Let <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> be an arbitrary non-empty set, and let <mml:math alttext="$\mathcal L$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#8466;</mml:mi></mml:math>, <mml:math alttext="\[\mathcal L_1  \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>, <mml:math alttext="\[\mathcal L_2  \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>
 be lattices of subsets of <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>
containing 
<mml:math alttext="$\phi$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#981;</mml:mi></mml:math> and <mml:math alttext="$X$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>. <mml:math alttext="\[ \mathcal A\left(\mathcal L \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119964;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> designates the algebra generated by <mml:math alttext="$\mathcal L$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#8466;</mml:mi></mml:math> and <mml:math alttext="\[ M\left(\mathcal L \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, these finite, non-trivial,
non-negative finitely additive measures on <mml:math alttext="\[ \mathcal A\left(\mathcal L \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119964;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. <mml:math alttext="\[ I\left(\mathcal L \right) \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> denotes those elements of <mml:math alttext="\[ M\left(\mathcal L \right) \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> which assume
only the values zero and one. In terms of a <mml:math alttext="\[ \mu  \in M\left(\mathcal L \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> or <mml:math alttext="\[ I\left(\mathcal L \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, various outer measures are introduced.
Their properties are investigated. The interplay of 
measurability, smoothness of <mml:math alttext="\[ \mu  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi></mml:math>, regularity of <mml:math alttext="\[ \mu  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi></mml:math> and
lattice topological properties on these outer measures is also investigated.Finally, applications of these outer measures to separation type 
properties between pairs of
lattices <mml:math alttext="\[ \mathcal L_1  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math>, <mml:math alttext="\[ \mathcal L_2  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> where <mml:math alttext="\[ \mathcal L_1   \subset \mathcal L_2  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>&#8834;</mml:mo><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> are developed. In 
terms of measures from <mml:math alttext="\[ I\left(\mathcal L \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>I</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#8466;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, necessary and sufficient
conditions are established for <mml:math alttext="\[ \mathcal L_1  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> to semi-separate <mml:math alttext="\[ \mathcal L_2  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>, for <mml:math alttext="\[ \mathcal L_1  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> to separate <mml:math alttext="\[ \mathcal L_2 \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>, and finally for <mml:math alttext="\[ \mathcal L_2  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> to coseparate <mml:math alttext="\[ \mathcal L_2  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#8466;</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>.

Maurice C. Figueres

Lattice separation, coseparation and regular measures

The notion of weakly quasi continuous functions introduced 
by Popa and Stan [1].
In this paper, the authors obtain the further properties 
of such functions and introduce weak<mml:math alttext="$*$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>*</mml:mo></mml:math> quasi
continuity which is weaker than semi continuity [2] 
but independent ofweak quasi continuity.

Jin Han Park

Hoe Young Ha

A note on weakly quasi continuous functions

The collection of fuzzy subsets of a set 
<mml:math alttext="\[ X \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> forms a complete lattice
that extends the complete lattice <mml:math alttext="\[ \mathcal P\left( X \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> of crisp subsets of <mml:math alttext="\[ X \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>. In this paper, we
interpret this extension as a special case of the &#147;fuzzification&#148; 
of an arbitrary
complete lattice <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. We show how to construct a complete lattice <mml:math alttext="\[ F\left( {A,L} \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>L</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>
 &#150;the <mml:math alttext="$L$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-fuzzificatio
of <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>, where <mml:math alttext="$L$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>
 is the valuation lattice&#150; that extends <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> while
preserving all suprema and infima. The &#147;fuzzy&#148; 
objects in <mml:math alttext="\[ F\left( {A,L} \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mo>,</mml:mo><mml:mi>L</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>
 may be interpreted as the sup-preserving maps from <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>
 to the dual of <mml:math alttext="$L$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>. In particular,
each complete lattice coincides with its <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>-fuzzification, where <mml:math alttext="$2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math> is  the twoelement
lattice. Some familiar fuzzifications (fuzzy subgroups, fuzzy subalgebras,
fuzzy topologies, etc.) are special cases of our construction. Finally, 
we show that
the binary relations on a set <mml:math alttext="\[ X \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> may be seen as the fuzzy subsets of <mml:math alttext="\[ X \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> with
respect to the valuation lattice <mml:math alttext="\[ \mathcal P\left( X \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#119979;</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>.

Arturo A. L. Sangalli

Lattices of fuzzy objects

In this paper we prove the existence and uniqueness of weak solutions of the mixed
problem for the nonlinear hyperbolic-parabolic equation
<mml:math display="block" alttext="\[ \left( {K_1 \left( {x,t} \right)u'} \right)^\prime   + K_2 \left( {x,t} \right)u' + A\left( t \right)u + F\left( u \right) = f \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>K</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8242;</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>K</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi></mml:math>
with null Dirichlet boundary conditions and 
zero initial data, where <mml:math alttext="\[ F\left( s \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is a continuous function such
that <mml:math alttext="\[ sF\left( s \right) \geqslant 0 \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>s</mml:mi><mml:mi>F</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#8805;</mml:mo><mml:mn>0</mml:mn></mml:math>, <mml:math alttext="\[ \forall s \in R \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&#8704;</mml:mo><mml:mi>s</mml:mi><mml:mo>&#8712;</mml:mo><mml:mi>R</mml:mi></mml:math> and <mml:math alttext="\[ \left\{ {A\left( t \right);t \geqslant 0} \right\} \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>;</mml:mo><mml:mi>t</mml:mi><mml:mo>&#8805;</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math> is a family of operators of <mml:math alttext="\[ L\left( {H_0^1 \left( \Omega  \right);H^{ - 1} \left( \Omega  \right)} \right) \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>H</mml:mi><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#937;</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>;</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#937;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>.
 For the
existence we apply the Faedo-Galerkin method with 
an unusual a priori estimate and a result of
W. A. Strauss. Uniqueness is proved only for some 
particular classes of functions <mml:math alttext="$F$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>.

Jorge  Ferreira

On weak solutions of semilinear hyperbolic-parabolic equations

The process of changing a topology by some types of 
its local discrete expansion preserves
<mml:math alttext="$s$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>s</mml:mi></mml:math>-closeness, <mml:math alttext="$S$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math>-closeness, semi-compactness, semi-<mml:math alttext="\[ T_i  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>T</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>, semi-<mml:math alttext="\[ R_i  \] " xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>, <mml:math alttext="\[ i \in \left\{ {0,1,2} \right\} \]" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi><mml:mo>&#8712;</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math>, and extremely disconnectness.
Via some other forms of such above replacements one can have topologies which satisfy
separation axioms the original topology does not have.

M. E. Abd El-Monsef

A. M. Kozae

A. A. Abo Khadra

Application on local discrete expansion

In this paper we give some significative 
counterexamples to prove that some well known
generalizations of Lindelöf property are proper. Also we 
give some new results, we introduce and study
the almost regular-Lindelof spaces as a generalization of 
the almost-Lindelöf spaces and as a new and
significative application of the quasi-regular open subsets of [1].

Filippo Cammaroto

Grazia Santoro

Some counterexamples and properties on generalizations of Lindelöf spaces

In this note, we prove a representation
 theorem for bounded commutative <mml:math alttext="$\rm BCK$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>BCK</mml:mtext></mml:math>-algebras.

H. A. S. Abujabal

M. Aslam

A. B. Thaheem

A representation of bounded commutative             <svg style="vertical-align:-0.20473pt" height="11.6875" version="1.1" viewBox="0 0 33.200001 11.6875" width="33.200001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path id="x42" d="M372 352v-2q176 -27 176 -159q0 -99 -95 -151q-73 -40 -196 -40h-221v28q65 5 78.5 19t13.5 76v404q0 63 -12 77t-72 18v28h257q102 0 150 -34q26 -18 41 -48.5t15 -65.5q0 -111 -135 -150zM213 362h47q80 0 117.5 32t37.5 96q0 56 -33 92t-100 36q-41 0 -56 -10&#xA;q-13 -8 -13 -50v-196zM213 328v-203q0 -54 17 -72.5t66 -18.5q67 1 110.5 37t43.5 110q0 147 -192 147h-45z" /></g><g transform="matrix(.017,-0,0,-.017,10.381,11.363)"><path id="x43" d="M614 175l29 -10q-33 -109 -57 -154q-121 -26 -184 -26q-90 0 -160.5 29t-112.5 77t-63.5 105.5t-21.5 119.5q0 157 108 253t277 96q36 0 71.5 -5t69 -13.5t36.5 -8.5q15 -102 20 -150l-29 -8q-20 79 -66.5 114t-128.5 35q-119 0 -187.5 -86t-68.5 -207&#xA;q0 -140 73.5 -227.5t188.5 -87.5q73 0 119.5 37.5t86.5 116.5z" /></g><g transform="matrix(.017,-0,0,-.017,21.685,11.363)"><path id="x4B" d="M667 0h-153l-248 291q-28 32 -54 27v-190q0 -65 14.5 -80t77.5 -20v-28h-260v28q58 5 71 20t13 80v392q0 48 -6 66.5t-22 25.5t-59 10v28h258v-28q-59 -5 -73 -20.5t-14 -81.5v-175q29 6 70 45q101 97 167 178q23 27 20 37t-36 14l-23 3v28h240v-28q-57 -5 -86 -18.5&#xA;t-76 -56.5q-40 -35 -177 -175q97 -124 235 -272q39 -41 61.5 -54t59.5 -18v-28z" /></g></svg>-algebras


A strictly barrelled disk <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> in a Hausdorff locally convex space <mml:math alttext="$E$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi></mml:math> is
a disk such that the linear span of <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> with the topology of the Minkowski
functional of <mml:math alttext="$B$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> is a strictly barrelled space. Valdivia's closed graph theorems
are used to show that closed strictly barrelled 
disk in a quasi-<mml:math alttext="$(\rm LB)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>LB</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>-space is
bounded. It is shown that a locally strictly 
barrelled quasi-<mml:math alttext="$(\rm LB)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>LB</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>-space is
locally complete. Also, we show that a regular 
inductive limit of quasi-<mml:math alttext="$(\rm LB)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mtext>LB</mml:mtext></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>-spaces is locally complete if and only if each closed bounded disk is a strictly
barrelled disk in one of the constituents.

Carlos  Bosch

Thomas E. Gilsdorf

Strictly barrelled disks in inductive limits of 
            quasi-<svg style="vertical-align:-2.3205pt" height="15.0875" version="1.1" viewBox="0 0 31.025 15.0875" width="31.025" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><path id="x4C" d="M495 163l30 -6q-19 -99 -36 -157h-453v28q63 5 77.5 20t14.5 78v400q0 63 -13 77t-75 19v28h261v-28q-61 -5 -74.5 -19t-13.5 -77v-395q0 -41 5 -59t18 -25q24 -13 96 -13q85 0 112 31q30 33 51 98z" /></g><g transform="matrix(.017,-0,0,-.017,15.089,12.138)"><path id="x42" d="M372 352v-2q176 -27 176 -159q0 -99 -95 -151q-73 -40 -196 -40h-221v28q65 5 78.5 19t13.5 76v404q0 63 -12 77t-72 18v28h257q102 0 150 -34q26 -18 41 -48.5t15 -65.5q0 -111 -135 -150zM213 362h47q80 0 117.5 32t37.5 96q0 56 -33 92t-100 36q-41 0 -56 -10&#xA;q-13 -8 -13 -50v-196zM213 328v-203q0 -54 17 -72.5t66 -18.5q67 1 110.5 37t43.5 110q0 147 -192 147h-45z" /></g><g transform="matrix(.017,-0,0,-.017,25.085,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg>-spaces

Research on Wiener type spaces was initiated by N.Wiener in [15]. A number of authors worked on these spaces or some special cases of these spaces. A kind of generalization of the Wiener's definition was given by H.Feichtinger in [2] as a Banach spaces of functions (or measures, distributions) on locally compact groups that are defined by means of the global behaviour of certain local properties of their elements. In the present paper we discussed Wiener type spaces using the spaces <mml:math alttext="$A_{w,\omega }^{p,q} \left( G \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>A</mml:mi><mml:mrow><mml:mi>w</mml:mi><mml:mo>,</mml:mo><mml:mi>&#969;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> and <mml:math alttext="$F_{w,\omega }^{p,q} \left( G \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>F</mml:mi><mml:mrow><mml:mi>w</mml:mi><mml:mo>,</mml:mo><mml:mi>&#969;</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> (c. f. [8]) as a local component, and <mml:math alttext="$L_\nu ^r \left( G \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>L</mml:mi><mml:mi>&#957;</mml:mi><mml:mi>r</mml:mi></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> as a global component, where <mml:math alttext="$w$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>w</mml:mi></mml:math> and <mml:math alttext="$\nu $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#957;</mml:mi></mml:math> are Beurling weights on <mml:math alttext="$G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> and <mml:math alttext="$\omega $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#969;</mml:mi></mml:math> is a Beurling weight on <mml:math alttext="$\hat G$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mi>G</mml:mi><mml:mo>&#710;</mml:mo></mml:mover></mml:math> (c. f. [13]).

R. H. Fischer

A. T. G&#252;rkanli

T. S. Liu

On a family of Wiener type spaces

We study the rate of approximation to functions in <mml:math alttext="$L^p $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:mrow></mml:math> and, in particular, in <mml:math alttext="${\text{Lip}}\left( {\alpha ,p} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>Lip</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> by weighted means of their Walsh-Fourier series, where <mml:math alttext="$\alpha  &#62; 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>&#62;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> and <mml:math alttext="$1 \leqslant p \leqslant \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>p</mml:mi><mml:mo>&#8804;</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:math>. For the case <mml:math alttext="$p = \infty $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>&#8734;</mml:mi></mml:mrow></mml:math>, <mml:math alttext="$L^p $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup></mml:mrow></mml:math> is interpreted to be <mml:math alttext="$C_W $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>C</mml:mi><mml:mi>W</mml:mi></mml:msub></mml:mrow></mml:math>, the collection of uniformly <mml:math alttext="$W$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi></mml:math> continuous functions over the unit interval <mml:math alttext="$\left[ {0,1} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>. We also note that the weighted mean kernel is quasi-positive, under fairly general conditions.

F.  M&#243;ricz

B. E. Rhoades

Approximation by weighted means of Walsh-Fourier series

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International Journal of Mathematics and Mathematical Sciences publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.

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International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences