Asian Journal of Mathematics & Statistics

Year: 2013 | Volume: 6 | Issue: 1 | Page No.: 12-22
DOI: 10.3923/ajms.2013.12.22

Approximation of Signals by Product Summability Transform

Vishnu Narayan Mishra, Huzoor H. Khan, Kejal Khatri, Idrees A. Khan and Lakshmi N. Mishra

Abstract: The theory of approximation is a very extensive field which has various applications in pure and applied mathematics. Broadly speaking, Signals are treated as functions of one variable and images are represented by functions of two variables. The present study deals with the new theorem on the degree of approximation of a Signal associated with Fourier series and belonging to the generalized weighted W(L_r, ξ(t)) (r≥1, t>0)- class by product summability (C, 1) (E, q) method, where ξ (t) is non-negative and non-decreasing function of t. The main result obtained in this study generalizes some well-known results in this direction. The class W(L_rξ(t)) (r≥1, t>0), we have used here in the main theorem includes the Lip (ξ(t)), Lip (α, r) and Lip α classes.

Keywords: degree of approximation, Fourier series and product summability (C, 1) (E, q) transform

INTRODUCTION

Khan (1974) has studied the degree of approximation of a function belonging to Lip (α, r) and W (L_r, ξ(t)) classes by Norlund and generalized Norlund means. Working in the same direction, Mittal et al. (2006), Mittal and Mishra (2008), Mishra (2009), Mishra et al. (2011), Mishra and Mishra (2012) and Mishra et al. (2012) have studied the degree of approximation of a 2π periodic signal belonging to W (L_r, ξ(t)) and other classes through trigonometric Fourier approximation by positive linear operators. Recently, Rhoades et al. (2011) have determined very interesting result on the degree of approximation of a function belonging to Lip α class by Hausdorff means. But nothing seems to have been done so far to obtain the degree of approximation of a Signal associated with Fourier series and belonging to the generalized weighted W (L_r, ξ(t))-class by product summability (C, 1) (E, q) method. The generalized weighted class W (L_r, ξ(t)) (r≥1) is generalization of Lip α, Lip (α, r) and Lip (ξ(t), r) classes. Therefore, in the present paper, a new theorem on the degree of approximation of signals belonging to the generalized weighted W (L_r, ξ(t)), r≥1 class by (C, 1) (E, q) product summability means of Fourier series with a proper set of conditions has been proved.

Let f(x) be a 2π-periodic signal (function) and let fεL₁(0, 2π) = L₁. Then the Fourier series of a function (signal) f at any point x is given by:

with partial sums s_n(f;x)-a trigonometric polynomial of degree (or order) n, of the first (n+1) terms of the Fourier series of f.

A signal (function) fεLip α, for 0<α≤1, if |f(x+t)-f(x)| = O(t^α).

A signal fεLip (α, r) for r≥1, 0<α≤1, (Khan, 1974; McFadden, 1942), if:

Given a positive increasing function ξ(t) and an integer r≥1, fεLip (ξ(t), r), (Mittal et al., 2011), Mishra et al. (2011) if:

In case ξ(t) = t^α, 0<α≤1, then Lip(ξ(t), r) coincides with the class Lip(α, r). If r→∞ in Lip(α, r) class then this class reduces to Lipα.

For a given positive increasing function ξ(t), an integer r≥1, fεW(L_r, ξ(t)) (Khan, 1982), if:

We note that, if β = 0 then the weighted class W(L_r, ξ(t)) coincides with the class Lip(ξ(t), r) and if ξ(t) = t^α then Lip(ξ(t), r) class coincides with the class Lip(α, r). Lip(α, r)→Lipα for r→∞.

Also we observe that:

The L_r-norm of a signal f : R→R is defined by:

The L_∞-norm of a function f: R→R is defined by ||f||_∞ = sup{|f(x)|: xεR} and the degree of approximation E_n(f, x) is given by Zygmund (1959):

in terms of n, where:

is a trigonometric polynomial of degree n. This method of approximation is called Trigonometric Fourier Approximation (TFA) Mishra et al. (2012):

Let:

be a given infinite series with the sequence of nth partial sums {s_n}. If:

then an infinite series:

with the partial sums s_n is said to be (E, q) summable to the definite number s (Hardy, 1949).

An infinite series:

is said to be (C, 1) summable to s if:

The (C, 1) transform of the (E, q) transform E^q_n defines the Cesáro-Euler (C, 1) (E, q) transform of the partial sums s_n of the series i.e., the product summability (C, 1) (E, q) is obtained by superimposing (C, 1) summability on (E, q) summability.

Thus, if:

where, E^q_n denote the (E, q) transform of s_n, then an infinite series with the partial sums s_n is said to be summable (C, 1) (E, q) means or simply summable (C, 1) (E, q) to the definite number s and we can write:

We note that (C, 1) and (C, E)^q_n are also trigonometric polynomials of degree (or order) n.

The (C, 1) summability method is regular and the regularity condition of (C, 1) (E, q) method is as follows:

method is regular method is regular method is regular.

Riesz-Hölder Inequality states that if r and s be non-negative extended real numbers such that 1/r+1/s. If fεL^r(a, b) and gεL^s(a, b) then f. gεL¹(a, b) and:

Equality holds if and only if, for some non-zero constants A and B, we have A|f|^r = B|g|^sa.e.

Second mean value theorem for integration states that if G: (a, b)→R is a positive monotonically decreasing function and φ: (a, b)→R is an integrable function, then ∃ a number xε(a, b) such that:

Here G(a+0) stands for the existence of which follows from the conditions. Note that it is essential that the interval (a, b] contains b. A variant not having this requirement is:

If G: (a, b)→R is a monotonic (not necessarily decreasing and positive) function and φ: (a, b)→R is an integrable function, then ∃ a number Xε(a, b) such that:

We shall use the following notations:

Furthermore C will denote an absolute positive constant, not necessarily the same at each occurrence.

MAIN RESULT

Various investigators such as Qureshi (1981, 1982), Khan (1974), Qureshi and Neha (1990a), Qureshi and Neha (1990b) discussed the degree of approximation of signals belonging to Lipα, Lip(α, r), Lip(ξ(t), r) and W(L_r, ξ(t))-classes of an infinite series through trigonometric Fourier approximation using different summability matrices with monotone rows.

In the present study, we determine the degree of approximation for the signals f of weighted W(L_r, ξ(t)) (r≥1)-class by using product summabilities (C, 1) (E, q) means of its Fourier series. We prove:

Theorem 1: If f: R→R is a 2π-periodic, Lebesgue integrable and belonging to weighted W(L_r, ξ(t)) (r≥1)-class, then the degree of approximation of f(x) by (C, 1) (E, q) means of its Fourier series is given by:

provided ξ(t) is positive increasing function of t satisfying the following conditions:

and

where, δ is an arbitrary number such that s(1-δ+β)-1>0, s the conjugate index of r, r^-1+s^-1 = 1, conditions (9) (10) hold uniformly in x and (C, 1)^q_n are (C, 1) (E, q) means of Fourier series (1).

Note 1: Condition (11) implies ξ(π/n)≤πξ(1/n), for (π/n)≤(1/n) i.e. (n/π)ξ(π/n)≤nξ(1/n).

Note 2: The product transform (C, 1) (E, q) plays an important role in signal theory as a double digital filter (Mittal and Singh, 2008) and the theory of machines in mechanical engineering (Mishra et al., 2012).

Lemmas: In order to prove our theorem 1, we require the following lemma:

Proof of lemma 1: Using sin nt≤n sin t for 0<t<π/n then:

This completes the proof of lemma 1.

Proof of lemma 2: Using sin(t/2)≥(t/π) and sin kt≤1 for π/n<t<π, we obtain:

This completes the proof of lemma 2.

Proof of theorem 1: It is well known from Titchmarsh (1939) that the nth partial sum s_n of Fourier series (1) at t = x may be written as:

so that, (E, q) means (transform) of s_n(f, x) are given by:

Now, the (C, 1) (E, q) transform of s_n(f, x) is given by:

Where:

Therefore, we have:

Using Hölder’s inequality, condition (9), note 1, Lemma 1, the fact that:

r^-1+s^-1 second mean value theorem for integrals, we find:

Now by Hölder’s inequality, conditions (10), lemma 2, the fact that:

r^-1+s^-1 we obtain:

Now putting:

Since ξ(t) is a positive increasing function, so:

is a positive increasing function and using second mean value theorem for integrals:

Combining I₁ and I₂ yields:

Now, using the L_r-norm, we get:

This completes the proof of our theorem 1.

APPLICATIONS

It is well known that the theory of approximation i.e., TFA which originated from a theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 131 years. These approximations have assumed important new dimensions due to their wide applications in signal analysis, in general and in digital signal processing in particular, in view of the classical Shannon sampling theorem. Broadly speaking, signals are treated as function of one variable and images are represented by functions of two variable.

The theory of approximation is a very extensive field which has various applications. From the point of view of the applications, Sharper estimates of infinite matrices (Mittal et al., 2011), are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix valued functions and enables to investigate perturbations of matrix valued functions and compare them. The following corollaries can be derived from our main Theorem 1.

Corollary 1: If β = 0 and ξ(t) = t^α, 0<α≤1, then the weighted W(L_r, ξ(t)) (r≥1)-class reduces to Lip(α, r)-class and the degree of approximation of a function f(x)εLip(α, r) is given by:

Proof of corollary 1: From our theorem 1 for β = 0, we have:

Thus we get:

This completes the proof of corollary 1.

Corollary 2: If ξ(t) = t^α, 0<α<1 and r→∞ in corollary 1, then f(x)εLipα and:

Proof of corollary 2: For r = ∞ in (16) we obtain:

Thus we get:

This completes the proof of corollary 2 which is theorem of Lal and Kushwaha (2009).

Corollary 3: If f: R→R is a 2π-periodic, Lebesgue integrable and belonging to weighted W(L_r, ξ(t)) (r≥1)-class, then the degree of approximation of f(x) by (C, 1) (E, q) means of its Fourier series is given by:

provided ξ(t) is positive increasing function of t satisfying the conditions (9) (10) uniformly in x (11) and (C, E)¹_n are (C, 1) (E, 1) means of Fourier series (1).

Proof of corollary 3: An independent proof of the corollary can be derived by taking q = 1 along the same lines as in our theorem 1.

Note 3: If we put β = 0 in our corollary 3 then f(x)εLip(ξ(t) r) and hence a theorem of Lal and Singh (2002) becomes particular case of our theorem 1.

Remarks
Example 1: Consider the infinite series:

The nth partial sum of (18) is given by:

and so:

Therefore the series (18) is not (E, 1) summable. Also the series (18) is not (C, 1) summable. But since {(-1)ⁿ} is (C, 1) summable, the series (18) is (C, 1) (E, 1) summable. Therefore the product summability (C, 1) (E, 1) is more powerful than the individual methods (C, 1) and (E, 1). Consequently (C, 1) (E, 1) mean gives better approximation than the individual methods (C, 1) and (E, 1).

CONCLUSION

The present study has obtained some results pertaining to the degree of approximation of signals (functions) belonging to the various classes have been reviewed. Further, a proper set of conditions have been discussed to rectify the errors and applications pointed out in Notes 1 and 2. These results are quite general in nature and reduce to corresponding various spaces of functions and their several special cases. Thus these results can be applied to various problems of Mathematical Analysis, Mathematical Physics, Electronics and Communication Technology and other Engineering branches etc.

ACKNOWLEDGMENTS

The authors are highly thankful to the anonymous reviewer for the careful reading, their critical remarks, valuable comments and several useful suggestions leading to overall improvements and the better presentation of the present paper. The authors are also grateful to all the members of editorial board of Asian Journal of Mathematics and Statistics (Science Alert New York, USA) for their kind cooperation during communication.

REFERENCES