Approximation of Integrable Functions by Wavelet Expansions

Walter (J Approx Theory 80:108–118, 1995), Xiehua (Approx Theory Appl 14(1):81–90, 1998) and Lal and Kumar (Lobachevskii J Math 34(2):163–172, 2013) established results on pointwise and uniform convergence of wavelet expansions. Working in this direction new more general theorems on degree of pointwise approximation by such expansions have been proved.


Introduction
In this paper we use the following notations.
L p (R) (1 < p < ∞) denotes the space of measurable, integrable with p−th power functions f . The norm of f ∈ L p (R) is written by f L p (R) . l 2 (Z) is the vector space of square-summable functions. A multiresolution approximation of L 2 (R) (see [3]) is a sequence (V j ) j∈Z of closed subspaces of L 2 (R) such that the following hold: There exists an isomorphism I from V 0 onto l 2 (Z) which commutes with the action of Z.
We consider orthonormal bases of wavelets in L 2 (R) (see [1]). These are functions of the form ψ j,k (x) = 2 for some fixed function ψ and are in turn based on a scaling function ϕ. In addition to the ladder of approximation spaces we have the orthogonal complement of each in the next higher one on the ladders i.e., The translates of the scaling function ϕ are an orthonormal basis of V 0 while the translates of ψ are an orthogonal basis of W o (see [1]). The same holds for their dilations in V j and W j . Hence each f ∈ L 2 (R) has a representation where a n,k , b j,k are expansion coefficients of f. f n is said to be the partial sums of the wavelet expansion. It is given by the reproducing kernels (see [4]).
The pointwise modules of continuity of the function f at point x are given by for fixed x.
where "O" depends only on C.
We also note the following well known result of Daubechies.
Theorem B [2, Theorem 9.1.6.]. Let ϕ be a continuous function such that ϕ and ϕ satisfy (1). If ϕ n,k constitute an orthonormal basis on L 2 (R), then the {ϕ n,k : n, k ∈ Z} also constitute an unconditional basis for all the spaces L p (R) Here, we will give more general and precise estimates of the pointwise convergence of wavelet expansions.

Statement of the Results
We determine the degree of approximation of functions by wavelet expansions.

Theorem 1. Assume that a continuous function ϕ is such that ϕ and ϕ satisfy
at every point x and with a positive constant K dependent on C.
Proof. From (1) it is easy to say that Following Walter [10], Meyer [6] and Novikov, Protasov, and we have Next, using (2) Results Math Observing that we obtain Collecting our partial estimates the result follows.
where w x is a function of modulus of continuity type, then by the monotonicity of wx(δ) δ and α > 2, we obtain

Theorem 2. Under the assumptions of Theorem 1, we have
Similarly as above Thus we obtain the desired estimate.

Corollary 2. Under the assumptions of Theorem 1 and if
The same order of approximation one can find in the paper of Skopina [9]. We also show our estimates in the following summation forms.