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Error Estimation of the Function by Using Product Means of the Conjugate Fourier Series

Aradhana Dutt Jauhari* and Pankaj Tiwari

Division of Mathematics, Department of Basic Sciences, Galgotias University, Greater Noida, G. B. Nagar, U.P., India

Abstract

The purpose of the current chapter is to attain the best result on a new way and the best approximation of different classes of the work, which has been discussed by different mathematician under different summability means. Here, we are presenting the theorems established under means of the CFS of a signal, belongs to Some known and unknown results have been proven by many mathematicians. But this is a new and unique way of proving a new result.

Keywords: Generalized Zygmund class product means, Degree of Approximation (DoA), Conjugate Fourier Series (CFS)

1.1 Introduction

Let, us take San - an infinite series and sn - the sequence of partial sums. Also, {pn} and {qn} are the sequences of positive real numbers s.t.-

1.1.1 Definition 1

Sequence-to-sequence of the transformation:

defines, mean of {sn}.

If the San is summable to s by (E, s) method regular ([12] Hardy, 1949).

1.1.2 Definition 2

Sequence-to-sequence transformation:

where,

Rn = po qn + p1 qn - 1 + ? + pn qo (?), p-1 = q-1 = R-1 then the series San is summable to s. Regularity conditions [12]-

1. where M > 0 and independent of n.

1.1.3 Definition 3

The transform (E, s) over summability and is given by [7]-

and San is said to summable to s by the product means

Remark. If we put qn = 1 in equation (1.3) then summability mean reduces to mean and for pn = 1 it becomes to mean.

Now we take Fourier and its CFS (Conjugate Fourier series) as-

The partial sum is given by,

Let known as space function which is 2p - periodic and also integrable. Norm ||.||r is defined as,

also,

again, C2p- The Banach space of 2p- periodic functions, defined on [0, 2p] under the sup. norm.

Where, 0 < a = 1,

the function space [11].

For g ? Lr [0, 2p], r = 1,

For g ? C2p and r = 8

also,

Now,

also, Banach space with the norm ||.||(a), r can be defined as the space Z(a), r = 1, 0 < a = 1

The class function

Where, w > 0, continuous function having sub-linear property, that is

1. w (0) = 0,
2. w (t1 + t2) = w (t1) + w (t2)

Let w: [0, 2p] Rs.t.w (t) > 0 for 0 = t < 2p.

Clearly is a norm on

As we know Lr (r = 1) is complete, the space is also complete Banach space under the norm

Used notations in this paper:

1.2 Theorems

Many mathematicians and the researcher studies and worked on Error Estimation of FS and its CFS in different Lip class, Weightage class as well as Zygmund class using various product means. Amongst them some are Nigam [1-3], Singh [4], Deer [8], Lal [5], Pradhan [6, 7] and Mishra [9, 10] etc.

We are going to prove a new result for conjugate Fourier series belonging to means.

1.2.1 Theorem 1

Let 2p-periodic function and Lebesgue integrable in [0, 2p] and belonging to then using the product mean and the DoA of the function is given by-

Where w(t) and v(t) denoted the Zygmund modulus of continuity s.t. positive.

1.2.2 Theorem 2

Let is 2p - periodic function and Lebesgue integrable in [0, 2p] and belonging to then using the product mean the DoA of is given by-

1.3 Lemmas

1.3.1 Lemma 1

Proof.

1.3.2 Lemma 2

Proof. Since, using Jordan lemma and sinnt = 1

Taking, first term of (1.5)-

Taking, next term of (1.5), by Abel's lemma-

Combining (1.5), (1.6), and (1.7) we get

1.3.3 Lemma 3

Let, where 0 < t = p,

1.4 Proof of the Theorems

1.4.1 Proof of the Theorem 1

Using Riemann-Lebesgue theorem,

Further,

Using GMI, we have

Now, using Lemma 1 and Lemma 3 with monotonicity of with the respect to t,

Using, 2nd MVT of integral, we have

Using, Lemma 2 and Lemma 3-

From (1.8), (1.9) and (1.10) we have

Clearly,

Applying Minikowski's inequality,

Using Lemma 2 and Lemma 3, we get

From (1.11) and (1.12), we have

We write H1 in terms of H3 and H2, H3 in terms of H4.

In show of monotonicity of v (t) for 0 < t = p, we have

Therefore, we can write (1.14)

Again, using monotonicity of v (t)

By using the fact that we have

Therefore, we write

Now combining, (1.15) and (1.17) we have-

Hence,

Proof of the theorem 1 is completed.

1.4.2 Proof of the Theorem 2

We have,

we have assumed that the is positive and decreasing in t. Hence,

Proof of the theorem 2 is completed.

1.5 Corollaries

1.5.1 Corollary 1

If we put (E, 1), (C, 1) mean in place of mean, in the theorem-1, the DoA of the function (C, 1) mean-

Of the CFS is,

Proof: [8]

1.5.2 Corollary 2

If we put mean in place of mean in the Theorem 1, the DoA of the function by (E, 1), (C, 1) mean

of CFS is,

Proof: [8]

1.6 Example

In this example, we are showing (E, s) and (Nörlund (Np)) summability of the partial sums of a FS is better behaved than the sequence of partial sum sn (x).

Let

with f (x + 2p) = f (x), ? x.

Fourier series of

The nth partial sum sn (x) of FS is -

Figure 1.1 [10]: The graph of f (x) function and the Fourier series expansion for n = 10

If Nörland method Np has increasing weights {pn}, its means Np is given by

We observed that converge to f (x) faster than sn (x) in

Figure 1.1 Radial basis neural network architecture. Reprint with copyright permission from [11].

1.7 Conclusion

The proved results in this paper are an attempt to study the approximations of the functions in by using product means of conjugate Fourier series, which generalize several results. Further, the result can be extended for other functions belonging to

References

1. 1. Nigam, H.K., Best approximation conjugate of a function in generalized Zygmund class. Tamkang J. Math., 50, 4, 417-428, Dec. 2019. doi:10.5556/j. tkjm.50.2019.3006.
2. 2. Nigam, H.K. and Sharma, K., On (e, 1) (c, 1) summability of fourier series and its conjugate series. Int. J. Pure Appl. Math., 82, 3, 365-375, ISSN: 1311-8080, 2013.
3. 3. Nigam, H.K., On approximation in generalized Zygmund class. De Gruyter. Demonstratio Mathematica, 52, 370-387, 2019. http://doi.org/10.1515/dem-2019-0022.
4. 4. Singh, M.V., Mittal, M.L., Rhoades, B.E., Approximation of function in the generalized Zygmund class using hausdroff means. J. Inequal. Appl., 2017, 101, 2017. DOI 10.1186/s13660-017-1361-8.
5. 5. Lal, S. and Shireen, Best approximations of functions of generalized Zygmund class by Matrix- Euler Summability means of fourier series. Bulletin of Mathematical Analysis & Applications, ISSN: 1821-1291, 5, 4, 1-13. 13p, 2013. URL: http//www.bmathaa.org.
6. 6. Pradhan, T., Paikray, S., Das, A., Dutta, H., Approximation of signals in generalised Zygmund class via...