Discrete Operators in Quotient Spaces

Let D(k, h)=(h(k−1/2), h(k+1/2)) be a partition of the real axis with step h and integer . Consider a family of operators Th defined on functions by the

relation

(6.1.1)

where π is the operator introduced in Definition 5.1.1; u(k)=πu(x, h); the numbers γ(i) have the form

(6.1.2)

It follows from (6.1.2) that γ(0)=0, γ(−i)=−γ(i). Using (5.1.9), we get

(6.1.3)

where is the discrete Fourier transform (see Definition 5.1.4) of the grid function u(k)= πa(x, h), and the function S1(φ) is given by

(6.1.4)

Since γ(k) is odd, it follows from (6.1.4) that

Note that the definition of γ(k) implies the inequalities

(6.1.5)

Moreover, for the kernel

the following estimates hold

(6.1.6)

Let us show that the function S1(φ) is bounded on the interval [π, π]. Take δ>0, say δ=100−1, and consider an arbitrary φ such that |φ|≥δ. Then, using the

Abel transformation, we find that

(6.1.7)

Now, let |φ|<δ. For φ=0, we have S1(φ)=0, and therefore, we may assume that . The sets K(N)={φ: (2N)−1<|φ|≤N−1} cover the set K={φ: 0<|φ|≤1}, and thus, for any fixed

there exists N such that . Let . Consider the function

For the term S1(N−1, φ) we obtain the estimate

(6.1.8)

Using the Abel transformation, we get

(6.1.9)

Since , it follows from (6.1.9) that

|S2(N, p, φ)|≤C2. (6.1.10)

From (6.1.9), it follows that the series is convergent for any . Its convergence for φ=0 is obvious. Therefore, it is convergent for any . The estimates (6.1.7), (6.1.8), (6.1.10) imply that S1(φ) is bounded on [−π, π]. Definition 6.1.1. The operators Th defined by (6.1.1) are said to form a family of discrete one-dimensional singular operators. Proposition 6.1.1. The family of discrete one-dimensional singular operators Th from M(r, h) to M(r, h) is uniformly bounded for any h.