广义Walsh变式与一极值问题

<正> 设p为大于1的整数,t为非负实数,t的p进表示为

THE GENERALIZED WALSH TRANSFORM AND AN EXTREMUM PROBLEM

Let be the p-adic generalized Walsh func tions, where are the expresSions of p-adic numbers of bers of t,x,respectively The Walsh transform f of ∈L_(o,∞_~2 is defined by f(t)= and the Walsh transform on L_(o.∞) is defined similarly. It is known that for Walsh transform many basic properties analogous to Fourier transform are valid. In this paper we shall consider other properties. Introducing the function class of W_N -type and the concept of generalized logical derivatives, we Deal with the descent order of finite Walsh transform and obtain certain asymptotic behavoir of L(t). We Also give the necessary and sufficient conditions of functions which may be expressed by the form of (1). Finally, we study what role the function φ(x) and its derivative will play in the solution of an extremum problem, and the condition characterizing the best solution is obtained. Thus a model on a logical differencial-integral equation in connection with an interesting variation problem is immediately in sight.