The degree of approximation of functions belonging to Lip*α*, Lip(\alpha ,r), Lip(\xi (t),r) and W({L}_{r},\xi (t)), (r\ge 1)-classes through trigonometric Fourier approximation using different summability matrices with monotone rows has been proved by various investigators like Khan [1], Mittal *et al*. [2, 3], Mittal, Rhoades and Mishra [4], Qureshi [5], Chandra [6], Leindler [7], Rhoades *et al*. [8]. Recently Lal [9] has proved a theorem on the degree of approximation of a function *f* belonging to the Lip*α* (0<\alpha \le 1) class by {C}^{1}\cdot {N}_{p} summability method of its Fourier series. Lal [9] has assumed monotonicity on the generating sequence \{{p}_{n}\}. The approximation of a function \tilde{f}(x), conjugate to a 2*π* periodic function to f\in Lip\alpha (0<\alpha \le 1) using product ({C}^{1}\cdot {N}_{p})-summability has not been studied so far. In this paper, we obtain a new theorem on the degree of approximation of a function \tilde{f}, conjugate to a 2*π* periodic function f\in Lip\alpha (0<\alpha \le 1) class without monotonicity condition on the generating sequence \{{p}_{n}\}.

Let {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n} be a given infinite series with the sequence of *n* th partial sums \{{s}_{n}\}. Let \{{p}_{n}\} be a non-negative sequence of constants, real (**R**) or complex, and let us write

{P}_{n}=\sum _{k=0}^{n}{p}_{k}\ne 0\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge 0,{p}_{-1}=0={P}_{-1}\text{and}{P}_{n}\to \mathrm{\infty}\text{as}n\to \mathrm{\infty}.

The sequence to sequence transformation {t}_{n}^{N}={\sum}_{\nu =0}^{n}{p}_{n-\nu}{s}_{\nu}/{P}_{n} defines the sequence \{{t}_{n}^{N}\} of Nörlund means of the sequence \{{s}_{n}\}, generated by the sequence of coefficients \{{p}_{n}\}. The series {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n} is said to be {N}_{p} summable to the sum *s* if {lim}_{n\to \mathrm{\infty}}{t}_{n}^{N} exists and is equal to a finite number *s*. In the special case, in which

{p}_{n}=\left(\begin{array}{c}n+\alpha -1\\ \alpha -1\end{array}\right)=\frac{(n+\alpha )}{(n+1)(\alpha )}\phantom{\rule{1em}{0ex}}(\alpha >0),

the Nörlund summability {N}_{p} reduces to the familiar {C}^{\alpha} summability.

The product of {C}^{1} summability with a {N}_{p} summability defines {C}^{1}\cdot {N}_{p} summability. Thus the {C}^{1}\cdot {N}_{p} mean is given by {t}_{n}^{CN}=\frac{1}{n+1}{\sum}_{k=0}^{n}{P}_{k}^{-1}{\sum}_{\nu =0}^{k}{p}_{k-\nu}{s}_{\nu}.

If {t}_{n}^{CN}\to s as n\to \mathrm{\infty}, then the infinite series {\sum}_{n=0}^{\mathrm{\infty}}{a}_{n} or the sequence \{{s}_{n}\} is said to be summable {C}^{1}\cdot {N}_{p} to the sum *s* if {lim}_{n\to \mathrm{\infty}}{t}_{n}^{CN} exists and is equal to *s*.

\begin{array}{rl}{s}_{n}\to s\phantom{\rule{1em}{0ex}}& \Rightarrow \phantom{\rule{1em}{0ex}}{N}_{p}({s}_{n})={t}_{n}^{N}={P}_{n}^{-1}\sum _{\nu =0}^{n}{p}_{n-\nu}{s}_{\nu}\to s,\phantom{\rule{1em}{0ex}}\text{as}n\to \mathrm{\infty},{N}_{p}\text{method is regular,}\\ \Rightarrow \phantom{\rule{1em}{0ex}}{C}^{1}({N}_{p}({s}_{n}))={t}_{n}^{CN}\to s,\phantom{\rule{1em}{0ex}}\text{as}n\to \mathrm{\infty},{C}^{1}\text{method is regular,}\\ \Rightarrow \phantom{\rule{1em}{0ex}}{C}^{1}\cdot {N}_{p}\phantom{\rule{1em}{0ex}}\text{method is regular.}\end{array}

Let f(x) be a 2*π*-periodic function and Lebesgue integrable. The Fourier series of f(x) is given by

f(x)\sim \frac{{a}_{0}}{2}+\sum _{n=1}^{\mathrm{\infty}}({a}_{n}cosnx+{b}_{n}sinnx)\equiv \sum _{n=0}^{\mathrm{\infty}}{A}_{n}(x),\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge 0,

(1.1)

with (n+1)th partial sum {s}_{n}(f;x) called the trigonometric polynomial of degree (order) *n* of the first (n+1) terms of the Fourier series of *f*.

The conjugate series of Fourier series (1.1) is given by

\sum _{n=1}^{\mathrm{\infty}}({b}_{n}cosnx-{a}_{n}sinnx)\equiv \sum _{n=1}^{\mathrm{\infty}}{B}_{n}(x).

(1.2)

A function f(x)\in Lip\alpha if

f(x+t)-f(x)=O\left(\right|{t}^{\alpha}\left|\right)\phantom{\rule{1em}{0ex}}\text{for}0\alpha \le 1,t0.

{L}_{\mathrm{\infty}}-norm of a function f:R\to R is defined by {\parallel f\parallel}_{\mathrm{\infty}}=sup\{|f(x)|:x\in R\}.

The degree of approximation of a function f:R\to R by the trigonometric polynomial {t}_{n} of order *n* under the sup norm {\parallel \phantom{\rule{0.25em}{0ex}}\parallel}_{\mathrm{\infty}} is defined by [10]

{\parallel {t}_{n}-f\parallel}_{\mathrm{\infty}}=sup\left\{\right|{t}_{n}(x)-f(x)|:x\in R\}

and {E}_{n}(f) of a function f\in {L}_{r} is given by {E}_{n}(f)={min}_{n}{\parallel {t}_{n}-f\parallel}_{r}.

The conjugate function \tilde{f}(x) is defined for almost every *x* by

\begin{array}{rl}\tilde{f}(x)& =-\frac{1}{2\pi}{\int}_{0}^{\pi}\psi (t)cott/2\phantom{\rule{0.2em}{0ex}}dt\\ =\underset{h\to 0}{lim}(-\frac{1}{2\pi}{\int}_{h}^{\pi}\psi (t)cott/2\phantom{\rule{0.2em}{0ex}}dt)\phantom{\rule{1em}{0ex}}\text{(see [11, Definition~1.10])}.\end{array}

We note that {t}_{n}^{N} and {t}_{n}^{CN} are also trigonometric polynomials of degree (or order) *n*.

Abel’s transformation: The formula

\sum _{k=m}^{n}{u}_{k}{v}_{k}=\sum _{k=m}^{n-1}{U}_{k}({v}_{k}-{v}_{k+1})-{U}_{m-1}{v}_{m}+{U}_{n}{v}_{n},

(1.3)

where 0\le m\le n, {U}_{k}={u}_{0}+{u}_{1}+{u}_{2}+\cdots +{u}_{k}, if k\ge 0, {U}_{-1}=0, which can be verified, is known as Abel’s transformation and will be used extensively in what follows.

If {v}_{m},{v}_{m+1},\dots ,{v}_{n} are non-negative and non-increasing, the left-hand side of (1.3) does not exceed 2{v}_{m}{max}_{m-1\le k\le n}|{U}_{k}| in absolute value. In fact,

\begin{array}{rl}\left|\sum _{k=m}^{n}{u}_{k}{v}_{k}\right|& \le max|{U}_{k}|\{\sum _{k=m}^{n-1}({v}_{k}-{v}_{k+1})+{v}_{m}+{v}_{n}\}\\ =2{v}_{m}max|{U}_{k}|.\end{array}

(1.4)

We write throughout the paper

\begin{array}{r}{\psi}_{x}(t)=\psi (t)=f(x+t)-f(x-t),\\ {(\tilde{CN})}_{n}(t)=\frac{1}{2\pi (n+1)}\sum _{k=0}^{n}{P}_{k}^{-1}\sum _{\nu =0}^{k}{p}_{\nu}\frac{cos(k-v+1/2)t}{sint/2},\end{array}

(1.5)

\tau =[1/t], where *τ* denotes the greatest integer not exceeding 1/t, {P}_{\tau}=P[1/t], \mathrm{\Delta}{p}_{k}={p}_{k}-{p}_{k+1}.