Generalized Absolute Convergence of Trigonometric Fourier Series

Abstract

Recently, Moricz and Veres generalized the classical results of Bernstein, Szasz, Zygmund and others related to the absolute convergence of single and multiple Fourier series. In this paper, we have extended this result for single Fourier series of functions of the classes $\Lambda BV(\mathbb {\overline{T}})$ and $\Lambda BV^{(p)}(\mathbb {\overline{T}})$.

1 Introduction

The classical result of Zygmund, for the absolute convergence of Fourier series if a function of bounded variation on $\overline{T}$, where $\mathbb {T}=[-\pi ,\pi )$ is the torus, is generalized in many ways and many interesting results are obtained for different generalized absolute convergence of Fourier of functions of different generalized classes (see [1, 4]). In 2006, Gogoladze and Meskhia [1] obtained sufficient conditions for the generalized absolute convergence of a single Fourier series. Moricz and Veres [2] obtained sufficient conditions for the generalized absolute convergence of single and multiple Fourier series of functions of the classes $BV^{(p)}(\mathbb {\overline{T}})$ and $BV^{(p)}(\mathbb {\overline{T}}^{N})$, respectively (also see [5]). In this paper, generalizing such results for single Fourier series, we have obtained sufficient conditions for the generalized absolute convergence of single Fourier series of functions of the classes $\Lambda BV(\mathbb {\overline{T}})$ and $\Lambda BV^{(p)}(\mathbb {\overline{T}})$.

In the sequel, $\mathbb {L}$ is the class of non-decreasing sequence $\Lambda =\{\lambda _i\}~ (i=1,2,\ldots )$ of positive numbers such that $\sum _{i}\frac{1}{\lambda _{i}}$ diverges, a real number $p\ge 1$ and C represents a constant vary time to time.

2 Notations and Definitions

For a complex valued, 2$\pi$-periodic, function $f\in L^1(\overline{\mathbb {T}})$, its Fourier series is defined as

$$ f(x)~\sim ~\sum _{m\in \mathbb {Z}} \hat{f}(m)e^{imx}, \quad x\in \overline{\mathbb {T}}, $$

where

$$ \hat{f}(m)~=~\left( \frac{1}{2\pi }\right) \int _{\overline{\mathbb {T}}}~f(x)e^{-imx}~dx $$

denotes the mth Fourier coefficient of f.

For $p\ge 1$, the p-integral modulus of continuity of f over $\overline{\mathbb {T}}$ is define as

$$ \omega ^{(p)}(f;\delta ):={sup \atop 0<h\le \delta }\parallel T_hf-f\parallel _{p}, $$

where $T_hf(x)=f(x+h)$ for all x and $\parallel (^.)\parallel _{p}$ denotes the $L^p$-norm over $\overline{\mathbb {T}}$. $p=\infty$ gives the modulus of continuity $\omega (f;\delta )$ of f.

Following the definition in [1], a sequence $\gamma =\{\gamma _m:~m\in \mathbb {N}\}$ of nonnegative numbers is said to belongs to the class $\mathcal {A}_\alpha$ for some $\alpha \ge 1$ if

$$\begin{aligned} \left( \sum _{m\in \mathcal {D}_\mu } \gamma _{m}^\alpha \right) ^{1/\alpha }\le \kappa 2^{\mu (1-\alpha )/\alpha }\sum _{m\in \mathcal {D}_{\mu -1}} \gamma _m,~~~\mu \in \mathbb {N}, \end{aligned}$$

(2.1)

where

$$\begin{aligned} \mathcal {D}_0:=\{1\};~~~\mathcal {D}_\mu :=\{2^{\mu -1}+1,2^{\mu -1}+2,\ldots ,2^{\mu }\},~~~\mu \in \mathbb {N}; \end{aligned}$$

(2.2)

and the constant $\kappa$ does not dependent on $\mu$. Without the loss of generality, we assume that $\kappa \ge 1$.

Note that,

$$\begin{aligned} \mathcal {A}_{\alpha _2}\subset \mathcal {A}_{\alpha _1},\quad where\quad 1\le \alpha _1< \alpha _2<\infty . \end{aligned}$$

(2.3)

If a sequences $\gamma$ is such that

$$\begin{aligned} \max \{\gamma _m:m\in \mathcal {D}_\mu \} \le \kappa \ \min \{\gamma _m:m\in \mathcal {D}_{\mu -1}\},\quad \mu \in \mathbb {N}, \end{aligned}$$

(2.4)

then $\gamma \in \mathcal {A}_{\alpha }$ for every $\alpha \ge 1$. This inequality was introduced by Ul’yanov [3]. Moreover, Moricz and Veres [2] observed that, if a sequence $\gamma =\{\gamma _m\}$ is of the form

$$ \gamma _m=m^\tau w(m),~~~m\in \mathbb {N}, $$

where $\tau \in \mathbb {R}$ and $w:\mathbb {R}_+\rightarrow \mathbb {R}_+$ is a slowly varying function, that is,

$$\begin{aligned} {\lim \atop x\rightarrow \infty } \frac{w(\lambda x)}{w(x)} = 1,~~~for~every~0<\lambda <\infty , \end{aligned}$$

(2.5)

then $\gamma \in \mathcal {A}_\alpha$ for every $\alpha \ge 1$.

For convenience in writing, put

$$\begin{aligned} \gamma _{-m}:=\gamma _{m},\quad m\in \mathbb {N}. \end{aligned}$$

(2.6)

Definition 2.1

Given $\Lambda =\{\lambda _{n}\}\in \mathbb {L}$. A complex valued function f defined on an interval $I:=[a,b]$ is said to be of $p-\Lambda$-bounded variation (that is, $f\in \Lambda BV^{(p)} (I))$ if

$$ V_{\Lambda _{p}}(f,I)=\sup _{\{\textit{I}_k\}} \left( \sum _k\frac{|f(\textit{I}_{k})|^p}{\lambda _{k}}\right) ^{1/p}<\infty , $$

where $\{\textit{I}_{k}\}$ is a finite collections of non-overlapping subintervals $I_k=[a_k,b_k]\subset [a,b]$ and $f(I_k)=f(b_k)-f(a_k)$.

Note that, for $p=1$ and $\Lambda =\{1\}$ (that is, $\lambda _n=1$, for all n,) the class $\Lambda BV^{(p)}(I)$ reduces to the class BV(I) (the class of functions of bounded variation). For $p=1$ the class $\Lambda BV^{(p)}(I)$ reduces to the class $\Lambda BV(I)$; and for $\Lambda =\{1\}$ the class $\Lambda BV^{(p)}(I)$ reduces to the class $BV^{(p)}(I)$ (the class of functions of p-bounded variation).

3 Results for Functions of Single Variable

Theorem 3.1

If $f\in \Lambda BV(\overline{\mathbb {T}})$ and $\gamma =\{\gamma _m\}\in \mathcal {A}_{2/(2-\beta )}$ for some $\beta \in (0,2)$ then

$$ \sum (\gamma ;f)_{\beta }=\sum _{\vert m\vert \ge 1}\gamma _{m}\vert \hat{f}(m)\vert ^{\beta }\le \kappa C\sum _{\mu =0}^\infty 2^{-\mu \beta /2}\Gamma _{\mu -1}~\left( \frac{(\omega (f;\frac{\pi }{2^\mu }))}{\sum _{i=1}^{2^\mu }\frac{1}{\lambda _i}}\right) ^{\beta /2}, $$

where $\kappa$ is from (2.1) corresponding to $\alpha =2/(2-\beta )$ and C is a constant,

$$\begin{aligned} \Gamma _{\mu } := \sum _{m \in \mathcal{{D}}_{\mu }}\gamma _{m}\quad for \quad \mu \in \mathbb {N},\quad and \quad \Gamma _{-1} := \Gamma _{0}=\{\gamma _{1}\} \end{aligned}$$

(3.1)

Corollary 3.2

Under the hypothesis of Theorem 3.1, we have

$$ \sum (\gamma ;f)_\beta \le \kappa C\sum _{m =1}^\infty m^{-\beta /2}\gamma _m~\left( \frac{(\omega (f;\frac{\pi }{m}))}{\sum _{i=1}^m\frac{1}{\lambda _i}}\right) ^{\beta /2}, $$

In the case when $\gamma _m\equiv 1$, it follows from the above Corollary that $\sum (1;f)_\beta :=\sum _{|m|\ge 1}|\hat{f}(m)|^\beta$

$$ \le C\sum _{m =1}^\infty m^{-\beta /2}~\left( \frac{(\omega (f;\frac{\pi }{m})}{\sum _{i=1}^m\frac{1}{\lambda _i}}\right) ^{\beta /2}. $$

This gives the result [6, Theorem1,with$n_k=k$, forall k,] as a particular case.

Above corollary can easily follow from the Theorem 3.1.

Theorem 3.3

If $f\in \Lambda BV^{(p)}(\overline{\mathbb {T}})$ and $\gamma =\{\gamma _m\}\in \mathcal {A}_{2/(2-\beta )}$ for some $\beta \in (0,2)$ then

$$ \sum (\gamma ;f)_\beta \le \kappa C\sum _{\mu =0}^\infty 2^{-\mu \beta /2}\Gamma _{\mu -1}~\left( \left( \frac{(\omega ^{((2-p)s+p)} (f;\frac{\pi }{2^\mu }))^{2r-p}}{\sum _{i=1}^{2^\mu }\frac{1}{\lambda _i}}\right) ^{1/r}\right) ^{\beta /2}, $$

where $\frac{1}{r}+\frac{1}{s}=1,\ \kappa$ is from (2.1) corresponding to $\alpha =2/(2-\beta )$ and C is a constant.

Corollary 3.4

Under the hypothesis of Theorem 3.3, we have

$$ \sum (\gamma ;f)_\beta \le \kappa C\sum _{m =1}^\infty m^{-\beta /2}\gamma _m~\left( \left( \frac{(\omega ^{((2-p)s+p)} (f;\frac{\pi }{m}))^{2r-p}}{\sum _{i=1}^m\frac{1}{\lambda _i}}\right) ^{1/r}\right) ^{\beta /2}, $$

In the case when $\gamma _m\equiv 1$, it follows from the above Corollary that

$$\begin{aligned} \sum (1;f)_\beta&:=\sum _{|m|\ge 1}|\hat{f}(m)|^\beta \\&\quad \le C\sum _{m =1}^\infty m^{-\beta /2}~\left( \left( \frac{(\omega ^{((2-p)s+p)} (f;\frac{\pi }{m}))^{2r-p}}{\sum _{i=1}^m\frac{1}{\lambda _i}}\right) ^{1/r}\right) ^{\beta /2}. \end{aligned}$$

This gives the result [4, Theorem1, with $n_k=k$, forall k,] as a particular case.

Above Corollary 3.4 can be easily follows from the Theorem 3.3.

Proof of Theorem 3.1 $f\in \Lambda BV(\overline{\mathbb {T}})$ implies that f is bounded over $\overline{\mathbb {T}}$ and hence $f\in L^2(\overline{\mathbb {T}})$. For given $h>0$, put $f_j=T_{jh}f-T_{(j-1)h}f$, then $\hat{f_j}(m)=2i\hat{f}(m)e^{im(j-\frac{1}{2}h)}\sin (\frac{mh}{2})$.

By Parseval’s equality, we get

$$ 4\sum _{m\in \mathbb {Z}}|\hat{f}(m)|^2\sin ^2\left( \frac{mh}{2}\right) =O(||f_j||_2^2). $$

Putting $h=\frac{\pi }{2^{\mu }}$, $\mu \in \mathbb {N}$, and observing that

$$ \frac{\pi }{4}<\frac{|m|\pi }{2^{\mu +1}}\le \frac{\pi }{2}~~~for~~~|m|\in \mathcal {D}_\mu ,~~~implies~~~\sin ^2\left( \frac{mh}{2}\right) >\frac{1}{2}. $$

Thus, we have

$$\begin{aligned} B&=\sum _{|m|\in \mathcal {D}_\mu }|\hat{f}(m)|^2 =O\left( ||f_j||_2^2\right) \nonumber \\&= O\left( \omega (f;h)\right) \left( \int _0^{2\pi }|f_j(x)| dx \right) . \end{aligned}$$

(3.2)

Multiplying both the sides of the above inequality by $\frac{1}{\lambda _j}$ and then summing over $j=1$ to $j=2^\mu$, we have

$$ B = O\left( \frac{\omega (f;h)}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) \left( \int _0^{2\pi }\sum _{j=1}^{2^{\mu }}\frac{(|f_j(x)|)}{\lambda _j}~dx\right) =O\left( \frac{\omega (f;h)}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) , $$

as $f\in \Lambda BV(\overline{\mathbb {T}})$ implies $\sum _{j=1}^{2^{\mu }}\frac{(|f_j(x)|)}{\lambda _j}=O(1)$.

Since $1=\frac{\beta }{2}+\frac{2-\beta }{2}$, by Holder’s inequality, for $\mu \ge 1$, we have

$$\begin{aligned} S_\mu&:=\sum _{|m|\in \mathcal {D}_\mu }\gamma _m|\hat{f}(m)|^\beta \le \left( \sum _{|m|\in \mathcal {D}_\mu }|\hat{f}(m)|^2\right) ^{\beta /2}\left( \sum _{|m|\in \mathcal {D}_\mu }\gamma _m^{2/(2-\beta )}\right) ^{(2-\beta )/2}\nonumber \\&\quad \le C \left( \frac{\omega (f;h)}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) ^{\frac{\beta }{2}} \left( \sum _{|m|\in \mathcal {D}_\mu }\gamma _m^{2/(2-\beta )}\right) ^{(2-\beta )/2}. \end{aligned}$$

(3.3)

Thus for $\mu \ge 1$,

$$ S_\mu \le C \kappa \left( 2^{-\mu \beta /2}~\Gamma _{\mu -1}\left( \frac{\omega (f;h)}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) ^{\frac{\beta }{2}}\right) . $$

If $\mu =0$, then from (3.3) it follows that

$$ S_0:=\gamma _1(|\hat{f}(-1)|^\beta +|\hat{f}(1)|^\beta )=O\left( \gamma _1 \left( \frac{\omega (f;\pi )}{\frac{1}{\lambda _1}}\right) \right) . $$

Hence, the result follows from

$$ \sum _{|m|\ge 1}\gamma _m|\hat{f}(m)|^\beta =\sum _{\mu =0}^\infty S_\mu . $$

Proof of Theorem 3.3. $f\in \Lambda BV^{(p)}(\overline{\mathbb {T}})$ implies that f is bounded over $\overline{\mathbb {T}}$ [4, in view of Lemma 1, p.771] and hence $f\in L^2(\overline{\mathbb {T}})$. Proceeding as in the proof of Theorem 3.1, we get (3.2).

Since $2=\frac{(2-p)s+p}{s}+\frac{p}{r}$, by using Holder’s inequality, we have

$$ ||f_j||_2^2\le \left( ||f_j||_p\right) ^{p/r} \left( \int _0^{2\pi }|f_j|^{(2-p)s+p} dx\right) ^{1/s}\le \left( ||f_j||_p\right) ^{p/r}\Omega _h^{1/r}, $$

where $\Omega _h^{1/r}=(\omega ^{(2-p)s+p}(f;h))^{2r-p}$.

This together with (3.2) implies

$$ B^r=\left( \sum _{|m|\in \mathcal {D}_\mu }|\hat{f}(m)|^2\right) ^r = O\left( \Omega _h~\int _0^{2\pi }|f_j(x)|^pdx\right) . $$

Multiplying both the sides of the above inequality by $\frac{1}{\lambda _j}$ and then summing over $j=1$ to $j=2^\mu$, we have

$$ B^r = O\left( \frac{\Omega _h}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) \left( \int _0^{2\pi }\sum _{j=1}^{2^{\mu }}\frac{(|f_j(x)|^p)}{\lambda _j}~dx\right) =O\left( \frac{\Omega _h}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) .$$

Thus

$$ B= O\left( \frac{\Omega _h}{\sum _{j=1}^{^{2^\mu }}\frac{1}{\lambda _j}}\right) ^{1/r}. $$

Now, proceeding as in the proof of the Theorem 3.1 the result follows.