In this paper, we establish a new multivariate Hermite sampling series involving samples from the function itself and its mixed and non-mixed partial derivatives of arbitrary order. This multivariate ...form of Hermite sampling will be valid for some classes of multivariate entire functions, satisfying certain growth conditions. We will show that many known results included in Commun Korean Math Soc, 2002, 17: 731–740, Turk J Math, 2017, 41: 387–403 and Filomat, 2020, 34: 3339–3347 are special cases of our results. Moreover, we estimate the truncation error of this sampling based on localized sampling without decay assumption. Illustrative examples are also presented.
Recently, Norvidas has introduced the general multidimensional Hermite sampling series, which involves samples from a function and its mixed and non-mixed partial derivatives. The convergence of this ...sampling series is slow unless the sample values |
f
(
x
)| rapidly decay as
|
x
j
|
→
∞
for all
1
≤
j
≤
n
. In this paper, we investigate a modified version of this sampling series that utilizes a multivariate Gaussian multiplier to approximate functions from two classes of multivariate analytic functions using a complex approach. The first class comprises entire functions of exponential type in
n
variables that fulfill a decay condition, while the second class includes analytic functions in
n
variables defined on a multidimensional horizontal strip. It has a significantly higher convergence rate compared to the general multidimensional Hermite sampling series.
In this paper, we deduce some hyperstability results for a generalized class of homogeneous Pexiderized functional equations, expressed as ∑ρ∈Γfxρ.y=ℓf(x)+ℓg(y), x,y∈M, which is inspired by the ...concept of Ulam stability. Indeed, we prove that function f that approximately satisfies an equation can, under certain conditions, be considered an exact solution. Domain M is a monoid (semigroup with a neutral element), Γ is a finite subgroup of the automorphisms group of M, ℓ is the cardinality of Γ, and f,g:M→G such that (G,+) denotes an ℓ-cancellative commutative group. We also examine the hyperstability of the given equation in its inhomogeneous version ∑ρ∈Γfxρ.y=ℓf(x)+ℓg(y)+ψ(x,y),x,y∈M, where ψ:M×M→G. Additionally, we apply the main results to elucidate the hyperstability of various functional equations with involutions.
The sinc-Gaussian sampling formula is used to approximate an analytic function, which satisfies a growth condition, using only finite samples of the function. The error of the sinc-Gaussian sampling ...formula decreases exponentially with respect to
N
, i.e.,
N
− 1/2
e
−
α
N
, where
α
is a positive number. In this paper, we extend this formula to allow the approximation of derivatives of any order of a function from two classes of analytic functions using only finite samples of the function itself. The theoretical error analysis is established based on a complex analytic approach; the convergence rate is also of exponential type. The estimate of Tanaka et al. (Jpan J. Ind. Appl. Math.
25
, 209–231
2008
) can be derived from ours as an immediate corollary. Various illustrative examples are presented, which show a good agreement with our theoretical analysis.
Recently, some authors have used the sinc-Gaussian sampling technique to approximate eigenvalues of boundary value problems rather than the classical sinc technique because the sinc-Gaussian ...technique has a convergence rate of the exponential order,
O
(
e
−
(
π
−
h
σ
)
N
/
2
/
N
)
, where
σ
,
h
are positive numbers and
N
is the number of terms in sinc-Gaussian technique. As is well known, the other sampling techniques (classical sinc, generalized sinc, Hermite) have a convergence rate of a polynomial order. In this paper, we use the Hermite-Gauss operator, which is established by Asharabi and Prestin (Numer. Funct. Anal. Optim. 36:419-437,
2015
), to construct a new sampling technique to approximate eigenvalues of regular Sturm-Liouville problems. This technique will be new and its accuracy is higher than the sinc-Gaussian because Hermite-Gauss has a convergence rate of order
O
(
e
−
(
2
π
−
h
σ
)
N
/
2
/
N
)
. Numerical examples are given with comparisons with the best sampling technique up to now,
i.e.
sinc-Gaussian.
The Hermite-Gauss sampling method is established to approximate the eigenvalues of the continuous Sturm-Liouville problems in 2016. In the present paper, we apply this method to approximate the ...eigenvalues of the Dirac system with transmission conditions at several points of discontinuity. This method gives us a higher accuracy results in comparison with the results of other sampling methods (classical sinc, regularized sinc, Hermite, sinc-Gaussian). The error of this method decays exponentially in terms of number of involved samples. Illustrative examples have been discussed to show the effectiveness of the presented method. We compare our results with the results of sinc-Gaussian sampling method which was the best sampling method before the presented method.
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