TY  - JOUR
AB  - In this paper, we present new regularized Shannon sampling formulas related to the special affine Fourier transform (SAFT). These sampling formulas use localized sampling with special compactly supported, continuous window functions, namely B-spline, sinh-type, and continuous Kaiser–Bessel window functions. In contrast to the known Shannon sampling series for SAFT, the regularized Shannon sampling formulas for SAFT possesses an exponential decay of the approximation error and are numerically robust in the presence of noise, if certain oversampling condition is fulfilled.
AU  - Filbir, F.
AU  - Tasche, M.*
C1  - 74880
C2  - 57649
TI  - Regularized shannon sampling formulas related to the special affine fourier transform.
JO  - J. Fourier Anal. Appl.
VL  - 31
IS  - 3
PY  - 2025
SN  - 1069-5869
ER  - 

TY  - JOUR
AB  - In this paper, we focus on the approximation of smooth functions f: [- π, π] → C, up to an unresolvable global phase ambiguity, from a finite set of Short Time Fourier Transform (STFT) magnitude (i.e., spectrogram) measurements. Two algorithms are developed for approximately inverting such measurements, each with theoretical error guarantees establishing their correctness. A detailed numerical study also demonstrates that both algorithms work well in practice and have good numerical convergence behavior.
AU  - Iwen, M.*
AU  - Perlmutter, M.*
AU  - Sissouno, N.
AU  - Viswanathan, A.*
C1  - 67152
C2  - 53514
CY  - 233 Spring Street, 6th Floor, New York, Ny 10013 Usa
TI  - Phase retrieval for L2([- π, π]) via the provably accurate and noise robust numerical inversion of spectrogram measurements.
JO  - J. Fourier Anal. Appl.
VL  - 29
IS  - 1
PB  - Springer Birkhauser
PY  - 2023
SN  - 1069-5869
ER  - 

TY  - JOUR
AB  - The angular synchronization problem of estimating a set of unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as an optimization problem on graphs involving the graph Laplacian matrix. We consider a general, weighted version of this problem, where the impact of the noise differs between different pairs of entries and some of the differences are erased completely; this version arises for example in ptychography. We study two common approaches for solving this problem, namely eigenvector relaxation and semidefinite convex relaxation. Although some recovery guarantees are available for both methods, their performance is either unsatisfying or restricted to the unweighted graphs. We close this gap, deriving recovery guarantees for the weighted problem that are completely analogous to the unweighted version.
AU  - Filbir, F.
AU  - Krahmer, F.*
AU  - Melnyk, O.
C1  - 61822
C2  - 50145
CY  - 233 Spring Street, 6th Floor, New York, Ny 10013 Usa
TI  - On recovery guarantees for angular synchronization.
JO  - J. Fourier Anal. Appl.
VL  - 27
IS  - 2
PB  - Springer Birkhauser
PY  - 2021
SN  - 1069-5869
ER  - 

TY  - JOUR
AB  - Let {phi(k)} be an orthonormal system on a quasi-metric measure space X, {l(k)} be a nondecreasing sequence of numbers with lim(k ->infinity)l(k) = infinity. A diffusion polynomial of degree L is an element of the span of {phi(k) : l(k) <= L}. The heat kernel is defined formally by K-t (x, y) = Sigma(infinity)(k=0) exp(-l(k)(2)t)phi(k)(x)phi(k)(y). If T is a (differential) operator, and both K-t and TyKt have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1 <= p <= infinity and diffusion polynomial P of degree L, parallel to TP parallel to(p) <= c(1)L(c)parallel to P parallel to(p). In particular, we are interested in the case when X is a Riemannian manifold, T is a derivative operator, and p not equal 2. In the case when X is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion polynomial valued operators.
AU  - Filbir, F.
AU  - Mhaskar, H.N.*
C1  - 6015
C2  - 28164
CY  - Boston, Inc.
SP  - 629-657
TI  - A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel.
JO  - J. Fourier Anal. Appl.
VL  - 16
IS  - 5
PB  - Birkhäuser
PY  - 2010
SN  - 1069-5869
ER  - 

TY  - JOUR
AB  - Wilson bases are constituted by trigonometric functions multiplied by translates of a window function with good time frequency localization. In this article we investigate the approximation of functions from Sobolev spaces by partial sums of the Wilson basis expansion. In particular, we show that the approximation can be improved if polynomials are reproduced. We give examples of Wilson bases, which reproduce linear functions with the lowest-frequency term only.
AU  - Bittner, K.
C1  - 10081
C2  - 21849
SP  - 85-108
TI  - Linear approximation and reproduction of polynomials by wilson bases. 
JO  - J. Fourier Anal. Appl.
VL  - 8
PY  - 2002
SN  - 1069-5869
ER  -