A.3 Classes of Functions

In this section, we list several classes of functions which are used below (cf [75], [63]).

A function f is defined on A = [ a, b] or on A = K, where K is a unit circle, satisfies the H lder condition with constant M and exponent ?, or belongs to the class H _?( M), M > 0, 0 < ? ? 1, if f( x') - f( x") ? Mx' - x" ^? for any x', x" ? A.

Class H _?, where ?( h) is a modulus of continuity, consists of all functions f ? C( A) with the property f( x ₁) - f( x ₂) ? M _?( x ₁ - x ₂), x ₁, x ₂ ? A.

Class W ^r( M) consists of functions f ? C( A) which have continuous derivatives f', f ?, ..., f ^{( r - 1)} on A, and a piecewise-continuous derivative f( ^r) on A satisfying max _{x ?[ a , b ]} f ^{( r )}( x) ? M.

Class ( M) , r = 1, 2 ..., 1 ? p ? ?, consists of functions f