\magnification=1200 \special{ps: xy 1.15in 1in} \hsize=5.3in \vsize=7.3in \baselineskip 16pt plus 6pt \mathsurround=2pt \def \S{{\cal S}} \def \Z{{\bf Z}} \def \T{{\cal T}} \def \rs#1#2{#1^{\raise2pt\hbox{$\scriptstyle#2\!$}}} \def \lo#1#2{#1_{\lower2pt\hbox{$\scriptstyle#2\!$}}} \def \rl#1#2#3{#1^{\raise1pt\hbox{$\scriptstyle#2\!$}} _{\lower2pt\hbox{$\scriptstyle#3\!$}}} \def \lop#1#2{#1_{\lower2pt\hbox{$\rm#2\!$}}} \def \ac{\vert\,I\,\vert} \def \sc{[\,I\,]} \def \acl{\ac_\l} \def \scl{\sc_\l} \def \g{\gamma} \def \a{\alpha} \def \l{\lambda} \def \D{\Delta} \def \Fp{{\cal F}_p} \def \Fi{{\cal F}_{\i}} \def \Fr{{\cal F}_r} \def \Lt{{\rs {L}{2}}} \def \Lp{{\rs {L}{p}}} \def \L{{\rs {L}{1}}} \def \lp{{\rs {\el}{p}}} \def \lr{{\rs {\el}{r}}} \def \ll{{\rs {\el}{1}}} \def \lt{{\rs {\el}{2}}} \def \Li{{\rs {L}{\infty}}} \def \li{{\rs {\el}{\infty}}} \def \Lc{L^1_c} \def \Ls{L^1_s} \def \Lh{\wh\L} \def \Lph{\wh\Lp} \def \K{{\cal K}} \def \W{{\cal W}} \def \q{{\sl q}_0} \def \c{{\bf c}_0} \def \bv{{\sl {bv}}} \def \bvp{{\rs {\sl bv}{p}}} \def \bvop{{\rs {{\sl {bv}}_0}{p}}} \def \bvt{{\rs {\sl {bv}}{2}}} \def \bvot{{\rs {{\sl {bv}}_0}{2}}} \def \hvp{{\rs {\sl {hv}}{p}}} \def \hlp{{\rs {\sl {hl}}{p}}} \def \dlp{{\rs {\sl {dl}}{p}}} \def \avp{{\rs {\sl {av}}{p}}} \def \hvr{{\rs {\sl {hv}}{r}}} \def \hv{{\rs {\sl {hv}}{1}}} \def \avr{{\rs {\sl {av}}{r}}} \def \av{{\rs {\sl {av}}{1}}} \def \avt{{\rs {\sl {av}}{2}}} \def \adt{{\rs {\sl {ad}}{2}}} \def \cvt{{\rs {\sl {cv}}{2}}} \def \dvt{{\rs {\sl {dv}}{2}}} \def \aat{{\rs {\sl {aa}}{2}}} \def \cat{{\rs {\sl {ca}}{2}}} \def \dat{{\rs {\sl {da}}{2}}} \def \ssl{{\cal S}^\lambda} \def \bs{\backslash} \def \wh{\widehat} \def \e{\epsilon} \def \ve{\varepsilon} \def \th{\theta} \def \vth{\vartheta} \def \l{\lambda} \def \vph{\varphi} \def \o{\omega} \def \lb{[\,} \def \rb{\,]} \def \lv{\vert\,} \def \lV{\Vert\,} \def \rv{\,\vert} \def \rV{\,\Vert} \def \pr{\prime} \def \es{\emptyset} \def \i{\infty} \def \el{\ell} \def \su{\subset} \def \aa{{\cal A}} \def \aal{{\cal A}^\lambda} \def \aam{{\cal A}^\mu} \def \uu{{\cal U}} \def \qed{\hfill\strut\vrule height4pt width4pt depth2pt} \bigskip \centerline{\bf New Integrability and $\L$--Convergence Classes for Even Trigonometric Series II} \bigskip \centerline{ M. Buntinas and N. Tanovi\'c--Miller\footnote*{Research partially supported by U.S.--Yugoslav Joint Fund (NSF JF 803). AMS (1980) subject classification: {\bf 42A16, 42A20 }}} \vskip 1.5cm {\bf Abstract.} If an even sequence $c=(c_k)_{k\in\Z}$ is constant on lacunary blocks, then $c\in\Lh$ if and only if $c$ is even and $c\in\avt$, i.e. $c$ is a null sequence and $\sum_{j=0}^\i\big(\sum_{k=2^j}^\i\lv\D c_k\rv^2{\rs {\big)}{1/2}}<\i$. This special class $\W$ is incomparable with the other known integrability and $\L$ convergence classes for even trigonometric series, such as Fomin's classes $\Fp ,\;p>1$, and their various extensions. It is known that $\W\cup\Fp\su\avt$ but it is not known whether $\avt\su\Lh$. In this paper we obtain new integrability and $\L$ convergence classes for even series, $\dvt$ and $\cvt$, subsuming both $\W$ and $\Fp$, $p>1$, as well as other known classes. In particular we prove that $\W\cup\Fp\su\dvt\su\cvt\su\Lh$ properly for all $p>1$. Furthermore we show that both $\dvt$ and $\cvt$ are normed linear spaces, $\dvt\su\bv_0\cap\avt$ properly, $\cvt\su\bvot$ but $\cvt\not\su\bv_0\cup\avt$. Here $\bv_0$ and $\bvot$ denote the spaces of all null sequences $c=(c_k)$ such that $\D c\in\ll$, respectively $\D c\in\lt$. \medskip \bigskip {\bf 1. Introduction, definitions and preliminaries} \bigskip Let $\Lp$, $p\ge1$, be the Banach space of all $2\pi$--periodic real or complex valued integrable functions with the norm $\lV f\rV_{L^p} = ({1\over{2\pi}}\int\lv f\rv^p)^{1/p}$, where the integral is taken over any interval of length $2\pi$. Let $C$ be the Banach space of all $2\pi$--periodic real or complex valued continuous functions with the norm $\lV f\rV_{C}=\sup_x\,\lv f(x)\rv$. For a nonnegative integer $n$, let $s_n(x)$ denote the $n$--th symmetric partial sum of the series $$\sum_{k\in \Z} c_k\, e^{ikx}.\eqno (1.1)$$ If (1.1) is a Fourier series of a function $f\in \L$ we shall write $\wh f(k)$ for the coefficients $c_k$, $\wh f$ for the sequence $(\wh f(k))_{k\in\Z}$ and $s_n f$ for the partial sums $s_n$. Let $\Lph=\lbrace \wh f:\;\,f\in\Lp\rbrace$. Clearly $\wh\Lp$ is a Banach space under the induced norm $\lV\wh f\rV_{\wh\Lp}:=\lV f\rV_{\Lp}$. One of the unpleasant facts in Fourier analysis is that there is no characterizations of $\wh\Lp$ in terms of sequences alone, except for $p=2$, in which case $\wh\Lt=\lt$. While for $\wh\Lp$, $p>1$, there are various useful inclusions, such as those implied by the Hausdorff--Young Theorem and other results, and while for $f\in\Lp$, $p>1,$ we have $\lV s_n f-f\rV_{\Lp}=o(1)$ as $(n\to\i)$, much less can be claimed for the case $p=1$. Concerning $\Lh$, all we can say in one direction is that $\Lh\su\c$ properly. In the other direction for example, $\K\su\Lh,$ where $\K$ is the class of all even convex null sequences due to Young. This result was extended by Kolmogorov to the space of quasiconvex null sequences $\q$, see [1 or 6]. Since $\q$ is the linear span of $\K,$ the conclusion that $\q\su\Lh$ also follows from Young's result. Furthermore $\q\su\bv_0\cap\Lh$, where $\bv_0$ denotes the space of null sequences of bounded variation. The problem of integrability amounts to finding subsets of $\Lh$ that can be described in terms of sequences alone, called {\it integrability classes}. Statements describing subsets of $\Lh$ in terms of the partials sums of the Fourier series or involving Dirichlet kernel, are of different nature and do not solve the integrability problem in a useful way. Another troublesome fact about $\L$ is that there exist functions $f\in\L$ for which $\lV s_n f-f\rV_{\L}\ne o(1)$ as $(n\to\i)$. By the mentioned results of Young and Kolmogorov, if $c\in\K$, respectively if $c\in\q$, then (1.1) is a Fourier series of an even function $f\in\L$ and $$\lV s_nf-f\rV_{\L}\,=\,o(1)\;\, {\rm if\;\, and\;\, only\;\, if}\;\, \wh f(n)\, \log\,\lv n\rv\,=\,o(1)\; (\lv n\rv\to\i). \eqno (1.2)$$ We shall say that $E$ is an {\it integrability and $\L$ convergence class} if $E$ is an integrability class and for each $c\in E$ the series (1.1), which is a Fourier series of a function $f\in\L$, satisfies (1.2). The class $\q$ has been extended to larger integrability and $\L$ convergence classes for even, odd and general trigonometric series by various authors including Sidon, Telya\-kov\-skii, Fomin, Stanojevi\'c and others, see [6 through 14] and the references cited there. More recent extensions have been obtained in [3, 4, 8, 9, 11 and 12]. Restricting our attention mainly to even trigonometric series, to motivate our results, we shall describe the development of these classes in several directions. Considering trigonometric series (1.1) with even coefficients $c=(c_k)$, i.e. satisfying $c_k=c_{-k}$ for all $k$, clearly (1.1) can be identified with the cosine series $$a_0/2\;+\;\sum_{k=1}^\i\,a_k\,\cos\,kx\eqno(1.3)$$ where $a_k=c_k+c_{-k}$. As usual $D_n$ will denote the real Dirichlet kernel, i.e. $$D_n(x)=1/2\,+\,\sum_{k=1}^n\,\cos\,kx .$$ For a real or complex valued sequence $c=(c_k)$, let $\D c=(\D c_k)$ where $\D c_k=c_k-c_{k+1}$ and let $\D^2 c=\D(\D c)$. We refer to the following standard sequence spaces of two--way even sequences: $\c$-- the space of all null sequences and $\li$-- the space of all bounded sequences, under the norm $\lV c{\lo {\rV}{\li}}=\sup_k\lv c_k\rv$; $\lp ,\,1\le p<\i$, -- the space of all sequences $(c_k)$ such that $\lV c{\lo {\rV}{\lp}}=\big(\sum\,{\lv c_k\rv}^p{\rs {\big)}{1/p}}$ is finite; $\bvp$-- the space of all sequences of bounded variation of index $p$, i.e. of all sequences $c$ such that $\D c\in\lp$ under the norm $\lV c{\rV}_{bv^p}=\lV c{\lo {\rV}{\li}}+\lV\D c{\lo {\rV}{\lp}}$; ${\sl q}$-- the space of all bounded quasiconvex sequences, i.e., of all $(c_k)$ such that $\lV c{\rV}_{q}=\sum\,k\,\lv\D^2 c_k\rv\,+\,\sup_k\,\lv c_k\rv<\i$; $\bvop=\bvp\cap\c$ and $\q={\sl q}\cap\c$. All of these are Banach spaces under their respective norms. The same notation is used for the corresponding spaces of one--way or general two--way sequences. Given an increasing sequence of positive integers $(k_j)_0^\i$ let $h_j=\,\lbrace k_j,k_j+1,\cdots$, $k_{j+1}-1\rbrace$ be the block of integers corresponding to the j--th gap, $j=1,2,\dots$ and $h_0=\lbrace 0,1,\dots,k_1-1\rbrace$. For one--way sequence $c=(c_k)$ let $h^j c$ denote the corresponding $j$--th section, i.e. $h^j c=\sum_{k\in h_j}c_k\,e^k$ where $e^k$ is the sequence with $1$ at the $k$--th place and $0$ elsewhere. In particular if $k_j=2^j$ we shall write $d_j$ and $d^j c$ instead of $h_j$ and $h^j c$, respectively. We now recall the definitions of some of the recent integrability and $\L$ convergence classes for even trigonometric series (1.1), respectively for cosine series (1.3). We shall describe them in terms of two--way even sequences $(c_k)$, but the same definitions apply to one--way sequences $(a_k)$. Let $p\ge 1$. We say that a sequence $c=(c_k)$ belongs to the Fomin's class $\Fp$ if and only if $c\in\c$ and $$\sum_{j=0}^\i\,2^{j/q}\,{\lV d^j\D c\rV}_\lp\,<\i.\eqno(1.4)$$ Here and throughout the paper let $1/p+1/q=1$. All statements should be properly interpreted for $p=1$ and $p=\i$ Fomin [7], proved that {\it for $11$, is an integrability and $\L$ convergence class for even trigonometric series} (1.1). Both $\Fp$ and $\hvp$ increase as $p$ decreases. Recalling that by [3, Theorem 1] $\hvp\su\bv_0$ for all $p\ge1$ from [11, Theorem 3] and [3, Theorem 4] it follows that: $$\K\su\q\su\S\T=\Fi\su\Fp\su\Fp^*\su\hvp\su\bv_0\cap\Lh\eqno(1.6)$$ holds properly for all $p>1$. Furthermore, $\Fp$ are linear spaces, while $\Fp^*$ and $\hvp$ are not [4]. The corresponding classes of general two--way sequences contained in $\Lh$ have been characterized in [8 and 12]. In particular, introducing certain new classes $\dlp$ and $\hlp$, incomparable with $\hvp$ and containing sequences outside of $\bv_0$, another proper extension of Fomin's classes was obtained in [12], namely: $$\Fp\su\dlp\su\hlp\su\Lh\eqno(1.7)$$ holds properly for all $p>1$. All of these are both integrability and $\L$ convergence classes for even trigonometric series. In regard to integrability, an earlier result due to Telyakovskii [13] describes a class $\T$ also contained in $\bv_0\cap\Lh$ and larger than $\Fp$, $p>1$. In this paper we consider new extensions of the Fomin's classes, different from those mentioned above. For $p\ge 1$ let $\avp$ be the set of all even sequences $c=(c_k)$ such that $(c_k)\in\c$ and $$\sum_{j=0}^\i\,\big(\sum_{k=2^j}^\i{\lv\D c_k\rv}^p{\rs {\big)}{1/p}}<\i.\eqno(1.8)$$ By [3, Theorem 7] we have: $$\av\su\hvp\su\avp\;\,{\rm for\;\,all\;\,}p>1.\eqno(1.9)$$ We say that a sequence $c\in\W$ if and only if $c\in\c$ and there exists a lacunary sequence $(k_j)$ such that $c_{k+1}=c_{k_{j+1}}$ for $k\in h_j,$ (i.e. $c$ is constant on lacunary blocks) and for which $$\sum_{j=0}^\i\,\log\,{k_{j+1}\over k_j}\,\big(\sum_{k=k_j}^\i{\lv\D c_k\rv}^2{\rs {\big)}{1/2}}<\i.$$ This class was considered in [3 and 5]. We have observed in [5] that $c\in\W$ if and only if $c\in\c$, $c$ is constant on lacunary blocks and $c\in\avt$. As a consequence of a result due to M. Weiss [15], it follows that: {\it For an even sequence $c$ which is constant on lacunary blocks, $c$ belongs to $\Lh$ if and only if $c\in\c$ and $c\in\avt$} [5]. Hence $\W\su\Lh$. The corresponding statement was also proved in [2, Theorem 2] and it was shown by Telyakovskii that $\W\not\su\T$. We have observed in [3] that $\W$ is not contained in $\hvp$ for all $p>1$ and similar examples show that it is also incomparable with $\hlp$. Hence $\W$ is not contained in any of the known integrability classes. From the necessity part of the above statement it clearly follows that: {\it $\avp$ is not an integrability class for all $p>2$}. Furthermore, $\W\su\avt$. By the results proved in [3] the classes $\avp$ increase with $p$ and $\hvp\su\avt$ for all $p>1$. Hence the space $\avt$ is particularly interesting and it is natural to ask whether $\avt$ is an integrability and $\L$ convergence class or to consider other related classes subsuming both $\W$ and $\Fp$, $p>1$. Introducing in Section 2, a subspace of $\avt$, denoted $\dvt$ and its natural extension $\cvt$, both of which are linear spaces containing $\W\cup\Fp$, we prove in Section 3 that $\cvt$ is an integrability and $\L$ convergence class for even trigonometric series. In particular we show that $\W\cup\Fp\su\dvt\su\cvt\su\Lh$ holds properly for all $p>1$, $\dvt\su\bv_o\cap\avt$ properly, but that $\cvt$ contains sequences outside of $\bv_0\cup\avt$. In view of the above comments, this indicates that $\cvt$ is a particularly large integrability and $\L$ convergence for even trigonometric series. \bigskip \medskip {\bf 2. Sequence space extensions of the classes $\W$ and $\Fp$} \bigskip \smallskip We begin with definitions of several new classes. A natural extension of the class $\W$ is the sequence space $\adt$ defined as follows. We say that a sequence $c\in\adt$ if and only if $c\in\c$ and $$\lV \D c {\rV}_{aa^2}:=\sum_{j=0}^\i\,\big(\sum_{r=j}^\i{\lV d^r\D c{\rl {\rV}{2}{\ll}}}{\rs {\big)}{1/2}}<\i.\eqno(2.1)$$ Clearly $\adt\su\avt$. Furthemore $\adt$ is a linear space and it is a normed space under the norm $\lV c {\rV}_{ad^2}=\lV c{\lo {\rV}{\li}}+\lV \D c {\rV}_{aa^2}$. Observing that for $r\ge j$ each block of integers $d_r$ can be divided into blocks of length $2^j$, the above condition (2.1) can be weakened to $$\sum_{j=0}^\i\,\Big(\sum_{r=j}^\i\;\sum_{\mu=0}^{2^{r-j}-1}\;\big(\sum_{k=2^r+\mu 2^j}^{2^r+(\mu+1)2^j-1}\lv \D c_k \rv{\rs {\big)}{2}}{\rs {\Big)}{1/2}}<\i.\eqno(2.2)$$ This leads to the following definition. We say that a sequence $c\in\dvt$ if and only if $c\in\c$ and $$\lV \D c {\rV}_{da^2}:=\sum_{j=0}^\i\,\Big(\;\sum_{r=1}^\i\;\big(\sum_{k=r 2^j}^{(r+1)2^j-1}\lv \D c_k \rv{\rs {\big)}{2}} {\rs {\Big)}{1/2}}<\i.\eqno(2.2')$$ Clearly (2.2') is just another form of expressing (2.2). Also clearly $\dvt$ is a normed linear space under the norm $\lV c {\rV}_{dv^2}=\lV c{\lo {\rV}{\li}}+\lV \D c {\rV}_{da^2}$ and $\dvt\su\avt$. Looking for integrability classes that subsume both $\W$ and $\Fp$ we will see that $\W\su\adt$, $\W\cup\Fp\su\dvt$, but that $\Fp\not\su\adt$. We now introduce another class, larger than $\dvt$ but containing sequences outside of $\avt$. Namely, observing that for $r\ge j$ every integer $k\in d_r$ can be written as $k=2^r+\mu 2^j+\nu$ where $\mu=0, 1, \dots, 2^{r-j}-1$ and $\nu=0, 1,\dots, 2^j-1$ , the above condition (2.2) can be weakened to, $$\sum_{j=0}^\i\,\max_{0\le\nu<2^j}\,\Big(\sum_{r=j}^\i\;\sum_{\mu=0}^{2^{r-j}-1}\;\lv c_{2^r+\mu 2^j+\nu}-c_{2^r+(\mu+1)2^j}{\rs {\rv}{2}}{\rs {\Big)}{1/2}}<\i.\eqno(2.3)$$ We say that a sequence $c\in\cvt$ if and only if $c\in\c$ and $$\lV \D c {\rV}_{ca^2}:=\sum_{j=0}^\i\,\max_{0\le\nu<2^j}\,\Big(\;\sum_{r=1}^\i\;\lv c_{r 2^j+\nu}-c_{(r+1)2^j}{\rs {\rv}{2}} {\rs {\Big)}{1/2}}<\i.\eqno(2.3')$$ The condition (2.3') is just another form of writing (2.3). In our discussions of the properties of $\dvt$ and $\cvt$ both forms of the respective conditions will be used. It may be observed immediately that $\cvt$ is a normed linear space under the norm $\lV c {\rV}_{cv^2}=\lV c{\lo {\rV}{\li}}+\lV \D c {\rV}_{ca^2}$. Furthemore, by standard arguments or by some known results on composed sequence spaces, it can be shown that {\it $\adt,\; \dvt$ and $\cvt$ are $BK$ spaces under the respective norms}, i.e. complete normed spaces with continuous coordinatewise functionals. For the other, not so obvious properties of the above spaces, we shall provide the proofs. We may also define the associated sequence spaces $\aat,\;\dat$ and $\cat$ of all one--way sequences $x=(x_k)$ such that $\lV x {\rV}_{aa^2}$, $\lV x {\rV}_{da^2}$ and $\lV x {\rV}_{ca^2}$ respectively are finite, where the corresponding definitions are obtained from (2,1), (2.2') and (2.3') replacing $\D c$ by $x$. Clearly these are also $BK$ spaces. We will show that each of these normed spaces has the property $AK$, i.e. that $\lV s^n x-x\rV=o(1)$ as $n\to\i$ for all $x$. Here $s^n x$ denotes the $n$--th section of the sequence $x$, i.e. $s^n x=\sum_{k=0}^{n}x_k\,e^k$. This property will be used in the proof of our main results in Section 3, observing that $c\in\adt$ if and only if $c\in\c$ and $\D c\in\aat$ and that similar statements hold for $\dvt$ and $\cvt$. \bigskip {\bf Theorem 1.} {\it The spaces $\adt$ and $\dvt$ satisfy the following proper inclusions: \item {i)} $\av\su\adt\su\dvt\su\bv_0\cap\avt$; \item {ii)} $\W\su\adt$, but $\Fp\not\su\adt$ for any $p>1$; \item {iii)} $\W\cup\Fp\su\dvt$ for all $p>1$.} \medskip {\bf Proof.} i) That $\av\su\adt\su\dvt\su\avt$ is almost immediate from the definitions. Namely, observing that: $$\sum_{k\in d_r}\lv\D c_k{\rs {\rv}{2}}\le\sum_{\mu=0}^{2^{r-j}-1}\;\big(\sum_{k=2^r+\mu 2^j}^{2^r+(\mu+1)2^j-1}\lv \D c_k \rv{\rs {\big)}{2}}\le \big(\sum_{k\in d_r}\lv\D c_k\rv{\rs {\big)}{2}}$$ the above inclusions follow from (1.8), (2.1) and (2.2). Now if $c\in\dvt$ then $c\in\c$ and (2.2). Leaving only the $r=j$ terms in the first inner sum of (2.2), we obtain $$\sum_{j=0}^\i\,\sum_{k=2^j}^{2^{j+1}-1}\lv \D c_k\rv<\i,$$ so that clearly $c\in\bv_0$. Hence, $\dvt\su\bv_0\cap\avt$. We shall illustrate now that all three inclusions are proper. To see that $\av\su\adt$ holds properly let $c=(c_k)$ be an even sequence defined by: $$c_0=1;\;\,c_{k+1}={1\over(r+1)}\;\,{\rm for}\;\, k\in d_r\;\,r=0,1,\dots.\eqno(2.4)$$ Then $\D c_{2^r}=1/r(r+1)$ for $r=1,2,\dots$ and $\D c_k=0$ otherwise. Hence, $$\sum_{j=0}^\i\,\big(\sum_{r=j}^\i{\lV d^r\D c{\rl {\rV}{2}{\ll}}}{\rs {\big)}{1/2}}<2\,\sum_{j=1}^\i{1\over j^{3/2}}<\i,$$ so that $c\in\adt$. However for this sequence (1.8) does not hold for $p=1$, i.e. $c\not\in\av$. To see that the second inclusion is proper consider the following example. Let $c=(c_k)$ be even and given by, $$c_k={1\over \log^{1/2}\,k}\;\;{\rm for}\;\;k=2,3,\dots; \;\;c_0=c_1=c_2\eqno(2.5)$$ Then a simple calculation shows that for $r=1,2,\dots,$ $${1\over 2^{r+1}(r+1)^{3/2}\log^{1/2}2}\le\lv\D c_k\rv\le {1\over 2^r r^{3/2}\log^{3/2}2}\;\,{\rm for}\;\, k\in d_r.\eqno(2.6)$$ Hence, $$\sum_{j=1}^\i\,\big(\sum_{r=j}^\i{\lV d^r\D c{\rl {\rV}{2}{\ll}}}{\rs {\big)}{1/2}}\ge{1\over 2\log^{1/2}2}\sum_{j=1}^\i\,\big(\sum_{r=j}^\i{1\over(r+1)^3}{\rs {\big)}{1/2}}\ge {1\over 4}\sum_{j=1}^\i{1\over j+1}=\i$$ so that $c\not\in\adt$. However, $$\displaylines{\sum_{j=1}^\i\,\Big(\sum_{r=j}^\i\;\sum_{\mu=0}^{2^{r-j}-1}\;\big(\sum_{k=2^r+\mu 2^j}^{2^r+(\mu+1)2^j-1}\lv \D c_k \rv{\rs {\big)}{2}}{\rs {\Big)}{1/2}}\cr\le{1\over \log^{3/2}2}\sum_{j=1}^\i\,\Big(2^j\,\sum_{r=j}^\i{1\over 2^r\,r^3}{\rs {\Big)}{1/2}}\le {2^{1/2}\over \log^{3/2}2}\sum_{j=1}^\i{1\over j^{3/2}}<\i\cr}$$ and therefore $c\in\dvt$. Thus $\adt\su\dvt$ holds properly. Next we construct a sequence illustrating that the third inclusion is proper. Let $c=(c_k)$ be even and given by $$c_0=1;\;\,c_{k+1}={(-1)^k\over2^{3r/4+1}}\;\,{\rm for}\;\, k=2^r,\dots,2^r+2^{[r/2]}-1;\;\,c_k=0\;\,{\rm otherwise}.\eqno(2.7)$$ Then, $$\lv\D c_k\rv={1\over 2^{3r/4}},\;\,{\rm for}\;\,k=2^r+1,\dots,2^r+2^{[r/2]}-2;\;\,\lv\D c_{2^r}\rv=\lv\D c_{2^r+2^{[r/2]}-1}\rv={1\over 2^{3r/4+1}}$$ and $\lv\D c_k\rv=0$ otherwise. Hence clearly, $$\sum_{r=0}^\i\sum_{k\in d_r}\lv\D c_k\rv\le\sum_{r=0}^\i{1\over 2^{r/4}}<\i$$ and therefore $c\in\bv_0$. Furthermore, $$\displaylines{\sum_{j=0}^\i\Big(\sum_{r=j}^\i\sum_{k\in d_r}\lv\D c_k{\rs {\rv}{2}}{\rs {\Big)}{1/2}}\cr \le\sum_{j=0}^\i\Big(\sum_{r=j}^\i{1\over 2^r}{\rs {\Big)}{1/2}}<2\,\sum_{j=0}^\i{1\over 2^{j/2}}<\i\cr}$$ i.e. (1.8) holds for $p=2$ and $c\in\avt$. Thus $c\in\bv_0\cap\avt$. We now verify that $c\not\in\dvt$. Namely by the above we have, $$\displaylines{\sum_{j=1}^\i\,\Big(\sum_{r=j}^\i\;\sum_{\mu=0}^{2^{r-j}-1}\;\big(\sum_{k=2^r+\mu 2^j}^{2^r+(\mu+1)2^j-1}\lv \D c_k \rv{\rs {\big)}{2}}{\rs {\Big)}{1/2}}\cr\ge\sum_{j=1}^\i\,\Big(\sum_{r=j}^\i2^{[r/2]-j}\,{2^{2j}\over 2^{3r/2+2}}{\rs {\Big)}{1/2}}> {1\over 2}\sum_{j=1}^\i\Big(2^j\,\sum_{r=j}^\i{1\over 2^{r+1}}{\rs {\Big)}{1/2}}=\i\cr}$$ so that (2.2) does not hold. This completes the proof of i). \smallskip ii) We have observed in [5] that $c\in\W$ implies that $c=\sum_{\nu=0}^n c^{(\nu)}$ where for each $\nu=0,1,\dots,n$ the sequence $c^{(\nu)}\in\avt$ and $\D c^{(\nu)}$ has at most one non--zero term in every dyadic block $d_r$. Consequently, $c=\sum_{\nu=0}^n c^{(\nu)}\in\adt$ since $\adt$ is a linear space and clearly $c^{(\nu)}\in\adt$ for each $\nu=0,1,\dots,n$. Thus $\W\su\adt$. A simple example shows that this inclusion is also proper. Namely, taking $c=(c_k)$ even with $c_k=1/(k+1)$ for $k=0,1,\dots$, we have $\sum_{k\in d_r}\lv\D c_k\rv\le 1/2^r$ and therefore $$\sum_{j=0}^\i\,\big(\sum_{r=j}^\i{\lV d^r\D c{\rl {\rV}{2}{\ll}}}{\rs {\big)}{1/2}}\le 2\,\sum_{j=0}^\i{1\over 2^j}<\i.$$ Hence clearly $c\in\adt$ but $c\not\in\W$. To verify that $\Fp\not\su\adt$ for all $p>1$ it suffices to consider the sequence given by (2.5). As it was shown already $c\not\in\adt$. Moreover from (2.6) it follows immediately that (1.4) is satisfied for all $p>1$ so that $c\in\Fp$. \smallskip iii) From i) and ii) it clearly follows that $\W\su\dvt$. We shall prove now that for all $p>1$, $\Fp\su\dvt$. Since the classes $\Fp$ decrease with $p$ increasing, we may assume that $c\in\Fp$ for some $10$ there exists $j_\e$ such that $$\sum_{j=j_\e}^\i\,\max_{0\le\nu<2^j}\,\Big(\;\sum_{r=1}^\i\;\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}<{\e\over 2}.\eqno(2.10)$$ Since $\Big(\;\sum_{r=1}^\i\;\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}$ is finite for each $\nu=0,1,\dots,2^j-1$ and $j=0,1,\dots,j_\e-1$, we can choose $r_j$ such that $r_j\ge r_{j-1}$ and such that $$\Big(\;\sum_{r=r_j}^\i\;\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}<{\e\over 4\,(j+1)^2}\;\;{\rm for}\;\;\nu=0,1,\dots,2^j-1.\eqno(2.11)$$ Define $n_\e=r_{j_\e}\,2^{j_\e}$. Then for all $n\ge n_\e$ we have, $$\left.\eqalign{&\lV x-s^n x {\rV}_{ca^2}=\sum_{j=0}^\i\,\max_{0\le\nu<2^j}\Big(\;\sum_{r=1}^\i\;\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,(x-s^n x)_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}\cr &\le\sum_{j=0}^{j_\e-1}\max_{0\le\nu<2^j}\Big(\sum_{r:\;r 2^j>n}\,\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}+\sum_{j=j_\e}^\i\,\max_{0\le\nu<2^j}\Big(\sum_{r=1}^\i\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}.\cr}\right.\eqno(2.12)$$ Observing that $n_\e=r_{j_\e}\,2^{j_\e}\ge r_j\,2^j$ for $j=0,1,\dots,j_\e-1$, from (2.11) it follows that for all $n\ge n_\e$, $$\displaylines{\sum_{j=0}^{j_\e-1}\,\max_{0\le\nu<2^j}\Big(\;\sum_{r:\;r 2^j>n}\,\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}\le\sum_{j=0}^{j_\e-1}\,\max_{0\le\nu<2^j}\Big(\;\sum_{r:\;r 2^j>n_\e}\,\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}\cr \le\sum_{j=0}^{j_\e-1}\,\max_{0\le\nu<2^j}\Big(\;\sum_{r=r_j}^\i\,\lv \sum_{k=r 2^j+\nu}^{(r+1)2^j-1}\,x_k{\rs {\rv}{2}} {\rs {\Big)}{1/2}}<\sum_{j=0}^{j_\e-1}\,{\e\over 4\,(j+1)^2}<{\e\over 2}.\cr}$$ Using (2.10) and the last estimate in (2.12) we conclude that $$\lV x-s^n x {\rV}_{ca^2}<\e\;\;{\rm for\;\;all}\;\;n\ge n_\e.$$ Consequently $\lV x-s^n x {\rV}_{ca^2}\to 0\;\;(n\to\i)$. \bigskip \medskip {\bf 3. The space $\cvt$ as an integrability and $\L$ convergence class} \bigskip We have shown in Section 2 that the linear spaces $\adt$, $\dvt$ and $\cvt$ are natural extensions of $\W$ and $\Fp$, $p>1$. Our goal here is to prove that they too are integrability and $\L$ convergence classes for even trigonometric series. By Theorems 1 and 2 it suffices to show that $\cvt$ is an integrability and $\L$ convergence class. The corresponding statements for $\adt$ and $\dvt$ will be deduced as corollaries of this result. \bigskip {\bf Theorem 4.} {\it Suppose that $c\in\cvt$. Then: \item {i)} The series {\rm (1.1)} converges a.e. to an even function $f$, \item {ii)} $f\in\L$ and {\rm (1.1)} is the Fourier series of $f$ and \item {iii)} {\rm (1.2)} holds.} \medskip {\bf Proof.} i) Let $c=(c_k)\in\cvt$. The partial sums $s_n(x)$ of the even series (1.1) can be expressed as $$s_n(x)=c_0\,+2\,\sum_{k=1}^n c_k\,\cos k\,x=2\,\Big\lbrace\sum_{k=0}^{n-1}\D c_k\,D_k(x)\,+\,c_n\,D_n(x)\Big\rbrace.\eqno(3.1)$$ By a standard argument we first observe that (1.1) converges a.e. if and only if the associated series $$\sum_{k=0}^\i\D c_k\,\sin(2k+1)\,x\eqno(3.2)$$ converges a.e. Namely, recalling that the Dirichlet kernel $(D_n(x))$ is bounded for $x\ne0\,(mod\;\,2\pi)$, from (3.1) and the assumption that $c\in\c$ it follows that (1.1) converges a.e. to a sum $f$ if and only if $$2\,\sum_{k=0}^\i\D c_k\,D_k(x)=\sum_{k=0}^\i\D c_k\,{\sin(k+1/2)\,x\over\sin x/2}$$ converges a.e. to $f$, if and only if (3.2) converges a.e. to a function $g$ related to $f$ by the equation $$f(x)=2\,\sum_{k=0}^\i\D c_k\,D_k(x)={g(x/2)\over \sin\,x/2}\;\;{\rm for}\;\;x\ne0\,(mod\;\,2\pi).\eqno(3.3)$$ Now by Theorem 2 i) $c\in\bvot$, i.e. $\D c\in\lt,$ and therefore there exists a function $g\in\Lt$ such that the associated series (3.2) is the Fourier series of $g$. Hence $g$ is an odd function and $$\wh{g_s}(2k+1)=\D c_k;\;\;\wh{g_s}(2k)=0\;\;{\rm for}\;\;k=0,1,2,\dots\,.\eqno(3.4)$$ Furthermore, by Carlson's theorem (3.2) converges a.e. to $g$. Consequently (1.1) converges a.e. to $f$ even, satisfying (3.3). \smallskip ii) We shall now prove that $f\in\L$. For this it will suffice to prove that $$\lV f{\lo {\rV}{\L}}={2\over\pi}\int_0^\pi\big\lv\,\sum_{k=0}^\i\D c_k\,D_k\,\big\vert < 10\,\lV\D c {\rV}_{ca^2}\eqno(3.5)$$ from which and the assumption that $c\in\cvt$ it clearly follows that $\lV f{\lo {\rV}{\L}}<\i$. Throughout this proof let $I_j=[\pi/2^{(j+1)},\pi/2^j]$. Then, $$\left.\eqalign{\int_{\pi/2^J}^\pi\big\lv\,\sum_{k=0}^\i\D c_k\,D_k\,\big\vert&=\sum_{j=0}^{J-1}\,\int_{I_j}\big\lv\,\sum_{k=0}^\i\D c_k\,D_k\,\big\vert\cr &\le\sum_{j=0}^{J-1}\,\int_{I_j}\big\lv\,\sum_{k=0}^{2^{j+1}-1}\D c_k\,D_k\,\big\vert\,+\,\sum_{j=0}^{J-1}\,\int_{I_j}\big\lv\,\sum_{k=2^{j+1}}^\i\D c_k\,D_k\,\big\vert\cr &\le\sum_{j=0}^{J-1}\,\sum_{r=0}^j\,\int_{I_j}\big\lv\,\sum_{k\in d_r}\D c_k\,D_k\,\big\vert\,+\,\sum_{j=0}^{J-1}\,\int_{I_j}\big\lv\,\sum_{r=j+1}^\i\sum_{k\in d_r}\D c_k\,D_k\,\big\vert\cr &=\;\;\sum\nolimits_{1}\;\quad+\;\quad\sum\nolimits_{2}.\cr}\right.\eqno(3.6)$$ To prove (3.5) we shall estimate each of the sums $\sum_{1}$ and $\sum_{2}$ as $J\to\i$. We first observe that by partial summation $$\left.\eqalign{\sum_{k\in d_r}\D c_k\,D_k(x)&=\sum_{k=2^r}^{2^{r+1}-1}\big(\sum_{\nu=k}^{2^{r+1}-1}\D c_\nu\big)\,\cos k\,x\,+\,\big(\sum_{\nu=2^r}^{2^{r+1}-1}\D c_\nu\big)\,D_{2^r-1}(x)\cr &=\sum_{k=2^r}^{2^{r+1}-1}\big(c_k-c_{2^{r+1}}\big)\,\cos k\,x\,+\,\big(c_{2^r}-c_{2^{r+1}}\big)\,D_{2^r-1}(x).\cr}\right.\eqno(3.7)$$ Using this in $\sum_{1}$ we have, $$\eqalign{\sum_{1}&=\sum_{j=0}^{J-1}\,\sum_{r=0}^j\,\int_{I_j}\big\lv\,\sum_{k\in d_r}\D c_k\,D_k(x)\,\big\vert\,dx\cr &\le\sum_{j=0}^{J-1}\,\sum_{r=0}^j\,(2^r\,\max_{k\in d_r}\lv c_k-c_{2^{r+1}}\rv+2^r\lv c_{2^r}-c_{2^{r+1}}\rv) \int_{I_j}dx\cr &\le 2\,\pi\,\sum_{j=0}^{J-1}\,\sum_{r=0}^j\,2^r\,\max_{k\in d_r}\lv c_k-c_{2^{r+1}}\rv\,{1\over 2^{j+1}}\cr &=2\,\pi\,\sum_{r=0}^{J-1}\,2^r\,\max_{k\in d_r}\lv c_k-c_{2^{r+1}}\rv\sum_{j=r}^{J-1}\,{1\over 2^{j+1}}\cr &\le 2\,\pi\,\sum_{r=0}^{J-1}\,\max_{0\le\nu<2^r}\lv c_{2^r+\nu}-c_{2^{r+1}}\rv.\cr}$$ Since, $$\sum_{j=0}^{J-1}\,\max_{0\le\nu<2^j}\lv c_{2^j+\nu}-c_{2^{j+1}}\rv\le\sum_{j=0}^\i\,\max_{0\le\nu<2^j}\,\Big(\;\sum_{r=1}^\i\;\lv c_{r 2^j+\nu}-c_{(r+1)2^j}{\rs {\rv}{2}} {\rs {\Big)}{1/2}} $$ from the above and (2.3') it follows that $$\sum\nolimits_{1}\le 2\,\pi\,\lV \D c {\rV}_{ca^2}\eqno(3.8)$$ for all $J=1,2,\dots$ . We now estimate the sum $$\sum\nolimits_{2}=\sum_{j=0}^{J-1}\,\int_{I_j}\big\lv\,\sum_{r=j+1}^\i\sum_{k\in d_r}\D c_k\,D_k\,\big\vert.$$ Observing that for $r=j+1, j+2,\dots$ each integer $k\in d_r$ can be expressed as $k=2^r+\mu 2^{j+1}+\nu$ where $\mu=0,1,\dots,2^{r-j-1}-1$ and $\nu=0,1,2,\dots,2^{j+1}-1$ the sums over the dyadic blocks can be written as: $$\sum_{k\in d_r}\D c_k\,D_k=\sum_{\mu=0}^{2^{r-j-1}-1}\,\sum_{\nu=0}^{2^{j+1}-1}\D c_{2^r+\mu 2^{j+1}+\nu}\,D_{2^r+\mu 2^{j+1}+\nu}.$$ Applying summation by parts to the inner sum, similarly as in (3.7), the last expression leads to $$\eqalign{\sum_{k\in d_r}\D c_k\,D_k&=\sum_{\mu=0}^{2^{r-j-1}-1}\sum_{\nu=0}^{2^{j+1}-1}\big(c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}\big)\cos(2^r+\mu 2^{j+1}+\nu)\,x\cr &+\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}}-c_{2^r+(\mu+1)2^{j+1}}\big)\,D_{2^r+\mu 2^{j+1}-1}(x)\cr}$$ $$\displaylines{=\sum_{\nu=1}^{2^{j+1}-1}\cos\nu\,x\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}\big)\,\cos(2^r+\mu2^{j+1})\,x\cr +\sum_{\nu=1}^{2^{j+1}-1}\sin\nu\,x\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}\big)\,\sin(2^r+\mu2^{j+1})\,x\cr +{1\over 2}\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}}-c_{2^r+(\mu+1)2^{j+1}}\big)\,\cos(2^r+\mu2^{j+1})\,x\cr +{1\over 2\,tg x/2}\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}}-c_{2^r+(\mu+1)2^{j+1}}\big)\,\sin(2^r+\mu2^{j+1})\,x.\cr}$$ Next we observe that (2.3), i.e. (2.3'), implies that for each $j$: $$\max_{0\le\nu<2^{j+1}}\Big(\sum_{r=j+1}^\i\;\sum_{\mu=0}^{2^{r-j-1}-1}\;\lv c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}{\rs {\rv}{2}}{\rs {\Big)}{1/2}}<\i.\eqno(3.9)$$ From the above equation, using H\"older's inequality it follows that for all $n\ge m\ge j+1$: $$\displaylines{\int_{I_j}\Big\lv\sum_{r=m}^n\sum_{k\in d_r}\D c_k\,D_k(x)\,\Big\vert\,dx\cr \le\!\!\sum_{\nu=1}^{2^{j+1}-1}\!\big({\pi\over 2^{j+1}}\big)^{1\over2}\!\Big(\int_{I_j}\Big\vert\sum_{r=m}^n\!\sum_{\mu=0}^{2^{r-j-1}-1}\!\!\big(c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}\big)\!\cos(2^r+\mu2^{j+1})x{\rs {\Big\vert}{2}}dx\Big)^{1\over2}\cr +\!\!\sum_{\nu=1}^{2^{j+1}-1}\!\big({\pi\over 2^{j+1}}\big)^{1\over2}\!\Big(\int_{I_j}\Big\vert\sum_{r=m}^n\!\sum_{\mu=0}^{2^{r-j-1}-1}\!\big(c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}\big)\!\sin(2^r+\mu2^{j+1})x{\rs {\Big\vert}{2}}dx\!\Big)^{1\over2}\cr +{1\over 2}\big({\pi\over 2^{j+1}}\big)^{1\over2}\Big(\int_{I_j}\Big\lv\sum_{r=m}^n\,\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}}-c_{2^r+(\mu+1)2^{j+1}}\big)\,\cos(2^r+\mu2^{j+1})\,x{\rs {\Big\vert}{2}}\,dx{\rs {\Big)}{1\over2}}\cr +{\pi\over 2}\big(\int_{I_j}{1\over x^2}\big)^{1\over2}\Big(\int_{I_j}\Big\lv\sum_{r=m}^n\,\sum_{\mu=0}^{2^{r-j-1}-1}\big(c_{2^r+\mu 2^{j+1}}-c_{2^r+(\mu+1)2^{j+1}}\big)\,\sin(2^r+\mu2^{j+1})\,x{\rs {\Big\vert}{2}}\,dx{\rs {\Big)}{1\over2}}.\cr}$$ Using a substitution $t=2^{j+1}x$ in the integrals appearing in the preceeding estimate and applying Parseval's formula we obtain: $$\eqalign{&\int_{I_j}\Big\lv\sum_{r=m}^n\sum_{k\in d_r}\D c_k\,D_k(x)\,\Big\vert\,dx\cr &\le2\,\pi^{1\over2}{1\over2^{j+1}}\sum_{\nu=1}^{2^{j+1}-1}\Big(\sum_{r=m}^n\;\sum_{\mu=0}^{2^{r-j-1}-1}\;\lv c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}{\rs {\rv}{2}}{\rs {\Big)}{1\over2}}\cr &+\pi^{1\over2}\Big(\sum_{r=m}^n\;\sum_{\mu=0}^{2^{r-j-1}-1}\;\lv c_{2^r+\mu 2^{j+1}}-c_{2^r+(\mu+1)2^{j+1}}{\rs {\rv}{2}}{\rs {\Big)}{1\over2}}.\cr}$$ Hence, $$\left.\eqalign{&\int_{I_j}\Big\lv\sum_{r=m}^n\sum_{k\in d_r}\D c_k\,D_k(x)\,\Big\vert\,dx\cr &\le3\,\pi^{1\over2}\max_{0\le\nu<2^{j+1}}\Big(\sum_{r=m}^n\sum_{\mu=0}^{2^{r-j-1}-1}\;\lv c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}{\rs {\rv}{2}}{\rs {\Big)}{1\over2}}\cr}\right.\eqno(3.10)$$ for all $n\ge m\ge j+1$. From (3.9) and (3.10) it clearly follows that $$\eqalign{&\int_{I_j}\Big\lv\sum_{r=j+1}^\i\sum_{k\in d_r}\D c_k\,D_k(x)\,\Big\vert\,dx\cr &\le3\,\pi^{1\over2}\max_{0\le\nu<2^{j+1}}\Big(\sum_{r=j+1}^\i\sum_{\mu=0}^{2^{r-j-1}-1}\;\lv c_{2^r+\mu 2^{j+1}+\nu}-c_{2^r+(\mu+1)2^{j+1}}{\rs {\rv}{2}}{\rs {\Big)}{1\over2}}.\cr}$$ Consequently by (2.3), i.e. (2.3'), we conclude that $$\sum\nolimits_{2}\le3\,\pi^{1\over2}\sum_{j=0}^J\max_{0\le\nu<2^j}\Big(\sum_{r=j}^\i\sum_{\mu=0}^{2^{r-j}-1}\;\lv c_{2^r+\mu 2^j+\nu}-c_{2^r+(\mu+1)2^j}{\rs {\rv}{2}}{\rs {\Big)}{1\over2}} \le 3\,\pi^{1\over2}\lV \D c {\rV}_{ca^2}\eqno(3.11)$$ for all $J=1,2,\dots$ . Putting (3.8) and (3.11) together in (3.6) and letting $J\to\i$ we obtain (3.5). This completes the proof that $f\in\L$. We now verify that (1.1) is the Fourier series of $f$ arguing directly, as in [3] or [11]. Namely, $f$ is related to $g\in\Lt$ by (3.3), where $g$ is odd and the Fourier coefficients of $g$ are given by (3.4). Hence, $f$ is even and $$\eqalign{c_0-c_n=\sum_{k=0}^{n_1}\D c_k&=\sum_{k=0}^{n-1}{2\over\pi}\int_0^\pi g(x)\,\sin(2k+1)x\,dx\cr &=\sum_{k=0}^{n-1}{1\over2\pi}\int_0^{2\pi} f(x)\,\big(\cos\,kx-\cos(k+1)x\big)\,dx\cr &=\sum_{k=0}^{n-1}\big(\wh{f_c}(k)-\wh{f_c}(k+1)\big)=\wh{f_c}(0)-\wh{f_c}(n).\cr}$$ Since $c_n\to 0$ and by Riemann--Lebesgue Lemma $\wh{f_c}(n)\to 0$, as $n\to\i$, from the above equality we have $c_0=\wh{f_c}(0)$ and consequently, $c_n=\wh{f_c}(n)$ for all $n=1,\,2,\dots$. Hence, $c_n=\wh{f_c}(n)$ for all $n\in\Z$ and therefore (1.1) is the Fourier series of $f$. \medskip iii)To show that (1.2) holds we first observe that by statement ii), (3.1) and (3.3), $$f(x)-s_n\,f(x)=2\,\sum_{k=n}^\i\D c_k\,D_k(x)-2\,c_n\,D_n(x).\eqno(3.12)$$ Furthermore, we observe that for the above $c\in\cvt$, the even sequences $c^{(n)}$ defined by the equations, $$c^{(n)}_k=c_n\;\;{\rm for}\;\;k=0,1,\dots,n-1\;\;{\rm and}\;\;c^{(n)}_k=c_k\;\;{\rm for}\;\;k=n,n+1,\dots\,$$ also belong to $\cvt$ for each nonnegative integer $n$, because clearly $c^{(n)}\in\c$, $\D c^{(n)}_k=0$ for $k1$.} \bigskip {\bf Remark 1.} The space $\adt$ is interesting because it is a simple and natural extension of the class $\W$, but not of $\Fp$. The space $\dvt$ extends both $\W$ and $\Fp$ within $\bv_0\cap\avt$. The largest space $\cvt$ contains sequences outside of both $\bv_0$ and $\avt$. \bigskip {\bf Remark 2.} Although the conclusions of Theorem 4 hold for $c\in\dvt$ as a corollary of Theorem 4, a direct proof of property i) for $c\in\dvt$ would not appeal to Carlson's Theorem, since by Theorem 1 $\dvt\su\bv_0$ and therefore in that case the associated series (3.2) converges uniformly to a function $g\in C$. \bigskip {\bf Remark 3.} The proper inclusions (3.13) complement other known extensions of the Fomin's classes $\Fp$ mentioned in the Introduction such as (1.6) and (1.7). \bigskip Considering the Banach spaces $E$ of all even functions $f\in\L$ such that $\wh f$ belongs to $\wh E=\;\,\adt,\;\,\dvt$ and $\cvt$ respectively, under the induced norm $\lV f{\lo {\rV}{E}}=\lV\wh f{\lo {\rV}{\wh E}}$, more can be concluded from the above results. \bigskip {\bf Theorem 6.} {\it Let $E$ be a function space such that $\wh E=\;\,\adt,\;\,\dvt$, and $\cvt$ respectively. Then $E$ is a Banach space and $$\lV s_nf-f\rV_{E}\,=\,o(1)\;\, {\rm if\;\, and\;\, only\;\, if}\;\, \wh f(n)\, \log\,\lv n\rv\,=\,o(1)\; (\lv n\rv\to\i). \eqno (3.13)$$} \smallskip {\bf Proof.} Since $\adt$, $\dvt$ and $\cvt$ are BK spaces clearly, in each case, $E$ is a Banach space under the induced norm. Furthermore, by the definitions of the corresponding norms it is trivial to see that $\lV D_n{\lo {\rV}{E}}\sim\,\log n$ as $n\to\i$. Recalling that by Theorem 3 each of the $BK$ spaces $\aat$, $\dat$ and $\cat$ has the property $AK$, we can use the same argument as in the proof of iii) of Theorem 4 to verify that (3.13) holds for each of these function spaces. \bigskip \medskip \centerline{\bf References} \bigskip \item{[1]} N.~K. Bary, {\sl Trigonometri\v ceskie Rjadi}, Gosudarstvennoe Izdateljstvo, Moscow, 1961. \smallskip \item{[2]} L.A. Balashov and S.A. Telyakovskii, Nekotorie svoistva lakunarnih rjadov i integrrujemost trigonometri\v ceskih rjadov, {\sl Trudi Mat. Inst. Akad. Nauk. SSSR \bf143} (1977) 32--41. \smallskip \item{[3]} M. Buntinas and N. Tanovi\'c--Miller, New integrability and $\L$ convergence classes for even trigonometric series. {\sl Radovi Mat. \bf6} 1,(1990), 149--170. \smallskip \item{[4]} M. Buntinas and N. Tanovi\'c--Miller, Integrability classes and Summability. {\sl Proc. Conf. on Approximation, Interpolation and Summability} Tel Aviv June 1990, (1991) to appear. \smallskip \item{[5]} D. \'Cerani\'c and N. Tanovi\'c--Miller, On some integrability and $\L$ convergence classes. To appear. \smallskip \item{[6]} R.~E. Edwards, {\sl Fourier series, Modern introduction}, Vols. 1 \& 2, Holt, Rinehart \& Winston, New York, 1967. \smallskip \item{[7]} G.~A. Fomin, Ob odnoi klase trigonometri\v ceskih rjadov, {\sl Mat. Zametki \bf23} (1978), 213--222. \smallskip \item{[8]} J.~F.Fournier and W.~M. Self, Some sufficient conditions for uniform convergence of Fourier series, {\sl J. Math. Anal. 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Telyakovskii, Ob odnom dostato\v cnom uslovii Sidona o integrrujemosti trigono\- metri\v ceskih rjadov, {\sl Mat. Zametki \bf14} (3) (1973) 317--328. \smallskip \item{[15]} M. Weiss, A theorem on lacunary trigonometric series, {\sl Proc. Conf. Harmonic Analysis, U. of Chicago}, (1967) 227--230. \smallskip \item{[16]} A. Zygmund, {\sl Trigonometric series}, Vols. I \& II, Cambridge University Press, New York, 1959. \vglue .5in \settabs 2\columns \+Department of Mathematical Sciences&Department of Mathematics\cr \+Loyola University of Chicago&University of Sarajevo\cr \+Chicago, IL USA&Sarajevo, Yugoslavia\cr \end