonin who showed that conditions (a) and (b) are sufficient for complete
interpolating sequences in $PW$. However, they are far from necessary in $PW$.
By contrast, in the shift-invariant spaces $V^{2}_{c}$ these conditions
provide a complete characterization.

To prove Theorem 1.3, we reduce the problem to the study of similar problems
in some Fock-type space of entire functions. After this reduction is done, one
can apply the description of complete interpolating sequences in this space
obtained in [4].

##  2\. Unitary equivalence with a Fock-type space

For a parameter $a>0$, we consider the Fock-type space of entire functions,
sometimes referred to as a small Fock space, defined by

$\mathcal{F}=\mathcal{F}_{a}=\bigg{\\{}F\ \text{entire}:\ \
\|F\|^{2}_{\mathcal{F}_{a}}=\frac{1}{2\sqrt{2\pi
a}}\int_{\mathbb{C}}|F(w)|^{2}e^{-\frac{1}{2a}(\log|w|)^{2}}dm_{2}(w)<\infty\bigg{\\}},$

where $m_{2}$ is the Lebesgue measure $\frac{1}{\pi}dxdy$. A simple
computation shows that for a function $F(w)=\sum_{n\geq 0}b_{n}w^{n}$ one has

$\|F\|^{2}_{\mathcal{F}_{a}}=\sum_{n\geq 0}|b_{n}|^{2}e^{2a(n+1)^{2}}.$

In what follows we assume $c\in\mathbb{C}$ to be fixed. To see the connection
between $\mathcal{F}_{a}$ and $V_{c}^{2}$, we note that

$\sum_{n\in\mathbb{Z}}c_{n}e^{-c(z-n)^{2}}=e^{-cz^{2}}\sum_{n\in\mathbb{Z}}c_{n}e^{-cn^{2}}e^{2cnz}.$

Introducing a new variable $w=e^{2cz}$, $V^{2}_{c}$ becomes unitarily
equivalent to the space of functions representable as

$\bigg{\\{}g(w)=\sum_{n\in\mathbb{Z}}d_{n}w^{n}:\ \
\|g\|^{2}:=\sum_{n\in\mathbb{Z}}|d_{n}|^{2}e^{2an^{2}}<\infty\bigg{\\}}.$

This space consists of functions analytic in $\mathbb{C}\setminus\\{0\\}$. It
is more convenient, however, to split this space into three parts,
corresponding to positive and negative powers of $w$, and the constant term.
Namely, for $f\in V^{2}_{c}$ we write

$\displaystyle f(z)=\sum_{n\in\mathbb{Z}}c_{n}e^{-c(z-n)^{2}}$
$\displaystyle=\sum_{n<0}c_{n}e^{-c(z-n)^{2}}+c_{0}e^{-cz^{2}}+\sum_{n>0}c_{n}e^{-c(z-n)^{2}}$
(1) $\displaystyle=f_{-}(z)+c_{0}e^{-cz^{2}}+f_{+}(z).$

With respect to this decomposition $V^{2}_{c}$ splits into a direct sum
$V^{2}_{c}=V^{2}_{-}\oplus e^{-cz^{2}}\mathbb{C}\oplus V^{2}_{+}$, which is,
in terms of the coefficients, simply
$\ell^{2}(\mathbb{Z})=\ell^{2}(\mathbb{N}_{-})\oplus\mathbb{C}\oplus\ell^{2}(\mathbb{N})$.

Next, for $w=e^{2cz}$, set

$F_{+}(w)=\sum_{n=1}^{\infty}c_{n}e^{-cn^{2}}w^{n-1},\qquad
F_{-}(w)=\sum_{n=1}^{\infty}c_{-n}e^{-cn^{2}}w^{n-1}\,,$

then in view of (1) we can write

$f(z)=e^{-cz^{2}}\big{(}w^{-1}F_{-}(w^{-1})+c_{0}+wF_{+}(w)\big{)}\,.$

It follows from the discussion above that $F_{+},F_{-}\in\mathcal{F}_{a}$ and

$\|F_{+}\|_{\mathcal{F}_{a}}=\Big{(}\sum_{n>0}|c_{n}e^{-cn^{2}}|^{2}e^{2an^{2}}\Big{)}^{1/2}=\|c\|_{2}=\|f_{+}\|_{V^{2}_{c}}\,,$

and $\|F_{-}\|_{\mathcal{F}_{a}}=\|f_{-}\|_{V_{c}^{2}}$. In addition every
function $F_{+},F_{-}\in\mathcal{F}_{a}$ can appear in this representation and

$wF_{+}(w)=e^{cz^{2}}f_{+}(z)\Big{|}_{e^{2cz}=w}.$

This representation of elements in $V_{c}^{2}$ makes it possible to reduce the
study of complete interpolating sequences in $V_{c}^{2}$ to the same problem
in the Fock-type space $\mathcal{F}_{a}$.

Let us introduce the notions of sampling and interpolation for the space
$\mathcal{F}_{a}$. Let $\varphi(z)=\frac{1}{4a}\log^{2}|z|$. In the following
we drop the reference to the parameter $a$, when it is not needed. We say that
the sequence $\\{w_{n}\\}$ is sampling for $\mathcal{F}$, if

(2)
$\sum_{n}(1+|w_{n}|^{2})e^{-2\varphi(w_{n})}|F(w_{n})|^{2}\asymp\|F\|^{2}_{\mathcal{F}},\qquad
F\in\mathcal{F}.$

Analogously, $\\{w_{n}\\}$ is interpolating for $\mathcal{F}$ if for every
sequence $\\{a_{n}\\}$ satisfying
$\sum_{n}(1+|w_{n}|^{2})e^{-2\varphi(w_{n})}|a_{n}|^{2}<\infty$ there exists
$F\in\mathcal{F}$ with $F(w_{n})=a_{n}$. Recall from [5, Lemma 2.7] that the
reproducing kernel $k_{w}$ of $\mathcal{F}$ has the norm

(3) $\|k_{w}\|_{\mathcal{F}}^{2}\asymp\frac{e^{2\varphi(w)}}{1+|w|^{2}}\,.$

Thus the weights in the sampling inequality are nothing but
$\|k_{w_{n}}\|^{-2}_{\mathcal{F}}$. So the norm equivalence (2) reads as
$\sum_{n}|\langle
f,k_{w_{n}}\rangle_{\mathcal{F}}|^{2}\|k_{w_{n}}\|_{\mathcal{F}}^{-2}\asymp\|f\|_{\mathcal{F}}^{2}$.
We may thus formulate the questions about sampling and interpolation in the
equivalent language of frames and Riesz sequences of normalized reproducing
kernels $\tilde{k}_{w_{n}}=k_{w_{n}}/\|k_{w_{n}}\|_{\mathcal{F}}$ in
$\mathcal{F}$. Namely, $\\{w_{n}\\}$ is sampling for $\mathcal{F}$ if and only
if $\\{\tilde{k}_{w_{n}}\\}$ is a frame in $\mathcal{F}$, while $\\{w_{n}\\}$
is interpolating if and only if $\\{\tilde{k}_{w_{n}}\\}$ is a Riesz sequence.
Finally, $\\{w_{n}\\}$ is a complete interpolating sequence if and only if
$\\{\tilde{k}_{w_{n}}\\}$ is a Riesz basis in $\mathcal{F}$.

We now formulate the description of complete interpolating sequences for
$\mathcal{F}_{a}$. First it was shown in [5] that the sequence
$\\{e^{2an}\\}_{n\geq 1}$ is a complete interpolating sequence for
$\mathcal{F}_{a}$. Then a full characterization of complete interpolating
sequences was obtained in [4]. Remarkably, this characterization is formulated
in terms of perturbations of the complete interpolating sequence
$\\{e^{2an}\\}_{n\geq 1}$, somewhat in the spirit of the theorems of Kadets
and Avdonin for complex exponentials (although this condition is only
sufficient in $PW$). Let $\\{w_{n}\\}_{n\geq 1}$ be a sequence such that
$0<|w_{n}|\leq|w_{n+1}|$ and let

$w_{n}=e^{2an}e^{\delta_{n}}e^{i\theta_{n}}$

with $\delta_{n},\theta_{n}\in\mathbb{R}$. In [4] the following result was
proved.

######  Theorem 2.1.

[4] A sequence $\\{w_{n}\\}_{n\geq 1}$ is a complete interpolating sequence
for $\mathcal{F}_{a}$111Note that in [4] the normalization for the small Fock
space is different; to formulate the results from [4] one needs to set
$\alpha=1/(4a)$., if and only if

  1. (i)

$\\{w_{n}\\}$ is $\mathcal{F}_{a}$-separated, i.e., there exists $\gamma>0$
such that $|w_{m}-w_{n}|\geq\gamma|w_{n}|$, $m\neq n$;

  2. (ii)

$(\delta_{n})_{n\geq 1}\in\ell^{\infty}$;

  3. (iii)

there exists $N\geq 1$ and $\delta>0$ such that

$\sup_{n\geq
1}\frac{1}{N}\Big{|}\sum_{k=n+1}^{n+N}\delta_{k}\Big{|}\leq\delta<a\,.$

Note that the result does not depend on the arguments of $w_{n}$, but only on
their moduli. For the description of complete interpolating sequences for a
more general class of Hilbert spaces of entire functions see [3].

##  3\. Proof of Theorem 1.3: Sufficiency

Assume that $\Lambda\subset\mathbb{R}$ is separated and that there exists an
enumeration $\Lambda=\\{\lambda_{m}\\}_{m\in\mathbb{Z}}$,
$\lambda_{m}=m+\delta_{m}$, $m\in\mathbb{Z}$, such that conditions (a) and (b)
of Theorem 1.3 are satisfied. We need to show that the mapping

(4) $J:\ (c_{n})_{n\in\mathbb{Z}}\mapsto(f(\lambda_{m}))_{m\in\mathbb{Z}}$

is an isomorphism of $\ell^{2}(\mathbb{Z})$ onto itself. Again we identify the
sequence $(c_{n})$ and the corresponding function $f\in V^{2}_{c}$.

Recall that $\varphi(w)=\tfrac{1}{4a}(\log|w|)^{2}$, and put

$w_{m}=e^{2c\lambda_{m}}=e^{2am}e^{2a\delta_{m}}e^{2ib(m+\delta_{m})}\,.$

The Avdonin-type condition (iii) of Theorem 2.1 reads as
$\frac{1}{N}\Big{|}\sum_{k=n+1}^{n+N}(2a\delta_{k})\Big{|}\leq\delta<a$, which
amounts precisely to the assumption (b) of Theorem 1.3. Thus each of the
sequences $\\{w_{m}\\}_{m\geq 1}$ and $\\{w_{-m}^{-1}\\}_{m\geq 1}$ is a
complete interpolating sequence for $\mathcal{F}_{a}$.

Claim 1. The mapping $U_{+}:(c_{n})_{n\geq
1}\mapsto(f_{+}(\lambda_{m}))_{m\geq 1}$ is an isomorphism from
$\ell^{2}(\mathbb{N})$ onto itself.  Indeed, since
$\varphi(w)=(\log|w|)^{2}/(4a)=\lambda^{2}$ whenever $w=e^{2c\lambda}$, we
have

(5)
$f_{+}(\lambda_{m})=e^{-c\lambda_{m}^{2}}w_{m}F_{+}(w_{m})=e^{-\varphi(w_{m})}e^{-ib\lambda_{m}^{2}}w_{m}F_{+}(w_{m}),$

and we conclude that $(f_{+}(\lambda_{m}))_{m\geq 1}\in\ell^{2}(\mathbb{N})$,
because $\\{w_{m}\\}$ is sampling, and that every sequence in
$\ell^{2}(\mathbb{N})$ can be obtained in this way from some function in
$F_{+}$, because $\\{w_{m}\\}$ is interpolating. Thus $U_{+}$ is one-to-one
and onto. Similarly the mapping
$U_{-}:(c_{n})_{n\leq-1}\mapsto(f_{-}(\lambda_{m}))_{m\leq-1}$ is an
isomorphism on $\ell^{2}(\mathbb{N}_{-})$.

Claim 2. The mapping $K_{+}:(c_{n})_{n\geq
1}\mapsto(f_{+}(\lambda_{m}))_{m\leq-1}$ is compact from
$\ell^{2}(\mathbb{N})$ to $\ell^{2}(\mathbb{N}_{-})$.  Since

$f_{+}(\lambda_{m})=\sum_{n\geq 1}c_{n}e^{-c(m+\delta_{m}-n)^{2}}\,,$

the matrix of $K_{+}$ has the entries $e^{-c(m+\delta_{m}-n)^{2}}$, $m<0,n>0$.
The boundedness of $\delta_{m}$ yields that
$\sum_{m<0,n>0}e^{-a(|m|+n-\delta_{m})^{2}}<\infty$, thus $K_{+}$ is a
Hilbert–Schmidt operator. Similarly, the operator
$K_{-}:(c_{n})_{n\leq-1}\mapsto(f_{-}(\lambda_{m}))_{m\geq 1}$ is compact from
$\ell^{2}(\mathbb{N}_{-})$ to $\ell^{2}(\mathbb{N})$.

Now we define the operator $U$ on
$\ell^{2}(\mathbb{Z})=\ell^{2}(\mathbb{N}_{-})\oplus\mathbb{C}\oplus\ell^{2}(\mathbb{N})$
by the formula

(6) $U:\ \big{(}(c_{n})_{n\leq-1};\,c_{0};\,(c_{n})_{n\geq
1}\big{)}\mapsto\big{(}(f_{-}(\lambda_{m}))_{m\leq-1};\,f(\lambda_{0});\,(f_{+}(\lambda_{m}))_{m\geq
1}\big{)}$

By Claim 1 $U$ is an isomorphism from $\ell^{2}(\mathbb{Z})$ onto itself. At
the same time, by Claim 2, the map
$J:\,(c_{n})_{n\in\mathbb{Z}}\mapsto(f(\lambda_{m}))_{m\in\mathbb{Z}}$ differs
from $U$ by a compact operator, since
$f(\lambda_{m})=f_{+}(\lambda_{m})+c_{0}e^{-c\lambda_{m}^{2}}+f_{-}(\lambda_{m})$.
Thus, to prove that $J$ is invertible it is sufficient to show that $J$ is
one-to-one, which is usually easier. In our context it means that $\Lambda$ is
a uniqueness set for the space $V^{2}_{c}$.

Claim 3. $\Lambda$ is a uniqueness set for the space $V^{2}_{c}$.  Assume that
$f\in V^{2}_{c}$ and $f(\lambda_{m})=0$ for all $m$. Then the function

$F(w)=w^{-1}F_{-}(w^{-1})+c_{0}+wF_{+}(w)$

vanishes on the set $w_{m}=e^{2c\lambda_{m}}$.

Since $W_{+}=\\{w_{m}\\}_{m\geq 1}$ is a complete interpolating sequence for
$\mathcal{F}$, there exists a so-called generating function $G_{+}$ for this
sequence. This is a function with simple zeros exactly at $w_{m}$, $m\geq 1$,
and no other zeros, such that $G_{+}\notin\mathcal{F}$, but
$\frac{G_{+}}{w-w_{m}}\in\mathcal{F}$ for all $m$. In fact, the system
$\big{\\{}\frac{G_{+}}{G^{\prime}_{+}(w_{m})(w-w_{m})}\\}_{m\geq 1}$ is the
biorthogonal system in $\mathcal{F}$ to the system of reproducing kernels
$\\{k_{w_{m}}\\}_{m\geq 1}$.

We will need several estimates for the generating functions. First, let
$G_{0}$ be the generating function for the sequence
$W_{0}=\\{e^{2am}\\}_{m\geq 1}$. It was shown in [5, Lemma 2.6] that

$|G_{0}(w)|\asymp e^{\varphi(w)}\frac{{\rm
dist}\,(w,W_{0})}{1+|w|^{3/2}},\qquad w\in\mathbb{C}.$

Using standard estimates of lacunary canonical products it is easy to show
(see [4, Section 3, p. 1373]) that for the generating function $G_{+}$ of the
perturbed sequence $W_{+}=\\{w_{m}\\}_{m\geq 1}$ with
$|w_{m}|=e^{2am+2a\delta_{m}}$ one has

$\frac{|G_{+}(w)|}{|G_{0}(w)|}\asymp\exp\left(-2a\sum_{k=1}^{m}\delta_{k}\right)\frac{{\rm
dist}\,(w,W_{+})}{{\rm dist}\,(w,W_{0})}$

whenever $e^{a(2m-1)}\leq|w|\leq e^{a(2m+1)}$. By condition (b) of Theorem
1.3, for sufficiently large $m$ we have
$\exp(-2a\sum_{k=1}^{m}\delta_{k})\geq\exp(-2a\delta m)\asymp|w|^{-\delta}$,
where $\delta<\frac{1}{2}$. Hence,

(7) $|G_{+}(w)|\gtrsim\frac{{\rm
dist}\,(w,W_{+})}{(1+|w|)^{\frac{3}{2}+\delta}}\,e^{\varphi(w)}.$

Analogously, we define the function $G_{-}$, the generating function of the
sequence $W_{-}=\\{w_{n}^{-1}\\}_{n\leq-1}$. Consequently, the zero set of the
function $(w-w_{0})G_{+}(w)G_{-}(w^{-1})$ is precisely
$\tilde{\Lambda}=\\{w_{n}:n\in\mathbb{Z}\\}$. Since by assumption $F$ vanishes
also on $\tilde{\Lambda}$, $F$ possesses the factorization

$F(w)=(w-w_{0})G_{+}(w)G_{-}(w^{-1})H(w)$

for some function $H$ analytic in $\mathbb{C}\setminus\\{0\\}$. We will show
that $H$ is an entire function which tends to $0$ at infinity, and thus must
be identically zero. Clearly, $w^{-1}F_{-}(w^{-1})\to 0$, as $|w|\to\infty$
and

$|F_{+}(w)|=|\langle
F_{+},k_{w}\rangle|\leq\|F_{+}\|_{\mathcal{F}}\,\|k_{w}\|_{\mathcal{F}}\lesssim\frac{e^{\varphi(w)}}{1+|w|}\,.$

by (3). Consequently,

(8) $|F(w)|\lesssim 1+|wF_{+}(w)|\lesssim|w|\frac{e^{\varphi(w)}}{1+|w|}\leq
e^{\varphi(w)},\qquad|w|\to\infty.$

Thus, using (7), (8), and the fact that $G_{-}(w^{-1})\to G_{-}(0)\neq 0$,
$|w|\to\infty$, we get

$\displaystyle|H(w)|$
$\displaystyle=\frac{|F(w)|}{|(w-w_{0})G_{+}(w)G_{-}(w)|}$
$\displaystyle\lesssim
e^{\varphi(w)}\cdot\frac{(1+|w|)^{\frac{3}{2}+\delta}}{|w|{\rm
dist}\,(w,W_{+})}\,e^{-\varphi(w)}\lesssim\frac{(1+|w|)^{\frac{1}{2}+\delta}}{{\rm
dist}\,(w,W_{+})},\qquad|w|\to\infty.$

It follows from the $\mathcal{F}$-separation of $W_{+}$ and the inclusion
$W_{+}\subset\mathbb{R}$ that there exists a sequence of radii
$r_{k}\to\infty$ such that ${\rm dist}\,(w,W_{+})\geq\gamma|w|$ when
$|w|=r_{k}$, with $\gamma>0$ independent on $k$. Since $\delta<\frac{1}{2}$,
we conclude that $\max_{|w|=r_{k}}|H(w)|\to 0$, as $k\to\infty$. The maximum
principle implies that also $H(w)\to 0$, as $|w|\to\infty$.

Analogously, replacing $w$ by $w^{-1}$ and using the estimate (7) for $G_{-}$,
we conclude that $H(w)\to 0$ as $w\to 0$. Consequently the singularity of $H$
at $0$ is removable, and $H$ is thus entire and bounded. Thus $H\equiv 0$ and
so $F\equiv 0$, as was to be shown. ∎

To conclude we formulate a simple, sufficient condition for complete
interpolating sets in the style of Kadets, see e.g. [19].

######  Corollary 3.1.

Assume that $|\delta_{n}|\leq\delta<\frac{1}{2}$ for all $n\in\mathbb{Z}$.
Then $\\{n+\delta_{n}\\}$ is a complete interpolating sequence for
$V^{2}_{c}$.

Note that in the Paley–Wiener space $PW$ the corresponding result holds with
$\delta<1/4$. The constant $1/2$ is sharp, as the set
$\\{n+1/2:n\in\mathbb{Z}\\}$ is not a sampling sequence for $V^{2}_{a}$ with
$a>0$ [13].

##  4\. Proof of Theorem 1.3: necessity

For the proof of necessity we need to reverse the argument of the sufficiency
part. Assume that $\Lambda$ is a complete interpolating sequence for
$V^{2}_{c}$. This means that the operator $J$ given by (4) is an isomorphism
of $\ell^{2}(\mathbb{Z})$ onto itself.

Since a complete interpolating sequence for $V^{2}_{c}$ is always separated,
the operators $K_{+}$ and $K_{-}$ defined in Claim 2 are compact, whence $U$
is a compact perturbation of the isomorphism $J$.

It follows that the kernel of $U$ has finite dimension and that the range of
$U$ has finite codimension:

(9) ${\rm dim}\,{\rm Ker}\,U<\infty\qquad\text{ and }\qquad{\rm codim}\,{\rm
Range}\,U<\infty\,.$

In the language of frame theory [10, Sec. 8.7] one speaks of the deficit and
the excess of the set of reproducing kernels. The condition
$\mathrm{Ker}\,U\neq\\{0\\}$ means that the reproducing kernels associated to
the sampling points $\lambda_{n},n\in\mathbb{Z}$, are not complete, but span a
proper subspace (the _deficit_). The finite codimension of the Range of $U$
means that there are linear dependencies of the reproducing kernels, in other
words, too many functions for a Riesz basis (the _excess_). The finiteness
condition of (9) implies that we can construct a Riesz basis of reproducing
kernels by adding and/or removing finitely many points.

In our context we will use an important property of the Fock space
$\mathcal{F}$. Assume that $F\in\mathcal{F}$ satisfies $F(w_{1})=0$ and
$w_{2}\in\mathbb{C},w_{2}\neq w_{1}$. Then the function
$\tilde{F}(z)=\frac{z-w_{2}}{z-w_{1}}F(z)$ is again in $\mathcal{F}$. This
property is called the _division property_ of $\mathcal{F}$.

Step 1. To start with the proof of necessity, assume that (9) holds. Recall
that $V_{+}^{2}=\\{f\in V^{2}_{c}:\,f=f_{+}\\}\cong\ell^{2}(\mathbb{N})$, and
consider the restriction $U_{+}=U|_{V_{+}^{2}}$ of $U$ to the subspace
$V_{+}^{2}$, given by $U_{+}(c_{n})_{n\geq 1}=(f_{+}(\lambda_{m}))_{m\geq 1}$.
From the properties of $U$ we conclude that the kernel
$\mathcal{N}=\mathrm{Ker}\,U_{+}$ in $V_{+}^{2}$ also is finite-dimensional
and that ${\rm Range}\,U_{+}$ is a closed subspace of $\ell^{2}(\mathbb{N})$
of finite codimension. Put $w_{m}=e^{2c\lambda_{m}}$, as before. If $U_{+}$ is
an isomorphism between $V^{2}_{+}$ and $\ell^{2}(\mathbb{N})$, then, by (5),
$\\{w_{m}\\}_{m\geq 1}$ is a complete interpolating sequence for
$\mathcal{F}$. If not, we distinguish two cases.

Case 1. Assume that $U_{+}$ has a nontrivial kernel $\mathcal{N}$. Then
$U_{+}|_{V_{+}^{2}\ominus\mathcal{N}}$ is an isomorphism on its image. Thus,

$\sum_{m\geq 1}|f_{+}(\lambda_{m})|^{2}\asymp\|f_{+}\|_{V_{+}^{2}}^{2},\qquad
f_{+}\in V_{+}^{2}\ominus\mathcal{N}.$

Note that $f\in\mathcal{N}$ (i.e. $f_{+}(\lambda_{m})=0$, $m\geq 1$) if and
only if $F_{+}(w_{m})=0$, $m\geq 1$. By (5), we have

$\sum_{m\geq
1}(1+|w_{m}|^{2})e^{-2\varphi(w_{m})}|F_{+}(w_{m})|^{2}\asymp\|F_{+}\|^{2}_{\mathcal{F}},\qquad
F_{+}\in\mathcal{F}\ominus\mathcal{M},$

where $\mathcal{M}$ is the finite-dimensional subspace in $\mathcal{F}$ which
consists of all functions vanishing on $\\{w_{m}\\}_{m\geq 1}$.

We will need the following simple lemma.

######  Lemma 4.1.

Let $\mathcal{M}=\\{F\in\mathcal{F}:\ F(w_{m})=0,m\geq 1\\}$ and $M={\rm
dim}\,\mathcal{M}<\infty$. Then there exists a set of points
$\\{\mu_{k}\\}_{1\leq k\leq M}$ which is a uniqueness set for $\mathcal{M}$,
while $\\{\mu_{k}\\}_{1\leq k\leq M-1}$ is a non-uniqueness set for
$\mathcal{M}$.

###### Proof.

We argue by induction on $M$. The base $M=1$ (i.e. $\mathcal{M}={\rm
Lin}\,\\{F_{1}\\}$) is trivial since one can take as $\mu_{1}$ any point such
that $F_{1}(\mu_{1})\neq 0$. Assume that we can prove the statement for $M-1$.
Take any point $\mu_{1}\in\mathbb{C}$ such that there is a function
$F_{0}\in\mathcal{M}$ with $F_{0}(\mu_{1})\neq 0$ and consider
$\mathcal{M}_{1}=\\{F\in\mathcal{M}:F(\mu_{1})=0\\}$. It is obvious that ${\rm
dim}\,\mathcal{M}=M-1$. By the induction hypothesis there exist points
$\mu_{2},\dots,\mu_{M}$ which form a uniqueness set for $\mathcal{M}_{1}$,
while $\\{\mu_{2},\dots,\mu_{M-1}\\}$ is a non-uniqueness set for
$\mathcal{M}_{1}$. Then the points $\\{\mu_{k}\\}_{1\leq k\leq M}$ have the
required property. ∎

Let $\\{\mu_{k}\\}_{1\leq k\leq M}$ be the set constructed in the lemma. Since
$\mathcal{M}$ is finite-dimensional, it follows that

$\sum_{k=1}^{M}(1+|\mu_{k}|^{2})e^{-2\varphi(\mu_{k})}|F_{+}(\mu_{k})|^{2}\asymp\|F_{+}\|^{2}_{\mathcal{F}},\qquad
F_{+}\in\mathcal{M}\,.$

Let $\tilde{W}=\\{w_{m}\\}_{m\geq 1}\cup\\{\mu_{k}\\}_{1\leq k\leq M}$ the
enlarged set. By construction, the sampling operator
$F_{+}\mapsto\big{(}F_{+}(\tilde{w_{k}})\big{)}_{w_{k}\in\tilde{W}}$ is one-
to-one and it has closed range of finite codimension in $\ell^{2}(\tilde{W})$.
Consequently, by [10, Thm. 8.29, (c)], $\tilde{W}$ is a sampling set for
$\mathcal{F}$.

By Lemma 4.1 the set $\tilde{W}\setminus\\{\mu_{M}\\}$ is a non-uniqueness set
for $\mathcal{F}$ and therefore there exists a function $H\in\mathcal{F}$,
such that $H(w_{m})=H(\mu_{k})=0$ for $m\in\mathbb{N}$ and $k=1,\dots,M-1$,
but $H(\mu_{M})\neq 0$. We now use the division property of $\mathcal{F}$ and
show that every set $\tilde{W}\setminus\\{v\\}$, where
$v\in\tilde{W}\setminus\\{\mu_{M}\\}$, is also a non-uniqueness set in
$\mathcal{F}$. Consider $\tilde{H}(w)=\frac{w-\mu_{M}}{w-v}H(w)$. Then
$\tilde{H}$ is non-zero in $\mathcal{F}$, and $\tilde{H}$ vanishes on
$\tilde{W}\setminus\\{v\\}$, but $\tilde{H}$ cannot vanish at $v$, because
$\tilde{W}$ is sampling on $\mathcal{F}_{+}$.

It follows that the frame of reproducing kernels $\\{k_{w}\\}_{w\in\tilde{W}}$
fails to be complete after removal of any of its members. We conclude that the
normalized reproducing kernels corresponding to the sampling set $\tilde{W}$
form an exact frame, i.e., a frame which fails to remain a frame after removal
of any of its vectors. By well-known results (see, e.g., [10, 19]) the
normalized reproducing kernels at $\tilde{W}$ form a Riesz basis in
$\mathcal{F}$, and, equivalently, $\tilde{W}$ is a complete interpolating
sequence for $\mathcal{F}$.

Case 2. Assume now that ${\rm Ker}\,U_{+}=0$, whence $U_{+}$ is an isomorphism
onto its image and so $\\{w_{m}\\}_{m\geq 1}$ is already sampling for
$\mathcal{F}$. Let ${\rm codim}\,{\rm Range}\,U_{+}=M$. This means that the
frame of reproducing kernels $\\{k_{w_{m}}\\}_{m\geq 1}$ has finite excess,
and therefore it becomes a Riesz basis after the removal of finitely many
elements, see [11, Thm. 2.4] and [10].

Repeating the argument for the negative part of $V^{2}_{c}$, we conclude that
the set $\\{w_{m}^{-1}\\}_{m\leq-1}$ either can be enlarged to a complete
interpolating set for $\mathcal{F}$ or becomes a complete interpolating set
after the removal of a finite set of points.

Step 2. It remains to show that after an appropriate enumeration the sequence
$\\{\lambda_{m}\\}$ will satisfy conditions (a) and (b) of Theorem 1.3. We
note that both these conditions and also the property of being a complete
interpolating sequence for $\mathcal{F}$ are stable with respect to moving a
finite number of points (without gluing them). Formally, we remove a finite
subset $F_{1}\subseteq\Lambda$ and add a finite set $F_{2}$, such that
$\mathrm{card}\,F_{1}=\mathrm{card}\,F_{2}$ and $F_{2}\cap\Lambda=\emptyset$.
After relabeling, the new set $\Lambda^{\prime}=(\Lambda\setminus F_{1})\cup
F_{2}$ satisfies conditions (a) and (b), if and only if $\Lambda$ satisfies
conditions (a) and (b).

In Step 1 we have shown that either (i) there exists a finite set
$F_{+}=\\{\mu_{j}\\}_{m=1,\dots,M}\\}\subset\mathbb{R}$ such that
$\tilde{W}_{+}=\\{w_{m}\\}_{m\geq 1}\cup\\{e^{2c\mu_{j}}\\}_{m=1,\dots,M}$ is
a complete interpolating set for $\mathcal{F}$ or (ii) we can remove a finite
set $J_{+}\subseteq\\{w_{m}\\}_{m\geq 1}$, such that
$\tilde{W}_{+}=\\{w_{m}\\}_{m\geq 1}\setminus J_{+}$ is a complete
interpolating set for $\mathcal{F}$. Using the original points $\lambda_{m}$
instead of $e^{2c\lambda_{m}}$ and calling the removed points also $F_{+}$,
the relabeled sequence $\\{\lambda_{m}\\}_{m\geq 0}\cup F_{+}$ or
$\\{\lambda_{m}\\}_{m\geq 0}\setminus F_{+}$ then satisfies a one-sided
version of (a) and (b) by Theorem 2.1.

We apply the same procedure to the sequence $\\{w_{m}^{-1}\\}_{m<0}$ and after
adding a finite set $F_{-}=\\{\nu_{j}\\}$, we obtain a complete interpolating
sequence $\tilde{W}_{-}=\\{w_{m}^{-1}\\}_{m<0}\cup\\{e^{-2c\nu_{j}}\\}$ or
$\tilde{W}_{-}=\\{w_{m}^{-1}\\}_{m<0}\setminus J_{-}$. Again a relabeling the
sequence $\\{\lambda_{m}\\}_{m<0}\cup F_{-}$ or
$\\{\lambda_{m}\\}_{m<0}\setminus F_{-}$ satisfies conditions (a) and (b).

By taking the union of both sequences and returning to
$\lambda_{m},\mu_{j},\nu_{j}$ (instead of $w_{m}$), we see that there exist
finite sets $F_{1}$ and $F_{2}$, satisfying $F_{1}\subseteq\Lambda$ and
$F_{2}\cap\Lambda=\emptyset$, such that $\Lambda^{\prime}=(\Lambda\setminus
F_{1})\cup F_{2}$ can be enumerated to satisfy (a) and (b). It is possible
that $F_{1}=\emptyset$ (we add points to both $\\{w_{m}\\}_{m\geq 0}$ and to
$\\{w_{m}\\}_{m<0}$) or $F_{2}=\emptyset$ (we remove points from both sets).

Hence, by sufficiency part of Theorem 1.3, $\Lambda^{\prime}$ is a complete
interpolating sequence for $V^{2}_{c}$. Note that
$\Lambda=(\Lambda^{\prime}\setminus F_{2})\cup F_{1}$. Let
$\mathrm{card}\,F_{1}=\mathrm{card}\,F_{2}+\ell$. If $\ell>0$, we have moved
$\mathrm{card}\,F_{2}$ points and added $\ell$ additional points. Then
$\Lambda$ fails to be a complete interpolating sequence for $V^{2}_{c}$ in
contradiction to the assumption. If $\ell<0$, we have moved
$\mathrm{card}\,F_{1}$ points and removed $\ell$ additional points. Again
$\Lambda$ fails to be a complete interpolating sequence.

We conclude that $\mathrm{card}\,F_{1}=\mathrm{card}\,F_{2}$ and $\Lambda$ is
obtained from $\Lambda^{\prime}$ by moving finitely many points. As observed
above, since $\Lambda^{\prime}$ satisfies conditions (a) and (b), so must the
original sequence $\Lambda$, as was to be proved. ∎

##  5\. Proof of the Density Results and Remarks

Finally we indicate how Theorems 1.1 and 1.2 follow from the characterization
of complete interpolating sequences in $V^{2}_{c}$. We use a simple
“combinatorial” argument from [17] in order to show that any separated
sequence $\Lambda$ with $D^{-}(\Lambda)>1$ contains a subsequence
$\Lambda^{\prime}=\\{m+\delta_{m}\\}_{m\in\mathbb{Z}}$ with $\delta_{m}$
satisfying Avdonin-type conditions (a) and (b) of Theorem 1.3. Analogously,
any separated $\Lambda$ with $D^{+}(\Lambda)<1$ can be enhanced to such a set
$\Lambda^{\prime}$. We refer to [4] for the precise details, as they are
almost identical.

The necessary density conditions for sampling and interpolation in a shift-
invariant space are well-known, see, e.g., [1]. Let us show that they can also
be deduced from the results for $\mathcal{F}$ in [4].

Let $\Lambda$ be an interpolating set for $V^{2}_{c}$. Then the mapping
$f\mapsto\\{f(\lambda_{m})\\}_{m\geq 1}$ acts from $V^{2}_{c}$ onto
$\ell^{2}(\mathbb{N})$. By Claim 2 of the proof of sufficiency part in Theorem
1.3 the map $f\mapsto\\{f_{-}(\lambda_{m})\\}_{m\geq 1}$ is a compact operator
from $V^{2}_{c}$ to $\ell^{2}(\mathbb{N})$, whence the operator
$f\mapsto\\{f_{+}(\lambda_{m})\\}_{m\geq 1}$ has the closed range of finite
codimension. Then it is easy to see that for some $m_{0}\geq 1$ the map
$f\mapsto\\{f_{+}(\lambda_{m})\\}_{m\geq m_{0}}$ will be onto. Hence, the set
$\\{\lambda_{m}\\}_{m\geq m_{0}}$ will be an interpolating set for $V^{2}_{+}$
and so $\\{w_{m}\\}_{m\geq m_{0}}$ will be interpolating for $\mathcal{F}$. We
conclude that $D^{+}(\Lambda)\leq 1$.

Now assume that $\Lambda$ is a sampling set for $V^{2}_{c}$ and, in
particular, for $V^{2}_{+}$. Since $\Lambda$ is a finite union of separated
sets, the sequence $\\{f_{+}(\lambda_{m})\\}_{m\leq-1}$ decays very fast and
it is easy to show that the set $\\{\lambda_{m}\\}_{m\geq m_{0}}$ will be
sampling for $V^{2}_{+}$.

### Further remarks.

1\. Let $V^{p}_{c}=\\{f(z)=\sum_{n\in\mathbb{Z}}c_{n}e^{-c(z-n)^{2}}:\
(c_{n})\in\ell^{p}(\mathbb{Z})\\}$ be the subspace generated by shifts of the
Gaussian with $\ell^{p}$ coefficients. This is a closed subspace of
$L^{p}(\mathbb{R})$. Again, a sequence
$\Lambda=\\{\lambda_{n}:n\in\mathbb{Z}\\}$ is called a sampling set for
$V_{c}^{p}$, if

$\sum_{n\in\mathbb{Z}}|f(\lambda_{n})|^{p}\asymp\|f\|^{p}_{V^{p}_{c}}\asymp\|c\|_{p}^{p},\qquad
f\in V_{c}^{p},$

and interpolating for $V^{p}_{c}$ if any $(a_{n})\in\ell^{p}(\mathbb{Z})$ can
be interpolated by a function $f\in V_{c}^{p}$. Based on the general theory of
sampling in shift-invariant spaces with a nice generator, we can assert that
_a set $\Lambda\subseteq\mathbb{R}$ is sampling for $V^{2}_{c}$, if and only
if $\Lambda$ is sampling for some $V^{p_{0}}_{c}$, if and only if $\Lambda$ is
sampling for $V^{p}_{c}$ for all $p,1\leq p\leq\infty$_ [8, Thm. 3.1]. A
similar statement holds for the interpolation property. Theorem 1.3 therefore
provides a characterization for complete interpolating sequences for all
spaces $V^{p}_{c}$.

2\. _Sign retrieval._ The sign retrieval problem asks whether a real-valued
function $f$ in some function space is uniquely determined by its unsigned
values $|f(\lambda)|$ on some set $\Lambda$. For $V^{2}_{a}$ with a real-
valued Gaussian generator, Theorem 1.3 implies the following result on sign
retrieval.

######  Corollary 5.1.

Let $\Lambda=\\{n/2+\delta_{n}:n\in\mathbb{Z}\\}$ be uniformly separated such
that $\\{\delta_{n}\\}$ satisfies the Avdonin-type conditions $(a)$ and $(b)$
with $\delta<1/4$. Then every $f\in V^{2}_{a}$ is uniquely determined by its
unsigned values $\\{|f(n/2+\delta_{n})|:n\in\mathbb{Z}\\}$.

The proof is the same as in [7], we simply replace sampling sets of density
$D^{-}(\Lambda)>2$ by a complete interpolating sequence for a dilated version
of $V^{2}_{a}$. Thus sign retrieval is possible even at the critical density.

3\. It would be interesting to obtain similar results for complete
interpolating sequences in shift-invariant spaces with other generators. Even
a sharp version of Corollary 3.1 would have important consequences for Gabor
frames.

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