On the Lp convergence of Lagrange interpolating entire functions of exponential type

Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals

“…Furthermore, one can give various error estimates for the so-called aliasing error |(S W f )(t) − f (t)|. It is also well known that continuity of f alone does not suffice for (1.3) to hold even if f has compact support; see [8,12,25,26,29,30,32], and also the overview paper [10]. The assertion (1.3), based on the inequality (1.2), has been termed the approximate sampling theorem in, e.g., [9].…”

Classical and approximate sampling theorems; studies in the LP(R) and the uniform norm

“…Since x j ∈(y k−1 ,y k ] j + the right-hand side of (17) is finite in view of (16). This gives (ii).…”

“…Observe that an approximate sampling theorem, namely that the limit relation (5) holds under suitable conditions (see below), was established in [16,5,17,1]. However, it was not shown in these papers that this result does indeed follow explicitly from the classical sampling theorem.…”

“…The function f * is an "upper encasing" step function majorizing f . It plays a similar role as do the majorizing functions occurring in the definitions of the spaces F p and p , respectively, in [16] or [5], although it is not necessarily non-increasing.…”

“…Property (b) gives a full proof of the representation of the integral R |f (u)| p du as a limit of an infinite Riemann sum of |f | p for a general family of partitions . It is a generalization of Lemma 13 of Rahman-Vértesi [16] for the particular partition w := { k w ; k ∈ Z}, the hypothesis there, namely f ∈ F p , also being weakened to our f ∈ p . Of course, the corresponding result for the integral N −N |f (u)| p du with f ∈ R loc (which is also needed in the proof) follows obviously from the definition of the Riemann integral.…”