Approximation of Discontinuous Signals by Exponential Sampling Series

Abstract

We analyze the behaviour of the exponential sampling series SwχfS_{w}^{\chi }f at jump discontinuity of the bounded signal f. We obtain a representation lemma that is used for analyzing the series SwχfS_{w}^{\chi }f and we establish approximation of jump discontinuity functions by the series Swχf.S_{w}^{\chi }f. The rate of approximation of the exponential sampling series SwχfS_{w}^{\chi }f is obtained in terms of logarithmic modulus of continuity of functions and the round-off and time-jitter errors are also studied. Finally we give some graphical representation of approximation of discontinuous functions by SwχfS_{w}^{\chi }f using suitable kernels.

Full text

Results Math (2022) 77:23
Online First
c
2021 The Author(s), under exclusive licence to
Springer Nature Switzerland AG
https://doi.org/10.1007/s00025-021-01551-x Results in Mathematics
Approximation of Discontinuous Signals by
Exponential Sampling Series
Sathish Kumar Angamuthu , Prashant Kumar, and
Devaraj Ponnaian
Abstract. We analyze the behaviour of the exponential sampling series
Sχ
wfat jump discontinuity of the bounded signal f. We obtain a repre-
sentation lemma that is used for analyzing the series Sχ
wfand we establish
approximation of jump discontinuity functions by the series Sχ
wf. The rate
of approximation of the exponential sampling series Sχ
wfis obtained in
terms of logarithmic modulus of continuity of functions and the round-off
and time-jitter errors are also studied. Finally we give some graphical
representation of approximation of discontinuous functions by Sχ
wfusing
suitable kernels.
Mathematics Subject Classification. 41A25, 26A15, 41A35.
Keywords. Exponential sampling series, discontinuous functions, logarith-
mic modulus of smoothness, rate of approximation, round-off and time
jitter errors.
1. Introduction and Preliminaries
Let R+denote the set of all positive real numbers and let χbe a real valued
function defined on R+.We say that χis a kernel if it satisfies the following
conditions:
(i) for every u∈R+,
+∞

k=−∞
χ(e−ku)=1,
(ii) for some ν>0,sup
u∈R+
+∞

k=−∞
|χ(e−ku)||k−log u|ν<+∞.
0123456789().: V,-vol

23 Page 2 of 22 S. K. Angamuthu et al. Results Math
Let Φ denote the set of all functions satisfying conditions (i) and (ii). For
ν∈N0=N∪{0},the algebraic moments of order νof the function χ∈Φis
defined by
mν(χ, u):=
+∞

k=−∞
χ(e−ku)(k−log u)ν,∀u∈R+.
In a similar way, we can define the absolute moment of order ν≥0ofthe
function χ∈Φatu∈R+as
Mν(χ, u):=
+∞

k=−∞
|χ(e−ku)||k−log u|ν.
We define Mν(χ):= sup
u∈R+
Mν(χ, u).For t∈R+,χ∈Φandw>0,the
exponential sampling series for a function f:R+→Ris defined by [6]
(Sχ
wf)(t)=
+∞

k=−∞
χ(e−ktw)f(ek
w).(1.1)
It is easy to see that the series Sχ
wfis well defined for f∈L∞(R+).Using the
above sampling series Sχ
wfone can reconstruct the functions which are not
Mellin band-limited. For Mellin band-limited, see [14]. Recently, Bardaro et
al. [5] pointed out that the study of Mellin band-limited functions is different
from that of Fourier band-limited functions. Mamedov was the first person
who studied the Mellin theory in [18] and then Butzer et.al. further developed
the Mellin theory and studied its approximation properties in [9–11,14].
The reconstruction using the exponential sampling formula was first
introduced by the work of Ostrowski, Bertero, Pike, in the setting of optical
physics (see [8,16,17,19]). The mathematical theory of the exponential sam-
pling formula was studied by Butzer and Jansche in [10]. The pointwise and
uniform convergence of the series Sχ
wffor continuous functions was analyzed
in [6] and the convergence of Sχ
wfwas studied in Mellin–Lebesgue spaces, see
[7]. Recently Bardaro et al. studied various approximation results using Mellin
transform which can be seen in [1,3–6]. To improve the rate of convergence, a
linear combination of Sχ
wfwas taken in [2].
The approximation of discontinuous functions by classical sampling oper-
ators was first initiated by Butzer et al. [13]. Further, the Kantorovich sampling
series for discontinuous signals was analyzed in [15]. Inspired by these works
and by [13,15] we analyze the behaviour of exponential sampling series (1.1)
as w→∞for discontinuity functions at the jump discontinuities, i.e., at a
point twhere the one-sided limits
f(t+ 0) := lim
p→0+f(t+p),

Approximation of Discontinuous Signals Page 3 of 22 23
and
f(t−0) := lim
p→0+f(t−p)
exists and are different. For a kernel χ, we define the functions
ψ+
χ(u):= 
k<log u
χ(ue−k),
and
ψ−
χ(u):= 
k>log u
χ(ue−k).
Now we recall the definition of the recurrent function. A function f:
R+→Cis said to be recurrent if f(x)=f(eax),for all x∈R+and for some
a∈R(see [12]). The fundamental interval of the above recurrent functions
can be taken as [1,e
a].
We observe that ψ+
χ(u)andψ−
χ(u) are recurrent functions with funda-
mental interval [1,e]. Indeed, we have
ψ+
χ(eu)= 
k<log u
χ(eue−k)= 
k−1<log u
χ(ue−(k−1))= 
˜
k<log u
χ(ue−˜
k)=ψ+
χ(u).
Similarly we have ψ+
χ(e−1u)=ψ+
χ(u).Similar analysis can be done for ψ−
χ(u).
Let Lp(R+),p∈[1,∞) be the set of all Lebesgue measurable and p-
integrable functions defined on R+.Let C(R+) denote the space of all real
valued bounded continuous functions on R+equipped with the supremum
norm f∞:= supx∈R+|f(x)|and for n≥1,C
(n)(R+) denotes the subspace
of C(R+) whose elements fare n-times continuously differentiable. We say
that a function f:R+→Ris log-uniformly continuous if the following hold:
for a given >0,there exists δ>0 such that |f(p)−f(q)|<whenever
|log p−log q|<δfor p, q ∈R+.The subspace consisting of all bounded log-
uniformly continuous functions on R+is denoted by C(R+).
For c∈R, we define the space
Xc={f:R+→C:f(·)(·)c−1∈L1(R+)}
equipped with the norm
fXc=f(·)(·)c−11=+∞
0
|f(y)|yc−1dy.
For f∈Xc,the Mellin transform is defined by

[f]M(s):=+∞
0
ys−1f(y)dy, (s=c+it, t ∈R).
A function f∈Xc∩C(R+),c ∈Ris said to be Mellin band-limited to the
interval [−κ, κ],if 
[f]M(c+it) = 0 for all |t|>κ, κ∈R+.

23 Page 4 of 22 S. K. Angamuthu et al. Results Math
The paper is organized as follows. In Sect. 2, we prove the representation
lemma for the exponential sampling series (1.1) and using this lemma we ana-
lyze the approximation of discontinuous functions by Sχ
wfin Theorems 2,3
and 5. Further we analyze the degree of approximation for the sampling series
(1.1) in terms of logarithmic modulus of smoothness in Sect. 3. In Sect. 4,we
study the round-off and time jitter errors for these sampling series. In Sect. 5,
we have given a construction of a family of Mellin band-limited kernels such
that χ(1) = 0 for which Sχ
wfconverge at any jump discontinuities. Further,
the convergence at discontinuity points of the sampling series Sχ
wfhas been
tested numerically and numerical results are provided in Tables 1,2and 3.
2. Approximation of Discontinuous Signals
For any given bounded function f:R+→R,we first prove the following
representation lemma for the exponential sampling series Sχ
wf. Throughout
this section we assume that the right and left limits of fat t∈R+exist and
are finite.
Lemma 1. For a given bounded function f:R+→Randafixedt∈R+,let
ht:R+→Rbe defined by
ht(x)=⎧
⎨
⎩
f(x)−f(t−0),if x<t
f(x)−f(t+0),if x>t
0,if x=t.
Then the following holds:
(Sχ
wf)(t)=(Sχ
wht)(t)+f(t−0) + ψ−
χ(tw)[f(t+0)−f(t−0)]
+χ(1)[f(t)−f(t−0)],
if wlog(t)∈Zand
(Sχ
wf)(t)=(Sχ
wht)(t)+f(t−0) + ψ−
χ(tw)[f(t+0)−f(t−0)],
if wlog(t)/∈Z.
Proof. Let w>0 such that wlog(t)∈Z. Then, we can write
(Sχ
wht)(t)= 
k<w log(t)
χ(e−ktw)(f(ek
w)−f(t−0))
+
k>w log(t)
χ(e−ktw)(f(ek
w)−f(t+ 0))
=
k<w log(t)
χ(e−ktw)f(ek
w)+ 
k≥wlog(t)
χ(e−ktw)f(ek
w)
−f(t−0) 
k<w log(t)
χ(e−ktw)

Approximation of Discontinuous Signals Page 5 of 22 23
−f(t+0) 
k>w log(t)
χ(e−ktw)−χ(1)f(t)
=(Sχ
wf)(t)−f(t−0) 
k<w log(t)
χ(e−ktw)
−f(t+0) 
k>w log(t)
χ(e−ktw)−χ(1)f(t).
Adding and subtracting f(t−0) 
k≥wlog(t)
χ(e−ktw) in the above equation and
rearranging all terms, we obtain
(Sχ
wf)(t)=(Sχ
wht)(t)+f(t−0) ⎛
⎝
k<w log(t)
χ(e−ktw)+ 
k≥wlog(t)
χ(e−ktw)⎞
⎠
+[f(t+0)−f(t−0)] 
k>w log(t)
χ(e−ktw)+χ(1)f(t)−f(t−0)χ(1)
=(Sχ
wht)(t)+f(t−0)
+∞

k=−∞
χ(e−ktw)+[f(t+0)
−f(t−0)] 
k>w log(t)
χ(e−ktw)χ(1)[f(t)−f(t−0)].
Hence using the condition that
+∞

k=−∞
χ(e−ku)=1,we can easily obtain
(Sχ
wf)(t)=(Sχ
wht)(t)+f(t−0) + ψ−
χ(tw)[f(t+0)−f(t−0)]
+χ(1)[f(t)−f(t−0)].
Now let wlog(t)/∈Zand w>0.Then repeating the same computations, we
easily obtain
(Sχ
wf)(t)=(Sχ
wht)(t)+f(t−0) + [f(t+0)−f(t−0)] 
k≥wlog(t)
χ(e−ktw)
=(Sχ
wht)(t)+f(t−0) + [f(t+0)−f(t−0)]ψ−
χ(tw).

Before proving the approximation of discontinuous functions by Sχ
wf, we
recall the following theorem proved in [6] for continuous functions on R+.
Theorem 1. Let f:R+→Rbe a bounded function and χ∈Φbe a contin-
uous kernel. Then (Sχ
wf)(t)converges to f(t)at any continuity point tof f.
Moreover, if f∈C(R+),then we have

23 Page 6 of 22 S. K. Angamuthu et al. Results Math
lim
w→∞ f−Sχ
wf∞=0.
Now we analyze the behaviour of the exponential sampling series at jump
discontinuity at t∈R+when wlog(t)∈Z.
Theorem 2. Let f:R+→Rbe a bounded signal and let t∈R+be a point
of jump discontinuity of f. Foragivenα∈R,the following statements are
equivalent:
(i) lim
w→∞
wlog(t)∈Z
(Sχ
wf)(t)=αf(t+0)+[1−α−χ(1)]f(t−0) + χ(1)f(t),
(ii) ψ−
χ(1) = α,
(iii) ψ+
χ(1) = 1 −α−χ(1).
Proof. First, we prove that (i)⇐⇒ (ii). In view of the representation Lemma 1,
we have
(Sχ
wf)(t)=(Sχ
wht)(t)+f(t−0) + ψ−
χ(tw)[f(t+0)−f(t−0)]
+χ(1)[f(t)−f(t−0)],
for any w>0 such that wlog(t)∈Z.Since htis bounded and continuous at t
and ht(t)=0,using Theorem 1, we obtain
lim
w→∞(Sχ
wht)(t)=0.
Thus, we have
lim
w→∞
wlog(t)∈Z
(Sχ
wf)(t)=f(t−0) + lim
w→∞
wlog(t)∈Z
ψ−
χ(tw)[f(t+0)−f(t−0)]
+χ(1)[f(t)−f(t−0)].
Now, we have
lim
w→∞
wlog(t)∈Z
ψ−
χ(tw) = lim
w→∞
wlog(t)∈Z⎛
⎝
k>w log(t)
χ(e−ktw)⎞
⎠.
Since wlog(t)∈Z,there exists n0such that wlog(t)=n0and ψ−
χis recurrent
with fundamental domain [1,e],we have
ψ−
χ(tw)=ψ−
χ(en0)=ψ−
χ(en0−1)=···=ψ−
χ(1).
Therefore, we have
ψ−
χ(tw)=ψ−
χ(1),∀w, t such that wlog(t)∈Z.

Approximation of Discontinuous Signals Page 7 of 22 23
Therefore, we have
lim
w→∞
wlog(t)∈Z
(Sχ
wf)(t)=ψ−
χ(1)f(t+0)+[1−ψ−
χ(1) −χ(1)]f(t−0) + χ(1)f(t).
Now (i)⇐⇒ αf(t+0)+[1−α−χ(1)]f(t−0) + χ(1)f(t)
=ψ−
χ(1)f(t+0)+[1−ψ−
χ(1) −χ(1)]f(t−0) + χ(1)f(t)
⇐⇒ ψ−
χ(1)(f(t+0)−f(t−0)) = α(f(t+0)−f(t−0))
⇐⇒ ψ−
χ(1) = α
⇐⇒ (ii) holds.
Since
+∞

k=−∞
χ(e−ktw)=1,we have
ψ+
χ(1) = 1 −χ(1) −ψ−
χ(1).
This implies that (ii)⇐⇒ (iii).Hence, the proof is completed. 
Next we analyze the behaviour of the exponential sampling series at jump
discontinuity at t∈R+when wlog(t)/∈Z.
Theorem 3. Let f:R+→Rbe a bounded signal and let t∈R+be a point
of jump discontinuity of f. Let α∈R.Then the following statements are
equivalent:
(i) lim
w→∞
wlog(t)/∈Z
(Sχ
wf)(t)=αf(t+0)+(1−α)f(t−0),
(ii) ψ−
χ(u)=α, for every u∈(1,e),
(iii) ψ+
χ(u)=1−α, for every u∈(1,e).
Proof. Using the representation result given in Lemma 1, we obtain
lim
w→∞
wlog(t)/∈Z
(Sχ
wf)(t)=f(t−0) + lim
w→∞
wlog(t)/∈Z
ψ−
χ(tw)[f(t+0)−f(t−0)]
(i)⇐⇒ αf(t+0)+(1−α)f(t−0) = f(t−0)
+lim
w→∞
wlog(t)/∈Z
ψ−
χ(tw)[f(t+0)−f(t−0)]
⇐⇒ α[f(t+0)−f(t−0)] = lim
w→∞
wlog(t)/∈Z
ψ−
χ(tw)[f(t+0)−f(t−0)]
⇐⇒ α= lim
w→∞
wlog(t)/∈Z
ψ−
χ(tw)
⇐⇒ α=ψ−
χ(u),∀u∈(1,e)
⇐⇒ (ii) holds.

23 Page 8 of 22 S. K. Angamuthu et al. Results Math
Let wlog(t)/∈Z.Then, we have
ψ+
χ(tw)+ψ−
χ(tw)=1.
Thus, we obtain
(ii)⇐⇒ ψ+
χ(u)=1−α, ∀u∈(1,e).

The assertion (i) implies (ii) in the above Theorem 3can also be seen
explicitly using the proof of the corresponding negation. This can be seen from
the following theorem.
Theorem 4. Let χbe a kernel such that ψ−
χ(u)is not constant on (1,e).Let
f:R+→Rbe a bounded signal with jump discontinuity t∈R+.Then (Sχ
wf)(t)
does not converge pointwise at t.
Proof. Suppose not. Then lim
w→∞
wlog(t)/∈Z
(Sχ
wf)(t)=, for some ∈R.By the
uniqueness of the limit and Lemma 1, we obtain
=f(t−0) + lim
w→∞ ψ−
χ(tw)[f(t+0)−f(t−0)].
Since f(t+0)−f(t−0) =0,we obtain
−f(t−0)
f(t+0)−f(t−0) = lim
w→∞ ψ−
χ(tw).
The above expression gives a contradiction. Indeed, if
lim
w→∞ ψ−
χ(tw)=C,
where Cis a constant, then it fails to satisfy that ψ−
χis recurrent and not a
constant, hence the theorem is proved. 
The convergence of exponential sampling series at jump discontinuities
for the cases wlog(t)∈Zand wlog(t)/∈Zare proved separately in Theorems 2
and 3. By imposing the additional condition that χ(1) = 0,these two cases
are treated together in the following theorem. Further, assuming that χis
continuous we obtain two more equivalent statements.
Theorem 5. Let f:R+→Rbe a bounded signal and let t∈R+be a point
of jump discontinuity of f. Let α∈R.Suppose that the kernel χsatisfies the
additional condition χ(1) = 0.Then, the following statements are equivalent:

Approximation of Discontinuous Signals Page 9 of 22 23
(i) lim
w→∞(Sχ
wf)(t)=αf(t+0)+(1−α)f(t−0),
(ii) ψ−
χ(u)=α, for every u∈[1,e),
(iii) ψ+
χ(u)=1−α, for every u∈[1,e).
Moreover, if in addition we assume that χis continuous on R+,then the above
statements are equivalent to the following statements:
(iv)
1
0
χ(u)u2kπi du
u=0,if k=0
α, if k=0,
(v)
∞
1
χ(u)u2kπi du
u=0,if k=0
1−α, if k=0.
Proof. Proceeding along the lines proof of Theorem 2and 3, we see that (i),(ii)
and (iii) are equivalent. Let χis continuous on R+.Let
χ0(u)= χ(u),for u<1
0,for u≥1.
Then, we have
ψ−
χ(u)= 
k>log u
χ(ue−k)=
k∈Z
χ0(ue−k).
Therefore, ψ−
χis recurrent continuous function with the fundamental interval
[1,e].Using Mellin–Poisson summation formula (see [11]), we obtain
ψ−
χ(u)=
+∞

k=−∞

[χ0]M(2kπi)u−2kπi =
+∞

k=−∞ 1
0
χ(u)u2kπi du
uu−2kπi.
Therefore, we obtain
ψ−
χ(u)=α, ∀u∈[1,e)
⇐⇒ 
[χ0]M(2kπi)=0,if k=0
α, if k=0
⇐⇒ 1
0
χ(u)u2kπi du
u=0,if k=0
α, if k=0.
This implies that (ii)⇐⇒ (iv).Finally using the condition

23 Page 10 of 22 S. K. Angamuthu et al. Results Math
+∞

k=−∞
χ(e−ku)=1⇐⇒ 
[χ]M(2kπi)=0,if k=0
1,if k=0
the equivalence between (iv) and (v) can be established easily. Thus the proof
is completed. 
Remark 1. Let f:R+→Rbe a bounded signal with a removable discontinuity
at t∈R+, i.e. f(t+0)=f(t−0) = . Then we have
(i) lim
w→∞
wlog(t)∈Z
(Sχ
wf)(t)=+χ(1)[f(t)−],
(ii) lim
w→∞
wlog(t)/∈Z
(Sχ
wf)(t)=,
(iii) If χ(1) = 0,then lim
w→∞(Sχ
wf)(t)=.
3. Degree of Approximation
In this section, we estimate the order of convergence of the exponential sam-
pling series by using the logarithmic modulus of continuity. Let f∈C(R+).
Then the logarithmic modulus of continuity is defined by
ω(f,δ):=sup{|f(p)−f(q)|: whenever |log p−log q|≤δ, δ ∈R+}.
The logarithmic modulus of continuity satisfies the following properties:
(a) ω(f,δ)→0,as δ→0.
(b) ω(f,cδ)≤(c+1)ω(f,δ),for every δ, c > 0.
Further properties of logarithmic modulus of continuity can be seen in [1]. In
the following theorem, we obtain the order of convergence for the exponential
sampling series when Mν(χ)<∞for 0 <ν<1.
Theorem 6. Let χ∈Φbe a kernel such that Mν(χ)<∞for 0<ν<1and
f∈C(R+). Then for sufficiently large w>0,the following hold:
|(Sχ
wf)(t)−f(t)|≤ω(f,w−ν)[Mν(χ)+2M0(χ)] + 2ν+1 f∞Mν(χ)w−ν,
for every t∈R+.
Proof. Let t∈R+be fixed. Then using the condition
+∞

k=−∞
χ(e−ktw)=1,we
obtain
|(Sχ
wf)(t)−f(t)|=
+∞

k=−∞
χ(e−ktw)(f(ek
w)−f(t))
≤⎛
⎜
⎝
k−wlog t<w
2
+
k−wlog t≥w
2
⎞
⎟
⎠χ(e−ktw)|f(ek
w)−f(t)|
:= I1+I2.

Approximation of Discontinuous Signals Page 11 of 22 23
Let 0 <ν<1.Then we have
ωf,
k
w−log t≤ωf,
k
w−log t
ν.
Therefore, using the above inequality and the property (b),we obtain
I1≤
k−wlog t<w
2χ(e−ktw)ωf, 
k
w−log t
ν
≤
k−wlog t<w
2χ(e−ktw)1+wν
k
w−log t
νω(f,w−ν)
≤
k−wlog t<w
2χ(e−ktw)ω(f, w−ν)
+
k−wlog t<w
2χ(e−ktw)wν
k
w−log t
ν
ω(f,w−ν)
≤ω(f,w−ν)
k−wlog t<w
2χ(e−ktw)
+ω(f,w−ν)
k−wlog t<w
2χ(e−ktw)k−wlog t
ν.
In view of the conditions M0(χ)andMν(χ),we easily obtain
I1≤ω(f,w−ν)[M0(χ)+Mν(χ)].
Now we estimate I2.Since k−wlog t≥w
2,we have
1
k−wlog t
ν≤2νw−ν,0<ν<1.
Hence, we obtain
I2≤2f∞
k−wlog t≥w
2χ(e−ktw)
≤2f∞
k−wlog t≥w
2
k−wlog t
ν
k−wlog t
νχ(e−ktw)
≤2ν+1f∞w−ν
k−wlog t≥w
2χ(e−ktw)k−wlog t
ν
≤2ν+1f∞w−νMν(χ)<∞.
On combining the estimates I1and I2,we get the desired estimate. 

23 Page 12 of 22 S. K. Angamuthu et al. Results Math
4. Round-Off and Time Jitter Errors
This section is devoted to analyze round off and time jitter errors connected
with exponential sampling series (1.1). The round-off error arises when the
exact sample values f(ek
w) are replaced by approximate close ones ¯
f(ek
w)in
the sampling series (1.1). Let ξk=f(ek
w)−¯
f(ek
w) be uniformly bounded by
ξ, i.e., |ξk|≤ ξ, for some ξ>0.We are interested in analyzing the error when
f(t) is approximated by the following exponential sampling series:
(Sχ
w¯
f)(t)=
+∞

k=−∞
χ(e−ktw)¯
f(ek
w).
The total round-off or quantization error is defined by
(Qξf)(t):=|(Sχ
wf)(t)−(Sχ
w¯
f)(t)|.
Theorem 7. For f∈C(R+),the following hold:
(i) (Qξf)C(R+)≤ξM0(χ)
(ii) f−Sχ
w¯
fC(R+)≤Cω f, 1
w+ξM0(χ),where C=M0(χ)+M1(χ).
Proof. The error in the approximation can be splitted as
|f(t)−(Sχ
w¯
f)(t)|≤|f(t)−(Sχ
wf)(t)|+(Qξf)(t):=I1+(Qξf)(t).
The term I1is the error arising if the actual sample value is used and the total
round-off or quantization error can be evaluated by
(Qξf)C(R+)=sup
t∈R+
+∞

k=−∞
χ(e−ktw)f(ek
w)−
+∞

k=−∞
χ(e−ktw)¯
f(ek
w)
=sup
t∈R+
+∞

k=−∞ 
ξkχ(e−ktw)
≤ξM0(χ).
In view of Theorem 4 [6, page no. 7], we have
I1≤M0(χ)ω(f,δ)+ ω(f, δ)
wδ M1(χ).
On combining the estimates I1and I2,we get
f−Sχ
w¯
fC(R+)≤M0(χ)+M1(χ)
wδ ω(f,δ)+ξM0(χ).
Choosing δ=1
w,we obtain
f−Sχ
w¯
fC(R+)≤Cω(f, 1
w)+ξM0(χ),
where C=M0(χ)+M1(χ).Hence, the proof is completed. 

Approximation of Discontinuous Signals Page 13 of 22 23
The time-jitter error occurs when the function f(t) being approximated
from samples which are taken at perturbed nodes, i.e., the exact sample values
f(ek
w) are replaced by f(ek
w+ek) in the sampling series (1.1). So we are
interested in analyzing time jitter error and the approximation behaviour when
f(t) is approximated by the sampling series
+∞

k=−∞
χ(e−ktw)f(ek
w+ek).We
assume that the values kare bounded by a small number , i.e., |k|≤ ,
for all k∈Zand for some >0.The total time jitter error is defined by
Jf(t):=
+∞

k=−∞
χ(e−ktw)fek
w−
+∞

k=−∞
χ(e−ktw)fek
w+ek
.
Theorem 8. For f∈C(1)(R+),the following hold:
(i) JfC(R+)≤efC(R+)M0(χ)
(ii) 



f(.)−
+∞

k=−∞
χ(e−k(.)w)f(ek
w+ek)


C(R+
)
≤Cω f, 1
w+efC(R+)
M0(χ),
where C=M0(χ)+M1(χ).
Proof. Applying the mean value theorem, the total time jitter error is esti-
mated by
Jf(t)=
+∞

k=−∞
χ(e−ktw)f(ek
w)−
+∞

k=−∞
χ(e−ktw)f(ek
w+ek)
≤
+∞

k=−∞ χ(e−ktw)f(ek
w)−f(ek
w+ek)
≤
+∞

k=−∞ χ(e−ktw)f(tk,w )|ek|.
Therefore, we obtain
JfC(R+)≤efC(R+)sup
t∈R+
+∞

k=−∞
|χ(e−ktw)|
≤efC(R+)M0(χ).
From the above estimates it is clear that the jitter error essentially depends on
the smoothness of function f. For f∈C(1)(R+),the associated approximation
error is estimated by

f(t)−
+∞

k=−∞
χ(e−ktw)f(ek
w+ek)

23 Page 14 of 22 S. K. Angamuthu et al. Results Math
≤
f(t)−
+∞

k=−∞
χ(e−ktw)fek
w
+
+∞

k=−∞
χ(e−ktw)fek
w−
+∞

k=−∞
χ(e−ktw)fek
w+ek
≤|f(t)−Sχ
wf(t)|+Jf(t).
Again using Theorem 4 [6, page no. 7], we have
|f(t)−Sχ
wf(t)|≤M0(χ)ω(f,δ)+ ω(f, δ)
wδ M1(χ).
Using the above estimate and Jf, we obtain





f(.)−
+∞

k=−∞
χ(e−k(.)w)fek
w+ek



C(R+)
≤M0(χ)ω(f,δ)+ ω(f, δ)
wδ M1(χ)+efC(R+)M0(χ).
Choosing δ=1
w,we obtain the desired result. 
5. Examples of the Kernels
In this section, we provide certain examples of the kernel functions which will
satisfy our assumptions. First we give the family of Mellin-B spline kernels.
The Mellin B-spline of order nis given by
¯
Bn(x):= 1
(n−1)!
n

j=0
(−1)jn
jn
2+ log x−jn−1
+,x∈R+
It can be easily seen that ¯
Bn(x) is compactly supported for every n∈N.The
Mellin transform of ¯
Bn(see [6]) is

[¯
Bn]M(c+it)=sin t
2
t
2n
,t=0,c=0.
The Mellin–Poisson summation formula [11] is defined by
(i)j
+∞

k=−∞
χ(ekx)(k−log u)j=
+∞

k=−∞
dj
dtj

[χ]M(2kπi)x−2kπi,for k∈Z.
We need the following lemma (see [6]).
Lemma 2. The condition
+∞

k=−∞
χ(e−kxw)=1is equivalent to

Approximation of Discontinuous Signals Page 15 of 22 23

[χ]M(2kπi)=1,if k=0
0,otherwise.
Moreover mj(χ, u)=0for j=1,2,...,n is equivalent to dj
dtj

[χ]M(2kπi)=0
for j=1,2,...,n and ∀k∈Z.
Using the above Lemma, we obtain

[¯
Bn]M(2kπi)=1,if k=0
0,otherwise.
Using Mellin’s-Poisson summation formula, it is easy to see that ¯
Bn(x) satisfies
the condition (i). As ¯
Bn(x) is compactly supported, the condition (ii) is also
satisfied. Next we consider the Mellin Jackson kernels. For x∈R+,β ∈N,γ ≥
1,the Mellin Jackson kernels are defined by
Jγ,β(x):=dγ,βx−csinc2βlog x
2γβπ,
where
d−1
γ,β := ∞
0
sinc2βlog x
2γβπdu
u.
One can easily verify that the Mellin Jackson kernels also satisfies conditions (i)
and (ii) (see [6]). We can analyze the convergence of the exponential sampling
series with jump discontinuity associated with these kernels only for the case
given in Theorem 2and we observe that χ(1) = 0. So Theorem 5can not be
applied for these kernels. In order to find examples of the kernels for which
the exponential sampling series converge at any jump discontinuity t∈R+of
the given bounded signal f:R+→R, we need to construct suitable kernels.
One such construction is given in the following theorem.
Theorem 9. Let χa,χ
bbe two continuous kernels supported respectively in the
intervals [e−a,e
a]and [e−b,e
b].Let α∈Rbe fixed. We define χ:R+→R+
by
χ(u):=(1−α)χa(2ue−a−1)+αχb(2ueb),u∈R+.
Then χis a kernel satisfying conditions (i), (ii) and χ(1) = 0.Moreover, the
corresponding exponential sampling series Sχ
wf,w > 0based upon χsatisfy (i)
of Theorem 5with parameter αfor a given bounded signal f:R+→Rat any
jump discontinuity t∈R+of f.
Proof. The Mellin transform of χ(u)is

[χ]M(s)=∞
0
(1 −α)χa(2te−a−1)ts−1dt +∞
0
αχb(2teb)ts−1dt
=(1−α)
[χa]M(s)e(1+a)
2s
+α
[χb]M(s)e−b
2s
.

23 Page 16 of 22 S. K. Angamuthu et al. Results Math
It is simple to check that χsatisfies condition (ii). Now we show that kernel
satisfies the condition (i). We obtain

[χ]M(2kπi)=(1−α)
[χa]M(2kπi)e(1+a)
22kπi
+α
[χb]M(2kπi)e−b
22kπi
.
As χaand χbsatisfies condition (i), we have

[χa]M(2kπi)=
[χb]M(2kπi)=0,if k=0
1,if k=0.
For suitable choices of aand b, we obtain

[χ]M(2kπi)=0,if k=0
1,if k=0.
Therefore, χsatisfies condition (i) and we can easily see that χ(1) = 0.Now,
we obtain
1
0
χ(u)u2kπi−1du =α1
0
χb(2ebu)u2kπi−1du
=α
[χb]M(2kπi)e−b2kπi
22kπi 
=0,if k=0
α, if k=0.
Therefore, the condition (iv) of Theorem 5is satisfied, hence the proof is
completed. 
Now we test numerically the approximation of discontinuous function
f(t)=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
11
2t2+1,t<
3
2
3,3
2≤t<7
2
2,7
2≤t<11
2
12
1+2t,t≥11
2
by exponential sampling series at jump discontinuities at t=3
2,t=7
2and
t=11
2.We consider a linear combination of Mellin B-spline kernels defined
by (see Fig. 1)
χ(t)=1
4¯
B2(2te−2)+3
4¯
B2(2te),

Approximation of Discontinuous Signals Page 17 of 22 23
Figure 1. Plot of the kernel χ(t)=1
4¯
B2(2te−2)+3
4¯
B2(2te).
Table 1. Approximation of fat the jump discontinuity point
t=3
2by the exponential sampling series Sχ
wfbased on χ(t)
for different values of w>0.
w5 10 20 50 100 200
Sχ
wf3.0036 2.8669 2.8059 2.7717 2.7608 2.7554
The theoretical limit of (Sχ
wf)3
2as w→∞is
3
4f3
2+0
+1
4f3
2−0=2.75
where ¯
B2is given by
¯
B2(t)=⎧
⎪
⎨
⎪
⎩
1−log t, 1<t<e
1 + log t, 1
e<t<1
0,otherwise.
Clearly the exponential sampling series Sχ
wfbased on χ(t) satisfies the condi-
tions (i), (ii) and χ(1) = 0.We also observe that the condition (i) of Theorem 5
is satisfied with α=3
4.From Theorems 5and 9, we have that at the disconti-
nuity points of f, the sampling series Sχ
wfconverges to 3
4f(t+0)+1
4f(t−0).

23 Page 18 of 22 S. K. Angamuthu et al. Results Math
Table 2. Approximation of fat the jump discontinuity point
t=7
2by the exponential sampling series Sχ
wfbased on χ(t)
for different values of w>0.
w5 10 20 50 100 200
Sχ
wf2.25 2.25 2.25 2.25 2.25 2.25
The theoretical limit of (Sχ
wf)7
2as w→∞is
3
4f7
2+0
+1
4f7
2−0=2.25
Table 3. Approximation of fat the jump discontinuity point
t=11
2by the exponential sampling series Sχ
wfbased on χ(t)
for different values of w>0.
w5 10 20 50 100 200
Sχ
wf1.0492 1.1420 1.1939 1.2271 1.2384 1.2442
The theoretical limit of (Sχ
wf)11
2as w→∞is
3
4f11
2+0
+1
4f11
2−0=1.25
The convergence of the sampling series Sχ
wfat discontinuity points t=3
2,
t=7
2and t=11
2of the function fhas been tested and numerical results
are presented in Tables 1,2and 3. The plots of the function ftogether with
its approximation Sχ
wfare shown in Figs. 2and 3respectively for w=5and
w=10.
We now consider discontinuous kernels: χd(t):=χc(t)+τ(t),t∈R+,
where
χc(t)=2
5¯
B2(te−2)+3
5¯
B2(te)
and
τ(t)=⎧
⎪
⎪
⎨
⎪
⎪
⎩
1,t=e, 1
e
−1,t=e2,1
e2
0,otherwise.

Approximation of Discontinuous Signals Page 19 of 22 23
Figure 2. Approximation of f(t)bySχ
wfbased on χ(t)=
1
4¯
B2(2te−2)+3
4¯
B2(2te)forw=5.
Figure 3. Approximation of f(t)bySχ
wfbased on χ(t)=
1
4¯
B2(2te−2)+3
4¯
B2(2te)forw= 10.

23 Page 20 of 22 S. K. Angamuthu et al. Results Math
Figure 4. Plot of the kernel χd(t)=χc(t)+τ(t).
Then, we observe that 
k∈Z
τ(e−ku)=0,and ψ−
χ(u) = 0 for every u∈[1,e).
It can be seen that χd(u) is not necessarily a continuous (see Fig. 4) and it is
easy to see that χd(u) satisfies the condition (i), (ii) and χd(u)=0,for every
u∈[1,e).Again from Theorem 5, we have that at the discontinuity points of
f, the sampling series Sχd(t)
wfconverges to 3
5f(t+0)+ 2
5f(t−0).
Acknowledgements
The first two authors are supported by DST-SERB, India Research Grant
EEQ/2017/000201. The third author P. Devaraj has been supported by DST-
SERB Research Grant MTR/2018/000559.
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23 Page 22 of 22 S. K. Angamuthu et al. Results Math
Sathish Kumar Angamuthu and Prashant Kumar
Department of Mathematics
Visvesvaraya National Institute of Technology Nagpur
Nagpur Maharashtra440010
India
e-mail: mathsatish9@gmail.com;
pranwd92@gmail.com
Devaraj Ponnaian
School of Mathematics
Indian Institute of Science Education and Research
Thiruvananthapuram
India
e-mail: devarajp@iisertvm.ac.in
Received: December 9, 2020.
Accepted: October 30, 2021.
Publisher’s Note Springer Nature remains neutral with regard to jurisdic-
tional claims in published maps and institutional affiliations.